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id="toc-Motivating_examples-sublist" class="vector-toc-list"> <li id="toc-Circle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Circle</span> </div> </a> <ul id="toc-Circle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sphere" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Sphere</span> </div> </a> <ul id="toc-Sphere-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Other curves</span> </div> </a> <ul id="toc-Other_curves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Mathematical definition</span> </div> </a> <ul id="toc-Mathematical_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Charts,_atlases,_and_transition_maps" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Charts,_atlases,_and_transition_maps"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Charts, atlases, and transition maps</span> </div> </a> <button aria-controls="toc-Charts,_atlases,_and_transition_maps-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Charts, atlases, and transition maps subsection</span> </button> <ul id="toc-Charts,_atlases,_and_transition_maps-sublist" class="vector-toc-list"> <li id="toc-Charts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charts"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Charts</span> </div> </a> <ul id="toc-Charts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atlases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atlases"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Atlases</span> </div> </a> <ul id="toc-Atlases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transition_maps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transition_maps"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Transition maps</span> </div> </a> <ul id="toc-Transition_maps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additional_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Additional structure</span> </div> </a> <ul id="toc-Additional_structure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Manifold_with_boundary" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Manifold_with_boundary"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Manifold with boundary</span> </div> </a> <button aria-controls="toc-Manifold_with_boundary-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Manifold with boundary subsection</span> </button> <ul id="toc-Manifold_with_boundary-sublist" class="vector-toc-list"> <li id="toc-Boundary_and_interior" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Boundary_and_interior"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Boundary and interior</span> </div> </a> <ul id="toc-Boundary_and_interior-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Construction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Construction</span> </div> </a> <button aria-controls="toc-Construction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Construction subsection</span> </button> <ul id="toc-Construction-sublist" class="vector-toc-list"> <li id="toc-Charts_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charts_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Charts</span> </div> </a> <ul id="toc-Charts_2-sublist" class="vector-toc-list"> <li id="toc-Sphere_with_charts" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sphere_with_charts"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Sphere with charts</span> </div> </a> <ul id="toc-Sphere_with_charts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Patchwork" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Patchwork"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Patchwork</span> </div> </a> <ul id="toc-Patchwork-sublist" class="vector-toc-list"> <li id="toc-Intrinsic_and_extrinsic_view" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Intrinsic_and_extrinsic_view"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.1</span> <span>Intrinsic and extrinsic view</span> </div> </a> <ul id="toc-Intrinsic_and_extrinsic_view-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n-Sphere_as_a_patchwork" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#n-Sphere_as_a_patchwork"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.2</span> <span><i>n</i>-Sphere as a patchwork</span> </div> </a> <ul id="toc-n-Sphere_as_a_patchwork-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Identifying_points_of_a_manifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identifying_points_of_a_manifold"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Identifying points of a manifold</span> </div> </a> <ul id="toc-Identifying_points_of_a_manifold-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gluing_along_boundaries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gluing_along_boundaries"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Gluing along boundaries</span> </div> </a> <ul id="toc-Gluing_along_boundaries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartesian_products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian_products"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Cartesian products</span> </div> </a> <ul id="toc-Cartesian_products-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Early_development" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_development"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Early development</span> </div> </a> <ul id="toc-Early_development-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Synthesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Synthesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Synthesis</span> </div> </a> <ul id="toc-Synthesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poincaré's_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poincaré's_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Poincaré's definition</span> </div> </a> <ul id="toc-Poincaré's_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology_of_manifolds:_highlights" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology_of_manifolds:_highlights"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Topology of manifolds: highlights</span> </div> </a> <ul id="toc-Topology_of_manifolds:_highlights-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Additional_structure_2" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Additional_structure_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Additional structure</span> </div> </a> <button aria-controls="toc-Additional_structure_2-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Additional structure subsection</span> </button> <ul id="toc-Additional_structure_2-sublist" class="vector-toc-list"> <li id="toc-Topological_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Topological manifolds</span> </div> </a> <ul id="toc-Topological_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differentiable_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differentiable_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Differentiable manifolds</span> </div> </a> <ul id="toc-Differentiable_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riemannian_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemannian_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Riemannian manifolds</span> </div> </a> <ul id="toc-Riemannian_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finsler_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finsler_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Finsler manifolds</span> </div> </a> <ul id="toc-Finsler_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Lie groups</span> </div> </a> <ul id="toc-Lie_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_types_of_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_types_of_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6</span> <span>Other types of manifolds</span> </div> </a> <ul id="toc-Other_types_of_manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Classification_and_invariants" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Classification_and_invariants"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Classification and invariants</span> </div> </a> <ul id="toc-Classification_and_invariants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surfaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Surfaces</span> </div> </a> <button aria-controls="toc-Surfaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Surfaces subsection</span> </button> <ul id="toc-Surfaces-sublist" class="vector-toc-list"> <li id="toc-Orientability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orientability"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Orientability</span> </div> </a> <ul id="toc-Orientability-sublist" class="vector-toc-list"> <li id="toc-Möbius_strip" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Möbius_strip"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1.1</span> <span>Möbius strip</span> </div> </a> <ul id="toc-Möbius_strip-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Klein_bottle" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Klein_bottle"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1.2</span> <span>Klein bottle</span> </div> </a> <ul id="toc-Klein_bottle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_projective_plane" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Real_projective_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1.3</span> <span>Real projective plane</span> </div> </a> <ul id="toc-Real_projective_plane-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Genus_and_the_Euler_characteristic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Genus_and_the_Euler_characteristic"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Genus and the Euler characteristic</span> </div> </a> <ul id="toc-Genus_and_the_Euler_characteristic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Maps_of_manifolds" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Maps_of_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Maps of manifolds</span> </div> </a> <button aria-controls="toc-Maps_of_manifolds-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Maps of manifolds subsection</span> </button> <ul id="toc-Maps_of_manifolds-sublist" class="vector-toc-list"> <li id="toc-Scalar-valued_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scalar-valued_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Scalar-valued functions</span> </div> </a> <ul id="toc-Scalar-valued_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_of_manifolds" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations_of_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Generalizations of manifolds</span> </div> </a> <ul id="toc-Generalizations_of_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-By_dimension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#By_dimension"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>By dimension</span> </div> </a> <ul id="toc-By_dimension-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Manifold</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" 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Available in 55 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-55" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">55 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit – Alemannic" lang="gsw" hreflang="gsw" data-title="Mannigfaltigkeit" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B9%D8%AF%D8%AF_%D8%B4%D8%B9%D8%A8" title="متعدد شعب – Arabic" lang="ar" hreflang="ar" data-title="متعدد شعب" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxobrazl%C4%B1" title="Çoxobrazlı – Azerbaijani" lang="az" hreflang="az" data-title="Çoxobrazlı" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B9%E0%A7%81%E0%A6%AD%E0%A6%BE%E0%A6%81%E0%A6%9C" title="বহুভাঁজ – Bangla" lang="bn" hreflang="bn" data-title="বহুভাঁজ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/To-i%C5%AB%E2%81%BF-th%C3%A9" title="To-iūⁿ-thé – Minnan" lang="nan" hreflang="nan" data-title="To-iūⁿ-thé" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Многообразие – Bulgarian" lang="bg" hreflang="bg" data-title="Многообразие" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Varietat_(matem%C3%A0tiques)" title="Varietat (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Varietat (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D1%83%D0%BC%D0%B0%D0%B9%D1%81%C4%83%D0%BD%D0%B0%D1%80%D0%BB%C4%83%D1%85" title="Нумайсăнарлăх – Chuvash" lang="cv" hreflang="cv" data-title="Нумайсăнарлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Varieta_(matematika)" title="Varieta (matematika) – Czech" lang="cs" hreflang="cs" data-title="Varieta (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Mangfoldighed_(matematik)" title="Mangfoldighed (matematik) – Danish" lang="da" hreflang="da" data-title="Mangfoldighed (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit – German" lang="de" hreflang="de" data-title="Mannigfaltigkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Muutkond" title="Muutkond – Estonian" lang="et" hreflang="et" data-title="Muutkond" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CE%BB%CE%B1%CF%80%CE%BB%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Πολλαπλότητα – Greek" lang="el" hreflang="el" data-title="Πολλαπλότητα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variedad_(matem%C3%A1ticas)" title="Variedad (matemáticas) – Spanish" lang="es" hreflang="es" data-title="Variedad (matemáticas)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Sterna%C4%B5o" title="Sternaĵo – Esperanto" lang="eo" hreflang="eo" data-title="Sternaĵo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Barietate_(matematika)" title="Barietate (matematika) – Basque" lang="eu" hreflang="eu" data-title="Barietate (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%DB%8C%D9%81%D9%84%D8%AF_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="منیفلد (هندسه) – Persian" lang="fa" hreflang="fa" data-title="منیفلد (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_(g%C3%A9om%C3%A9trie)" title="Variété (géométrie) – French" lang="fr" hreflang="fr" data-title="Variété (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Variedade_(xeometr%C3%ADa)" title="Variedade (xeometría) – Galician" lang="gl" hreflang="gl" data-title="Variedade (xeometría)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EC%96%91%EC%B2%B4" title="다양체 – Korean" lang="ko" hreflang="ko" data-title="다양체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B1%D6%87%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բազմաձևություն – Armenian" lang="hy" hreflang="hy" data-title="Բազմաձևություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Mnogostrukost" title="Mnogostrukost – Croatian" lang="hr" hreflang="hr" data-title="Mnogostrukost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lipatan_(matematika)" title="Lipatan (matematika) – Indonesian" lang="id" hreflang="id" data-title="Lipatan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variet%C3%A0_(geometria)" title="Varietà (geometria) – Italian" lang="it" hreflang="it" data-title="Varietà (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%94" title="יריעה – Hebrew" lang="he" hreflang="he" data-title="יריעה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mannifalt" title="Mannifalt – Luxembourgish" lang="lb" hreflang="lb" data-title="Mannifalt" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Daugdara" title="Daugdara – Lithuanian" lang="lt" hreflang="lt" data-title="Daugdara" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sokas%C3%A1g_(matematika)" title="Sokaság (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Sokaság (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vari%C3%ABteit_(wiskunde)" title="Variëteit (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Variëteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E6%A7%98%E4%BD%93" title="多様体 – Japanese" lang="ja" hreflang="ja" data-title="多様体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Mangfoldighet" title="Mangfoldighet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Mangfoldighet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Mangfald_i_matematikk" title="Mangfald i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Mangfald i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%88%E0%A8%A8%E0%A9%80%E0%A8%AB%E0%A9%8B%E0%A8%B2%E0%A8%A1" title="ਮੈਨੀਫੋਲਡ – Punjabi" lang="pa" hreflang="pa" data-title="ਮੈਨੀਫੋਲਡ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DB%8C%D9%86%DB%8C%D9%81%D9%88%D9%84%DA%88" title="مینیفولڈ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مینیفولڈ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Variet%C3%A0_topol%C3%B2gica" title="Varietà topològica – Piedmontese" lang="pms" hreflang="pms" data-title="Varietà topològica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozmaito%C5%9B%C4%87" title="Rozmaitość – Polish" lang="pl" hreflang="pl" data-title="Rozmaitość" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Variedade_(matem%C3%A1tica)" title="Variedade (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Variedade (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Varietate_(geometrie)" title="Varietate (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Varietate (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Многообразие – Russian" lang="ru" hreflang="ru" data-title="Многообразие" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Durthi" title="Durthi – Albanian" lang="sq" hreflang="sq" data-title="Durthi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Manifold" title="Manifold – Simple English" lang="en-simple" hreflang="en-simple" data-title="Manifold" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Varieta_(matematika)" title="Varieta (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Varieta (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mnogoterost" title="Mnogoterost – Slovenian" lang="sl" hreflang="sl" data-title="Mnogoterost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%81%D1%82%D1%80%D1%83%D0%BA%D0%BE%D1%81%D1%82" title="Многострукост – Serbian" lang="sr" hreflang="sr" data-title="Многострукост" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Mnogostrukost" title="Mnogostrukost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Mnogostrukost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Monisto" title="Monisto – Finnish" lang="fi" hreflang="fi" data-title="Monisto" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%A5ngfald_(matematik)" title="Mångfald (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Mångfald (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Manipoldo" title="Manipoldo – Tagalog" lang="tl" hreflang="tl" data-title="Manipoldo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A1%E0%B8%99%E0%B8%B4%E0%B9%82%E0%B8%9F%E0%B8%A5%E0%B8%94%E0%B9%8C" title="แมนิโฟลด์ – Thai" lang="th" hreflang="th" data-title="แมนิโฟลด์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87ok_katl%C4%B1" title="Çok katlı – Turkish" lang="tr" hreflang="tr" data-title="Çok katlı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%B2%D0%B8%D0%B4" title="Многовид – Ukrainian" lang="uk" hreflang="uk" data-title="Многовид" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90a_t%E1%BA%A1p" title="Đa tạp – Vietnamese" lang="vi" hreflang="vi" data-title="Đa tạp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%9C%D7%90%D7%9B%D7%98%D7%A2" title="פלאכטע – Yiddish" lang="yi" hreflang="yi" data-title="פלאכטע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B5%81%E5%BD%A2" title="流形 – Cantonese" lang="yue" hreflang="yue" data-title="流形" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B5%81%E5%BD%A2" title="流形 – Chinese" lang="zh" hreflang="zh" data-title="流形" data-language-autonym="中文" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">July 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Klein_bottle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/140px-Klein_bottle.svg.png" decoding="async" width="140" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/210px-Klein_bottle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Klein_bottle.svg/280px-Klein_bottle.svg.png 2x" data-file-width="250" data-file-height="480" /></a><figcaption>The <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> immersed in three-dimensional space</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Polar_stereographic_projections.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Polar_stereographic_projections.jpg/220px-Polar_stereographic_projections.jpg" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Polar_stereographic_projections.jpg/330px-Polar_stereographic_projections.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Polar_stereographic_projections.jpg/440px-Polar_stereographic_projections.jpg 2x" data-file-width="4096" data-file-height="2048" /></a><figcaption>The surface of the Earth requires (at least) two charts to include every point. Here the <a href="/wiki/Globe" title="Globe">globe</a> is decomposed into charts around the <a href="/wiki/North_pole" class="mw-redirect" title="North pole">North</a> and <a href="/wiki/South_Pole" title="South Pole">South Poles</a>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>manifold</b> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> that locally resembles <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> near each point. More precisely, an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional manifold, or <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-manifold</i> for short, is a topological space with the property that each point has a <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhood</a> that is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to an <a href="/wiki/Open_(topology)" class="mw-redirect" title="Open (topology)">open subset</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional Euclidean space. </p><p>One-dimensional manifolds include <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> and <a href="/wiki/Circle" title="Circle">circles</a>, but not <a href="/wiki/Lemniscate" title="Lemniscate">self-crossing curves such as a figure 8</a>. Two-dimensional manifolds are also called <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a>. Examples include the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, the <a href="/wiki/Sphere" title="Sphere">sphere</a>, and the <a href="/wiki/Torus" title="Torus">torus</a>, and also the <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> and <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>. </p><p>The concept of a manifold is central to many parts of <a href="/wiki/Geometry" title="Geometry">geometry</a> and modern <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of <a href="/wiki/Systems_of_equations" class="mw-redirect" title="Systems of equations">systems of equations</a> and as <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphs</a> of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. <a href="/wiki/CT_scan" title="CT scan">CT scans</a>). </p><p>Manifolds can be equipped with additional structure. One important class of manifolds are <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>; their <a href="/wiki/Differentiable_structure" class="mw-redirect" title="Differentiable structure">differentiable structure</a> allows <a href="/wiki/Calculus" title="Calculus">calculus</a> to be done. A <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> on a manifold allows <a href="/wiki/Distance" title="Distance">distances</a> and <a href="/wiki/Angle" title="Angle">angles</a> to be measured. <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic manifolds</a> serve as the <a href="/wiki/Phase_space" title="Phase space">phase spaces</a> in the <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian formalism</a> of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, while four-dimensional <a href="/wiki/Lorentzian_manifold" class="mw-redirect" title="Lorentzian manifold">Lorentzian manifolds</a> model <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p><p>The study of manifolds requires working knowledge of calculus and <a href="/wiki/Topology" title="Topology">topology</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivating_examples">Motivating examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=1" title="Edit section: Motivating examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Circle">Circle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=2" title="Edit section: Circle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_with_overlapping_manifold_charts.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Circle_with_overlapping_manifold_charts.svg/220px-Circle_with_overlapping_manifold_charts.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Circle_with_overlapping_manifold_charts.svg/330px-Circle_with_overlapping_manifold_charts.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Circle_with_overlapping_manifold_charts.svg/440px-Circle_with_overlapping_manifold_charts.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.</figcaption></figure> <p>After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>, <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 1, where the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system"><i>y</i>-coordinate</a> is positive (indicated by the yellow arc in <i>Figure 1</i>). Any point of this arc can be uniquely described by its <i>x</i>-coordinate. So, <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">projection</a> onto the first coordinate is a <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> and <a href="/wiki/Inverse_function" title="Inverse function">invertible</a> <a href="/wiki/Mapping_(mathematics)" class="mw-redirect" title="Mapping (mathematics)">mapping</a> from the upper arc to the <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a> (−1, 1): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54e617a6b0bf7528e91980df68a29d60ac6e9c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.853ex; height:3.009ex;" alt="{\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,}"></span> </p><p>Such functions along with the open regions they map are called <i><a href="/wiki/Atlas_(topology)#Charts" title="Atlas (topology)">charts</a></i>. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd20d8fcddbde345924b092a4d585d2eb37867c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.874ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}"></span> </p><p>Together, these parts cover the whole circle, and the four charts form an <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a> for the circle. </p><p>The top and right charts, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {top} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {top} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f44d5bf66863a5cf40b89dcd61cca395b3ce31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.063ex; height:2.343ex;" alt="{\displaystyle \chi _{\mathrm {top} }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {right} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {right} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e885dc63a8621eaa9aa9a8f33f64f8b7a1f41597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle \chi _{\mathrm {right} }}"></span> respectively, overlap in their domain: their intersection lies in the quarter of the circle where both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-coordinates are positive. Both map this part into the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>, though differently. Thus a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0717a218e62fa93acea22bca78b7c7e68115f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.045ex; height:3.509ex;" alt="{\displaystyle T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}}"></span> can be constructed, which takes values from the co-domain of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {top} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {top} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f44d5bf66863a5cf40b89dcd61cca395b3ce31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.063ex; height:2.343ex;" alt="{\displaystyle \chi _{\mathrm {top} }}"></span> back to the circle using the inverse, followed by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {right} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {right} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e885dc63a8621eaa9aa9a8f33f64f8b7a1f41597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle \chi _{\mathrm {right} }}"></span> back to the interval. If <i>a</i> is any number in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>, then: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>[</mo> <mi>a</mi> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ce5573c9fdbffd539c2d9a9f80fdd33ca68ae3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.164ex; margin-bottom: -0.174ex; width:27.727ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}}"></span> </p><p>Such a function is called a <a href="/wiki/Atlas_(topology)#Transition_maps" title="Atlas (topology)"><i>transition map</i></a>. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_manifold_chart_from_slope.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Circle_manifold_chart_from_slope.svg/220px-Circle_manifold_chart_from_slope.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Circle_manifold_chart_from_slope.svg/330px-Circle_manifold_chart_from_slope.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Circle_manifold_chart_from_slope.svg/440px-Circle_manifold_chart_from_slope.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.</figcaption></figure> <p>The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fca7161bff563f0090aefe795a0c5ef8a775165" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.774ex; height:5.009ex;" alt="{\displaystyle \chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f55ddc80e2443d65ff31fff224633ee80e523d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.154ex; height:5.009ex;" alt="{\displaystyle \chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}}"></span> </p><p>Here <i>s</i> is the slope of the line through the point at coordinates (<i>x</i>, <i>y</i>) and the fixed pivot point (−1, 0); similarly, <i>t</i> is the opposite of the slope of the line through the points at coordinates (<i>x</i>, <i>y</i>) and (+1, 0). The inverse mapping from <i>s</i> to (<i>x</i>, <i>y</i>) is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08d1a41825c29ff1f55d16dae15784d549e8179" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.668ex; margin-bottom: -0.337ex; width:12.163ex; height:13.009ex;" alt="{\displaystyle {\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}}"></span> </p><p>It can be confirmed that <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 1 for all values of <i>s</i> and <i>t</i>. These two charts provide a second atlas for the circle, with the transition map <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {1}{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\frac {1}{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1abc9b568a1b2b246f4b9697409287efe98267d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.937ex; height:5.176ex;" alt="{\displaystyle t={\frac {1}{s}}}"></span> (that is, one has this relation between <i>s</i> and <i>t</i> for every point where <i>s</i> and <i>t</i> are both nonzero). </p><p>Each chart omits a single point, either (−1, 0) for <i>s</i> or (+1, 0) for <i>t</i>, so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. </p> <div class="mw-heading mw-heading3"><h3 id="Sphere">Sphere</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=3" title="Edit section: Sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Sphere" title="Sphere">sphere</a> is an example of a surface. The <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> of <a href="/wiki/Implicit_equation" class="mw-redirect" title="Implicit equation">implicit equation</a> </p> <dl><dd><span class="texhtml"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> – 1 = 0</span></dd></dl> <p>may be covered by an atlas of six <a href="/wiki/Chart_(mathematics)" class="mw-redirect" title="Chart (mathematics)">charts</a>: the plane <span class="texhtml"><i>z</i> = 0</span> divides the sphere into two half spheres (<span class="texhtml"><i>z</i> > 0</span> and <span class="texhtml"><i>z</i> < 0</span>), which may both be mapped on the disc <span class="texhtml"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> < 1</span> by the projection on the <span class="texhtml"><i>xy</i></span> plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes. </p><p>As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart. </p><p>This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the <a href="/wiki/Earth" title="Earth">Earth</a> cannot have a plane representation consisting of a single <a href="/wiki/Map" title="Map">map</a> (also called "chart", see <a href="/wiki/Nautical_chart" title="Nautical chart">nautical chart</a>), and therefore one needs <a href="/wiki/Atlas" title="Atlas">atlases</a> for covering the whole Earth surface. </p> <div class="mw-heading mw-heading3"><h3 id="Other_curves">Other curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=4" title="Edit section: Other curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Conics_and_cubic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conics_and_cubic.svg/220px-Conics_and_cubic.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conics_and_cubic.svg/330px-Conics_and_cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conics_and_cubic.svg/440px-Conics_and_cubic.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>Four manifolds from <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a>: <span style="color:#bc1e47; font-size:100%; line-height:1;" title="#bc1e47">■</span> circles, <span style="color:#fec200; font-size:100%; line-height:1;" title="#fec200">■</span> parabola, <span style="color:#0081cd; font-size:100%; line-height:1;" title="#0081cd">■</span> hyperbola, <span style="color:#009246; font-size:100%; line-height:1;" title="#009246">■</span> cubic.</figcaption></figure> <p>Manifolds need not be <a href="/wiki/Connected_space" title="Connected space">connected</a> (all in "one piece"); an example is a pair of separate circles. </p><p>Manifolds need not be <a href="/wiki/Closed_manifold" title="Closed manifold">closed</a>; thus a line segment without its end points is a manifold. They are never <a href="/wiki/Countable_set" title="Countable set">countable</a>, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a <a href="/wiki/Parabola" title="Parabola">parabola</a>, a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>, and the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of points on a <a href="/wiki/Cubic_curve" class="mw-redirect" title="Cubic curve">cubic curve</a> <span class="nowrap"><i>y</i><sup>2</sup> = <i>x</i><sup>3</sup> − <i>x</i></span> (a closed loop piece and an open, infinite piece). </p><p>However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with four <a href="/wiki/Locally_connected_space" title="Locally connected space">components</a> (i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; <a href="/wiki/Homeomorphism" title="Homeomorphism">topological operations</a> always preserve the number of pieces. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=5" title="Edit section: Mathematical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Categories_of_manifolds" class="mw-redirect" title="Categories of manifolds">Categories of manifolds</a></div> <p>Informally, a manifold is a <a href="/wiki/Topological_space" title="Topological space">space</a> that is "modeled on" Euclidean space. </p><p>There are many different kinds of manifolds. In <a href="/wiki/Geometry_and_topology" title="Geometry and topology">geometry and topology</a>, all manifolds are <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifolds</a>, possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="New term. Same as transition maps? (June 2021)">clarification needed</span></a></i>]</sup>: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance, <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a> have homeomorphisms on overlapping neighborhoods <a href="/wiki/Diffeomorphic" class="mw-redirect" title="Diffeomorphic">diffeomorphic</a> with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole. </p><p>Formally, a (topological) manifold is a <a href="/wiki/Second_countable" class="mw-redirect" title="Second countable">second countable</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> that is locally homeomorphic to a Euclidean space. </p><p><i>Second countable</i> and <i>Hausdorff</i> are <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set</a> conditions; <i>second countable</i> excludes spaces which are in some sense 'too large' such as the <a href="/wiki/Long_line_(topology)" title="Long line (topology)">long line</a>, while <i>Hausdorff</i> excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in <a href="/wiki/Non-Hausdorff_manifold" title="Non-Hausdorff manifold">non-Hausdorff manifolds</a>). </p><p><i>Locally homeomorphic</i> to a Euclidean space means that every point has a neighborhood <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"></span> for some nonnegative integer <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>This implies that either the point is an <a href="/wiki/Isolated_point" title="Isolated point">isolated point</a> (if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"></span>), or it has a neighborhood homeomorphic to the <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open ball</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo><</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb77e171f09bb54e3d5bb5a3aa8ed8086f9140a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.796ex; height:3.176ex;" alt="{\displaystyle \mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.}"></span> This implies also that every point has a neighborhood homeomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> is homeomorphic, and even <a href="/wiki/Diffeomorphic" class="mw-redirect" title="Diffeomorphic">diffeomorphic</a> to any open ball in it (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span>). </p><p>The <span class="texhtml mvar" style="font-style:italic;">n</span> that appears in the preceding definition is called the <i>local dimension</i> of the manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called the <i>dimension</i> of the manifold. This is, in particular, the case when manifolds are <a href="/wiki/Connected_space" title="Connected space">connected</a>. However, some authors admit manifolds that are not connected, and where different points can have different <a href="/wiki/Dimension" title="Dimension">dimensions</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> If a manifold has a fixed dimension, this can be emphasized by calling it a <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="pure_manifold"></span><span class="vanchor-text">pure manifold</span></span></b>. For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of a sphere and a line in three-dimensional space is <i>not</i> a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is <a href="/wiki/Locally_constant" class="mw-redirect" title="Locally constant">locally constant</a>), each <a href="/wiki/Connected_space" title="Connected space">connected component</a> has a fixed dimension. </p><p><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf-theoretically</a>, a manifold is a <a href="/wiki/Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed space</a>, whose structure <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Charts,_atlases,_and_transition_maps"><span id="Charts.2C_atlases.2C_and_transition_maps"></span>Charts, atlases, and transition maps</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=6" title="Edit section: Charts, atlases, and transition maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas (topology)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Differentiable_manifold#Definition" title="Differentiable manifold">Differentiable manifold</a></div> <p>The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a manifold can be described using <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mathematical maps</a>, called <i>coordinate charts</i>, collected in a mathematical <i>atlas</i>. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. </p> <div class="mw-heading mw-heading3"><h3 id="Charts">Charts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=7" title="Edit section: Charts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Coordinate_chart" class="mw-redirect" title="Coordinate chart">Coordinate chart</a></div> <p>A <i>coordinate map</i>, a <i>coordinate chart</i>, or simply a <i>chart</i>, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure.<sup id="cite_ref-Morita_2-0" class="reference"><a href="#cite_note-Morita-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For a topological manifold, the simple space is a subset of some Euclidean space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> and interest focuses on the topological structure. This structure is preserved by <a href="/wiki/Homeomorphisms" class="mw-redirect" title="Homeomorphisms">homeomorphisms</a>, invertible maps that are continuous in both directions. </p><p>In the case of a differentiable manifold, a set of <i>charts</i> called an <i>atlas</i>, whose <i>transition functions</i> (see below) are all differentiable, allows us to do calculus on it. <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">Polar coordinates</a>, for example, form a chart for the plane <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> minus the positive <i>x</i>-axis and the origin. Another example of a chart is the map χ<sub>top</sub> mentioned above, a chart for the circle. </p> <div class="mw-heading mw-heading3"><h3 id="Atlases">Atlases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=8" title="Edit section: Atlases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas (topology)</a></div> <p>The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an <i><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a></i>. An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union is also an atlas. </p><p>The atlas containing all possible charts consistent with a given atlas is called the <i>maximal atlas</i> (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). </p> <div class="mw-heading mw-heading3"><h3 id="Transition_maps">Transition maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=9" title="Edit section: Transition maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, a <i>transition function</i> can be defined which goes from an open ball in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> to the manifold and then back to another (or perhaps the same) open ball in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. The resultant map, like the map <i>T</i> in the circle example above, is called a <i>change of coordinates</i>, a <i>coordinate transformation</i>, a <i>transition function</i>, or a <i>transition map</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Additional_structure">Additional structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=10" title="Edit section: Additional structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold. </p><p>This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> (that is, if they are <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>), the differential structure transfers to the manifold and turns it into a differentiable manifold. <a href="/wiki/Complex_manifold" title="Complex manifold">Complex manifolds</a> are introduced in an analogous way by requiring that the transition functions of an atlas are <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a>. For <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifolds</a>, the transition functions must be <a href="/wiki/Symplectomorphism" title="Symplectomorphism">symplectomorphisms</a>. </p><p>The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called <i>compatible</i>. </p><p>These notions are made precise in general through the use of <a href="/wiki/Pseudogroup" title="Pseudogroup">pseudogroups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Manifold_with_boundary">Manifold with boundary</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=11" title="Edit section: Manifold with boundary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Topological_manifold#Manifolds_with_boundary" title="Topological manifold">Topological manifold § Manifolds with boundary</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Boundary_of_Manifold_with_charts.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Boundary_of_Manifold_with_charts.png/280px-Boundary_of_Manifold_with_charts.png" decoding="async" width="280" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Boundary_of_Manifold_with_charts.png/420px-Boundary_of_Manifold_with_charts.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Boundary_of_Manifold_with_charts.png/560px-Boundary_of_Manifold_with_charts.png 2x" data-file-width="1576" data-file-height="1074" /></a><figcaption>A smooth 2-manifold: The interior chart with transition map <span class="texhtml"><i>φ</i><sub>1</sub></span> maps an open subset around an interior point to an open Euclidean subset, while the boundary chart with transition map <span class="texhtml"><i>φ</i><sub>2</sub></span> maps a closed subset around a boundary point to a closed Euclidean subset. The boundary is itself a <span class="texhtml">1</span>-manifold without boundary, so the chart with transition map <span class="texhtml"><i>φ</i><sub>3</sub></span> must map to an open Euclidean subset.</figcaption></figure> <p>A <b>manifold with boundary</b> is a manifold with an edge. For example, a sheet of paper is a <a href="/wiki/2-manifold" class="mw-redirect" title="2-manifold">2-manifold</a> with a 1-dimensional boundary. The boundary of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-manifold with boundary is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}"></span>-manifold. A <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a> (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a <a href="/wiki/1-manifold" class="mw-redirect" title="1-manifold">1-manifold</a>. A <a href="/wiki/Square" title="Square">square</a> with interior is also a 2-manifold with boundary. A <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a> (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. </p><p>In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-ball <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mi mathvariant="normal">Σ</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e100ff66f8a9c9011ad5412c710e01fbf4244e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.632ex; height:3.176ex;" alt="{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}}"></span>. Every boundary point has a neighborhood homeomorphic to the "half" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-ball <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mi mathvariant="normal">Σ</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc8fecee7bef6cb98b03607beaa10810a891342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.186ex; height:3.176ex;" alt="{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}}"></span>. Any homeomorphism between half-balls must send points with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d278bb750c26d4220fe951a98423a8e9cf354b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{1}=0}"></span> to points with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d278bb750c26d4220fe951a98423a8e9cf354b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{1}=0}"></span>. This invariance allows to "define" boundary points; see next paragraph. </p> <div class="mw-heading mw-heading3"><h3 id="Boundary_and_interior">Boundary and interior</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=12" title="Edit section: Boundary and interior"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> be a manifold with boundary. The <b>interior</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Int</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5926b38819033c310ba401e01fb80883fe9f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.866ex; height:2.176ex;" alt="{\displaystyle \operatorname {Int} M}"></span>, is the set of points in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> which have neighborhoods homeomorphic to an open subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. The <b>boundary</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8456299e2b600f44f5aa08920be090af1b35e013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.76ex; height:2.176ex;" alt="{\displaystyle \partial M}"></span>, is the <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Int</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5926b38819033c310ba401e01fb80883fe9f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.866ex; height:2.176ex;" alt="{\displaystyle \operatorname {Int} M}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. The boundary points can be characterized as those points which land on the boundary hyperplane <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n}=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{n}=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e933fef438fcb4c8ad97063ce5d5650c53537c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.618ex; height:2.843ex;" alt="{\displaystyle (x_{n}=0)}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{+}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8024dbc8cbeae85a715a4d414e9c06e2cd68e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.189ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} _{+}^{n}}"></span> under some coordinate chart. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a manifold with boundary of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Int</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5926b38819033c310ba401e01fb80883fe9f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.866ex; height:2.176ex;" alt="{\displaystyle \operatorname {Int} M}"></span> is a manifold (without boundary) of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8456299e2b600f44f5aa08920be090af1b35e013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.76ex; height:2.176ex;" alt="{\displaystyle \partial M}"></span> is a manifold (without boundary) of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=13" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. </p> <div class="mw-heading mw-heading3"><h3 id="Charts_2">Charts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=14" title="Edit section: Charts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_with_chart.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Sphere_with_chart.svg/170px-Sphere_with_chart.svg.png" decoding="async" width="170" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Sphere_with_chart.svg/255px-Sphere_with_chart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Sphere_with_chart.svg/340px-Sphere_with_chart.svg.png 2x" data-file-width="366" data-file-height="591" /></a><figcaption>The chart maps the part of the sphere with positive <i>z</i> coordinate to a disc.</figcaption></figure> <p>Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> is identified, and then an atlas covering this subset is constructed. The concept of <i>manifold</i> grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere: </p> <div class="mw-heading mw-heading4"><h4 id="Sphere_with_charts">Sphere with charts</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=15" title="Edit section: Sphere with charts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775afdf3449b1cfb06d12798796feb321467f0ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.988ex; height:3.343ex;" alt="{\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.}"></span> </p><p>The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. Consider the northern hemisphere, which is the part with positive <i>z</i> coordinate (coloured red in the picture on the right). The function <span class="texhtml mvar" style="font-style:italic;">χ</span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (x,y,z)=(x,y),\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (x,y,z)=(x,y),\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108dbaa2e46762e5b24a83558bca9c663056b0f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.947ex; height:2.843ex;" alt="{\displaystyle \chi (x,y,z)=(x,y),\ }"></span> </p><p>maps the northern hemisphere to the open <a href="/wiki/Unit_disc" class="mw-redirect" title="Unit disc">unit disc</a> by projecting it on the (<i>x</i>, <i>y</i>) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (<i>x</i>, <i>z</i>) plane and two charts projecting on the (<i>y</i>, <i>z</i>) plane, an atlas of six charts is obtained which covers the entire sphere. </p><p>This can be easily generalized to higher-dimensional spheres. </p> <div class="mw-heading mw-heading3"><h3 id="Patchwork">Patchwork</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=16" title="Edit section: Patchwork"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Surgery_theory" title="Surgery theory">Surgery theory</a></div> <p>A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold. </p><p>The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold. </p><p>This can be illustrated with the transition map <i>t</i> = <sup>1</sup>⁄<sub><i>s</i></sub> from the second half of the circle example. Start with two copies of the line. Use the coordinate <i>s</i> for the first copy, and <i>t</i> for the second copy. Now, glue both copies together by identifying the point <i>t</i> on the second copy with the point <i>s</i> = <sup>1</sup>⁄<sub><i>t</i></sub> on the first copy (the points <i>t</i> = 0 and <i>s</i> = 0 are not identified with any point on the first and second copy, respectively). This gives a circle. </p> <div class="mw-heading mw-heading4"><h4 id="Intrinsic_and_extrinsic_view">Intrinsic and extrinsic view</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=17" title="Edit section: Intrinsic and extrinsic view"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first construction and this construction are very similar, but represent rather different points of view. In the first construction, the manifold is seen as <a href="/wiki/Embedding" title="Embedding">embedded</a> in some Euclidean space. This is the <i>extrinsic view</i>. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space, it is always clear whether a vector at some point is <a href="/wiki/Tangential" class="mw-redirect" title="Tangential">tangential</a> or <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal</a> to some surface through that point. </p><p>The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the <i>intrinsic view</i>. It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a <a href="/wiki/Normal_bundle" title="Normal bundle">normal bundle</a>, but instead there is an intrinsic <a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">stable normal bundle</a>. </p> <div class="mw-heading mw-heading4"><h4 id="n-Sphere_as_a_patchwork"><i>n</i>-Sphere as a patchwork</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=18" title="Edit section: n-Sphere as a patchwork"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere"><i>n</i>-sphere</a> <b>S</b><sup><i>n</i></sup> is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An <i>n</i>-sphere <b>S</b><sup><i>n</i></sup> can be constructed by gluing together two copies of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. The transition map between them is <a href="/wiki/Inversion_in_a_sphere" class="mw-redirect" title="Inversion in a sphere">inversion in a sphere</a>, defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d700fbfa7db1a94b49659e7342fab0af214545a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.5ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}"></span> </p><p>This function is its own inverse and thus can be used in both directions. As the transition map is a <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a>, this atlas defines a smooth manifold. In the case <i>n</i> = 1, the example simplifies to the circle example given earlier. </p> <div class="mw-heading mw-heading3"><h3 id="Identifying_points_of_a_manifold">Identifying points of a manifold</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=19" title="Edit section: Identifying points of a manifold"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Orbifold" title="Orbifold">Orbifold</a> and <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">Group action (mathematics)</a></div> <p>It is possible to define different points of a manifold to be the same point. This can be visualized as gluing these points together in a single point, forming a <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a>. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, <a href="/wiki/Orbifold" title="Orbifold">orbifolds</a> and <a href="/wiki/CW_complex" title="CW complex">CW complexes</a> are considered to be relatively <a href="/wiki/Well-behaved" class="mw-redirect" title="Well-behaved">well-behaved</a>. An example of a quotient space of a manifold that is also a manifold is the <a href="/wiki/Real_projective_space" title="Real projective space">real projective space</a>, identified as a quotient space of the corresponding sphere. </p><p>One method of identifying points (gluing them together) is through a right (or left) action of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, which <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> on the manifold. Two points are identified if one is moved onto the other by some group element. If <i>M</i> is the manifold and <i>G</i> is the group, the resulting quotient space is denoted by <i>M</i> / <i>G</i> (or <i>G</i> \ <i>M</i>). </p><p>Manifolds which can be constructed by identifying points include <a href="/wiki/Torus#Topology" title="Torus">tori</a> and <a href="/wiki/Real_projective_space" title="Real projective space">real projective spaces</a> (starting with a plane and a sphere, respectively). </p> <div class="mw-heading mw-heading3"><h3 id="Gluing_along_boundaries">Gluing along boundaries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=20" title="Edit section: Gluing along boundaries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">Quotient space (topology)</a></div> <p>Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together. </p><p>Formally, the gluing is defined by a <a href="/wiki/Bijection" title="Bijection">bijection</a> between the two boundaries<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (February 2010)">dubious</span></a> – <a href="/wiki/Talk:Manifold#gluing_manifolds_with_boundary" title="Talk:Manifold">discuss</a></i>]</sup>. Two points are identified when they are mapped onto each other. For a topological manifold, this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly, for a differentiable manifold, it has to be a diffeomorphism. For other manifolds, other structures should be preserved. </p><p>A finite cylinder may be constructed as a manifold by starting with a strip [0,1] × [0,1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> may be obtained by gluing a sphere with a hole in it to a <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> along their respective circular boundaries. </p> <div class="mw-heading mw-heading3"><h3 id="Cartesian_products"><span class="anchor" id="Cartesian_products"></span> Cartesian products</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=21" title="Edit section: Cartesian products"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of manifolds is also a manifold. </p><p>The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the <a href="/wiki/Product_topology" title="Product topology">product topology</a>, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinders</a>, for example, as <b>S</b><sup>1</sup> × <b>S</b><sup>1</sup> and <b>S</b><sup>1</sup> × [0,1], respectively. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Red_cylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Red_cylinder.svg/220px-Red_cylinder.svg.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Red_cylinder.svg/330px-Red_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Red_cylinder.svg/440px-Red_cylinder.svg.png 2x" data-file-width="711" data-file-height="641" /></a><figcaption>A finite cylinder is a manifold with boundary.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=22" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History of manifolds and varieties</a></div> <p>The study of manifolds combines many important areas of mathematics: it generalizes concepts such as <a href="/wiki/Curve" title="Curve">curves</a> and surfaces as well as ideas from <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and topology. </p> <div class="mw-heading mw-heading3"><h3 id="Early_development">Early development</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=23" title="Edit section: Early development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Before the modern concept of a manifold there were several important results. </p><p><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a> considers spaces where <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> fails. <a href="/wiki/Giovanni_Gerolamo_Saccheri" class="mw-redirect" title="Giovanni Gerolamo Saccheri">Saccheri</a> first studied such geometries in 1733, but sought only to disprove them. <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>, <a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a> and <a href="/wiki/Nikolai_Ivanovich_Lobachevsky" class="mw-redirect" title="Nikolai Ivanovich Lobachevsky">Lobachevsky</a> independently discovered them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> and <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive <a href="/wiki/Curvature" title="Curvature">curvature</a>, respectively. </p><p>Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His <a href="/wiki/Theorema_egregium" class="mw-redirect" title="Theorema egregium">theorema egregium</a> gives a method for computing the curvature of a <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a> without considering the <a href="/wiki/Ambient_space" class="mw-redirect" title="Ambient space">ambient space</a> in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an <a href="/wiki/Intrinsic_and_extrinsic_properties" title="Intrinsic and extrinsic properties">intrinsic property</a>. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. </p><p>Another, more topological example of an intrinsic <a href="/wiki/Topological_property" title="Topological property">property</a> of a manifold is its <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a>. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> showed that for a convex <a href="/wiki/Polytope" title="Polytope">polytope</a> in the three-dimensional Euclidean space with <i>V</i> <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> (or corners), <i>E</i> edges, and <i>F</i> faces,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-E+F=2.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>2.</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V-E+F=2.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b650edf70725f0fec8b36e7ec9313ed61b6561" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.473ex; height:2.343ex;" alt="{\displaystyle V-E+F=2.\ }"></span>The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a <a href="/wiki/Topological_map" title="Topological map">topological map</a> with <i>V</i> vertices, <i>E</i> edges, and <i>F</i> faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Thus 2 is a topological invariant of the sphere, called its <b>Euler characteristic</b>. On the other hand, a <a href="/wiki/Torus" title="Torus">torus</a> can be sliced open by its 'parallel' and 'meridian' circles, creating a map with <i>V</i> = 1 vertex, <i>E</i> = 2 edges, and <i>F</i> = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariant</a>, which can be extended to higher dimensions using <a href="/wiki/Betti_number" title="Betti number">Betti numbers</a>. In the mid nineteenth century, the <a href="/wiki/Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a> linked the Euler characteristic to the <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Synthesis">Synthesis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=24" title="Edit section: Synthesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Investigations of <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a> and <a href="/wiki/Carl_Gustav_Jacobi" class="mw-redirect" title="Carl Gustav Jacobi">Carl Gustav Jacobi</a> on inversion of <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integrals</a> in the first half of 19th century led them to consider special types of complex manifolds, now known as <a href="/wiki/Abelian_variety" title="Abelian variety">Jacobians</a>. <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> further contributed to their theory, clarifying the geometric meaning of the process of <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> of functions of complex variables. </p><p>Another important source of manifolds in 19th century mathematics was <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">analytical mechanics</a>, as developed by <a href="/wiki/Sim%C3%A9on_Poisson" class="mw-redirect" title="Siméon Poisson">Siméon Poisson</a>, Jacobi, and <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>. The possible states of a mechanical system are thought to be points of an abstract space, <a href="/wiki/Phase_space" title="Phase space">phase space</a> in <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> and <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their <a href="/wiki/Generalized_coordinate" class="mw-redirect" title="Generalized coordinate">generalized coordinates</a>. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various <a href="/wiki/Conservation_laws" class="mw-redirect" title="Conservation laws">conservation laws</a> constrain it to more complicated formations, e.g. <a href="/w/index.php?title=Liouville_tori&action=edit&redlink=1" class="new" title="Liouville tori (page does not exist)">Liouville tori</a>. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>, one of the founders of topology. </p><p>Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name <i>manifold</i> comes from Riemann's original <a href="/wiki/German_language" title="German language">German</a> term, <i>Mannigfaltigkeit</i>, which <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">William Kingdon Clifford</a> translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a <i>Mannigfaltigkeit</i>, because the variable can have <i>many</i> values. He distinguishes between <i>stetige Mannigfaltigkeit</i> and <i>diskrete</i> <i>Mannigfaltigkeit</i> (<i>continuous manifoldness</i> and <i>discontinuous manifoldness</i>), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a>, Riemann constructs an <i>n-fach ausgedehnte Mannigfaltigkeit</i> (<i>n times extended manifoldness</i> or <i>n-dimensional manifoldness</i>) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a <i>Mannigfaltigkeit</i> evolved into what is today formalized as a manifold. Riemannian manifolds and <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a> are named after Riemann. </p> <div class="mw-heading mw-heading3"><h3 id="Poincaré's_definition"><span id="Poincar.C3.A9.27s_definition"></span>Poincaré's definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=25" title="Edit section: Poincaré's definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In his very influential paper, <a href="/wiki/Analysis_Situs_(paper)" title="Analysis Situs (paper)">Analysis Situs</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Henri Poincaré gave a definition of a differentiable manifold (<i>variété</i>) which served as a precursor to the modern concept of a manifold.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In the first section of Analysis Situs, Poincaré defines a manifold as the <a href="/wiki/Level_set" title="Level set">level set</a> of a <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a> function between Euclidean spaces that satisfies the nondegeneracy hypothesis of the <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>. In the third section, he begins by remarking that the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of a continuously differentiable function is a manifold in the latter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds' (<i>une chaîne des variétés</i>). </p><p>Poincaré's notion of a <i>chain of manifolds</i> is a precursor to the modern notion of atlas. In particular, he considers two manifolds defined respectively as graphs of functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0994fb69a6feada2c857693348ec209c50b66d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.055ex; height:2.843ex;" alt="{\displaystyle \theta (y)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta '\left(y'\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta '\left(y'\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df1b8fba8ab5321d5b3ffea5c5cd94a9d305efdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.817ex; height:3.009ex;" alt="{\displaystyle \theta '\left(y'\right)}"></span>. If these manifolds overlap (<i>a une partie commune</i>), then he requires that the coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> depend continuously differentiably on the coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span> and vice versa ('<i>...les <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> sont fonctions analytiques des <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span> et inversement</i>'). In this way he introduces a precursor to the notion of a <a href="#Charts">chart</a> and of a <a href="#Transition_maps">transition map</a>. </p><p>For example, the unit circle in the plane can be thought of as the graph of the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle y={\sqrt {1-x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac282fd415e57a16d0571c14bbaae6678f99565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.964ex; height:3.509ex;" alt="{\textstyle y={\sqrt {1-x^{2}}}}"></span> or else the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y=-{\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle y=-{\sqrt {1-x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173a110fb1529edfad3da406e90d3301904c9549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.773ex; height:3.509ex;" alt="{\textstyle y=-{\sqrt {1-x^{2}}}}"></span> in a neighborhood of every point except the points (1, 0) and (−1, 0); and in a neighborhood of those points, it can be thought of as the graph of, respectively, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x={\sqrt {1-y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x={\sqrt {1-y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26cb2b799254bd3bcbe835a608c79dd5cc0b1de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.969ex; height:3.509ex;" alt="{\textstyle x={\sqrt {1-y^{2}}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=-{\sqrt {1-y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=-{\sqrt {1-y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c23f2962ae7b30919617ba5b66866605e19f721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.778ex; height:3.509ex;" alt="{\textstyle x=-{\sqrt {1-y^{2}}}}"></span>. The circle can be represented by a graph in the neighborhood of every point because the left hand side of its defining equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee594b8851d760d0e2d44aba714907aca657b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.703ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-1=0}"></span> has nonzero gradient at every point of the circle. By the <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>, every <a href="/wiki/Submanifold" title="Submanifold">submanifold</a> of Euclidean space is locally the graph of a function. </p><p><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a <a href="/wiki/Topological_space" title="Topological space">topological space</a> that followed shortly. During the 1930s <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a> and others clarified the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundational</a> aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="/wiki/Lie_group" title="Lie group">Lie group</a> theory. Notably, the <a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding theorem</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> showed that the intrinsic definition in terms of charts was equivalent to Poincaré's definition in terms of subsets of Euclidean space. </p> <div class="mw-heading mw-heading3"><h3 id="Topology_of_manifolds:_highlights">Topology of manifolds: highlights</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=26" title="Edit section: Topology of manifolds: highlights"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two-dimensional manifolds, also known as a 2D <i>surfaces</i> embedded in our common 3D space, were considered by Riemann under the guise of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, and rigorously classified in the beginning of the 20th century by <a href="/wiki/Poul_Heegaard" title="Poul Heegaard">Poul Heegaard</a> and <a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a>. Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>. After nearly a century, <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a> proved the Poincaré conjecture (see the <a href="/wiki/Poincar%C3%A9_conjecture#Solution" title="Poincaré conjecture">Solution of the Poincaré conjecture</a>). <a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a>'s <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">geometrization program</a>, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by <a href="/wiki/Michael_Freedman" title="Michael Freedman">Michael Freedman</a> and in a different setting, by <a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Simon Donaldson</a>, who was motivated by the then recent progress in theoretical physics (<a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>), where they serve as a substitute for ordinary 'flat' <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. <a href="/wiki/Andrey_Markov_(Soviet_mathematician)" class="mw-redirect" title="Andrey Markov (Soviet mathematician)">Andrey Markov Jr.</a> showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, including <a href="/wiki/Generalized_Poincar%C3%A9_conjecture" title="Generalized Poincaré conjecture">analogues of the Poincaré conjecture</a>, had been done earlier by <a href="/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a>, <a href="/wiki/John_Milnor" title="John Milnor">John Milnor</a>, <a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a> and <a href="/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Sergei Novikov</a>. A very pervasive and flexible technique underlying much work on the <a href="/wiki/Differential_topology" title="Differential topology">topology of manifolds</a> is <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Additional_structure_2">Additional structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=27" title="Edit section: Additional structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Categories_of_manifolds" class="mw-redirect" title="Categories of manifolds">Categories of manifolds</a></div> <div class="mw-heading mw-heading3"><h3 id="Topological_manifolds">Topological manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=28" title="Edit section: Topological manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a></div> <p>The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space <a href="/wiki/Local_homeomorphism" title="Local homeomorphism">locally homeomorphic</a> to a Euclidean space. This means that every point has a neighbourhood for which there exists a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> (a <a href="/wiki/Bijection" title="Bijection">bijective</a> <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous function</a> whose inverse is also continuous) mapping that neighbourhood to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. These homeomorphisms are the charts of the manifold. </p><p>A <i>topological</i> manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any <i>particular</i> and <i>consistent</i> choice of such concepts.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In order to discuss such properties for a manifold, one needs to specify further structure and consider <a href="#Differentiable_manifolds">differentiable manifolds</a> and <a href="#Riemannian_manifolds">Riemannian manifolds</a> discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles. </p><p>Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> and <a href="/wiki/Second-countable_space" title="Second-countable space">second countable</a>. </p><p>The <i>dimension</i> of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number <i>n</i> in the definition). All points in a <a href="/wiki/Connected_space" title="Connected space">connected</a> manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension. </p> <div class="mw-heading mw-heading3"><h3 id="Differentiable_manifolds">Differentiable manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=29" title="Edit section: Differentiable manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable manifold</a></div> <p>For most applications, a special kind of topological manifold, namely, a <b>differentiable manifold</b>, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use <a href="/wiki/Calculus" title="Calculus">calculus</a> on a differentiable manifold. Each point of an <i>n</i>-dimensional differentiable manifold has a <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a>. This is an <i>n</i>-dimensional Euclidean space consisting of the <a href="/wiki/Tangent_vectors" class="mw-redirect" title="Tangent vectors">tangent vectors</a> of the curves through the point. </p><p>Two important classes of differentiable manifolds are <b>smooth</b> and <b><a href="/wiki/Analytic_manifold" title="Analytic manifold">analytic manifolds</a></b>. For smooth manifolds the transition maps are smooth, that is, infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are <a href="/wiki/Analytic_function" title="Analytic function">analytic</a> (they can be expressed as <a href="/wiki/Power_series" title="Power series">power series</a>). The sphere can be given analytic structure, as can most familiar curves and surfaces. </p><p>A <a href="/wiki/Rectifiable_set" title="Rectifiable set">rectifiable set</a> generalizes the idea of a piecewise smooth or <a href="/wiki/Rectifiable_curve" class="mw-redirect" title="Rectifiable curve">rectifiable curve</a> to higher dimensions; however, rectifiable sets are not in general manifolds. </p> <div class="mw-heading mw-heading3"><h3 id="Riemannian_manifolds">Riemannian manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=30" title="Edit section: Riemannian manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></div> <p>To measure distances and angles on manifolds, the manifold must be Riemannian. A <i>Riemannian manifold</i> is a differentiable manifold in which each <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> is equipped with an <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> in a manner which varies smoothly from point to point. Given two tangent vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>, the inner product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6704456ffbbed3155bc5dd40e03459129011c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.301ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle }"></span> gives a real number. The <a href="/wiki/Dot_product" title="Dot product">dot</a> (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length, <a href="/wiki/Angle" title="Angle">angles</a>, <a href="/wiki/Area" title="Area">areas</a> (or <a href="/wiki/Volume" title="Volume">volumes</a>), <a href="/wiki/Curvature" title="Curvature">curvature</a> and <a href="/wiki/Divergence" title="Divergence">divergence</a> of <a href="/wiki/Vector_field" title="Vector field">vector fields</a>. </p><p>All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all <span class="texhtml"><i>n</i></span>-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it. </p> <div class="mw-heading mw-heading3"><h3 id="Finsler_manifolds">Finsler manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=31" title="Edit section: Finsler manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler manifold</a></div> <p>A <b>Finsler manifold</b> allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">norm</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"></span>, in a manner which varies smoothly from point to point. This norm can be extended to a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>, defining the length of a curve; but it cannot in general be used to define an inner product. </p><p>Any Riemannian manifold is a Finsler manifold. </p> <div class="mw-heading mw-heading3"><h3 id="Lie_groups">Lie groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=32" title="Edit section: Lie groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lie_group" title="Lie group">Lie group</a></div> <p><b>Lie groups</b>, named after <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a>, are differentiable manifolds that carry also the structure of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> which is such that the group operations are defined by smooth maps. </p><p>A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a <a href="/wiki/Compact_space" title="Compact space">compact</a> Lie group is the circle: the group operation is simply rotation. This group, known as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {U} (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">U</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {U} (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39823af285de3e72d7d50f96fe5b8ba024017af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle \operatorname {U} (1)}"></span>, can be also characterised as the group of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> of <a href="/wiki/Absolute_value" title="Absolute value">modulus</a> 1 with multiplication as the group operation. </p><p>Other examples of Lie groups include special groups of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, which are all subgroups of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>, the group of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matrices with non-zero determinant. If the matrix entries are <a href="/wiki/Real_number" title="Real number">real numbers</a>, this will be an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}"></span>-dimensional disconnected manifold. The <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal groups</a>, the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a> of the sphere and <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hyperspheres</a>, are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(n-1)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(n-1)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1d96c185de1bffc1e78739934b09489f683efc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.926ex; height:2.843ex;" alt="{\displaystyle n(n-1)/2}"></span> dimensional manifolds, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> is the dimension of the sphere. Further examples can be found in the <a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">table of Lie groups</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_types_of_manifolds">Other types of manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=33" title="Edit section: Other types of manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <i><a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a></i> is a manifold whose charts take values in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> and whose transition functions are <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> on the overlaps. These manifolds are the basic objects of study in <a href="/wiki/Complex_geometry" title="Complex geometry">complex geometry</a>. A one-complex-dimensional manifold is called a <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a>. An <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional complex manifold has dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}"></span> as a real differentiable manifold.</li> <li>A <i><a href="/wiki/CR_manifold" title="CR manifold">CR manifold</a></i> is a manifold modeled on boundaries of domains in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>.</li> <li>'Infinite dimensional manifolds': to allow for infinite dimensions, one may consider <a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifolds</a> which are locally homeomorphic to <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>. Similarly, Fréchet manifolds are locally homeomorphic to <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a>.</li> <li>A <i><a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a></i> is a kind of manifold which is used to represent the phase spaces in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>. They are endowed with a <a href="/wiki/Differential_form" title="Differential form">2-form</a> that defines the <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a>. A closely related type of manifold is a <a href="/wiki/Contact_geometry" title="Contact geometry">contact manifold</a>.</li> <li>A <i><a href="/wiki/Combinatorial_manifold" class="mw-redirect" title="Combinatorial manifold">combinatorial manifold</a></i> is a kind of manifold which is discretization of a manifold. It usually means a <a href="/wiki/Piecewise_linear_manifold" title="Piecewise linear manifold">piecewise linear manifold</a> made by <a href="/wiki/Simplicial_complexes" class="mw-redirect" title="Simplicial complexes">simplicial complexes</a>.</li> <li>A <i><a href="/wiki/Digital_manifold" title="Digital manifold">digital manifold</a></i> is a special kind of combinatorial manifold which is defined in digital space. See <a href="/wiki/Digital_topology" title="Digital topology">digital topology</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Classification_and_invariants">Classification and invariants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=34" title="Edit section: Classification and invariants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></div> <p>Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds. </p><p>The classification of smooth closed manifolds is well understood <i>in principle</i>, except in <a href="/wiki/4-manifold" title="4-manifold">dimension 4</a>: in low dimensions (2 and 3) it is geometric, via the <a href="/wiki/Uniformization_theorem" title="Uniformization theorem">uniformization theorem</a> and the <a href="/wiki/Solution_of_the_Poincar%C3%A9_conjecture" class="mw-redirect" title="Solution of the Poincaré conjecture">solution of the Poincaré conjecture</a>, and in high dimension (5 and above) it is algebraic, via <a href="/wiki/Surgery_theory" title="Surgery theory">surgery theory</a>. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions. </p><p>Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a <a href="/wiki/Complete_set_of_invariants" title="Complete set of invariants">complete set of invariants</a>. </p><p>This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object. </p><p>However, one can determine if two manifolds are <i>different</i> if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as <b><a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariants</a></b>, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are <i>invariant</i> under different descriptions. </p><p>One could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. It is known that for manifolds of dimension 4 and higher, <a href="/wiki/Classification_of_manifolds#Computability" title="Classification of manifolds">no program exists</a> that can decide whether two manifolds are diffeomorphic. </p><p>Smooth manifolds have <a href="/wiki/Classification_of_manifolds#Enumeration_versus_invariants" title="Classification of manifolds">a rich set of invariants</a>, coming from <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a>, classic algebraic topology, and <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a>. The most familiar invariants, which are visible for surfaces, are <a href="/wiki/Orientability" title="Orientability">orientability</a> (a normal invariant, also detected by <a href="/wiki/Singular_homology" title="Singular homology">homology</a>) and <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> (a homological invariant). </p><p>Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the <a href="/wiki/Curvature_of_Riemannian_manifolds" title="Curvature of Riemannian manifolds">curvature of a Riemannian manifold</a> and the <a href="/wiki/Torsion_(differential_geometry)" class="mw-redirect" title="Torsion (differential geometry)">torsion</a> of a manifold equipped with an <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a>. This distinction between local invariants and no local invariants is a common way to distinguish between <a href="/wiki/Geometry_and_topology#Local_versus_global_structure" title="Geometry and topology">geometry and topology</a>. All invariants of a smooth closed manifold are thus global. </p><p><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic topology</a> is a source of a number of important global invariant properties. Some key criteria include the <i><a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a></i> property and orientability (see below). Indeed, several branches of mathematics, such as <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> and <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> theory, and the theory of <a href="/wiki/Characteristic_classes" class="mw-redirect" title="Characteristic classes">characteristic classes</a> were founded in order to study invariant properties of manifolds. </p> <div class="mw-heading mw-heading2"><h2 id="Surfaces">Surfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=35" title="Edit section: Surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Orientability">Orientability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=36" title="Edit section: Orientability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orientable_manifold" class="mw-redirect" title="Orientable manifold">Orientable manifold</a></div> <p>In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Given an <a href="/wiki/Basis_(linear_algebra)#Ordered_bases_and_coordinates" title="Basis (linear algebra)">ordered basis</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are <i><a href="/wiki/Orientability" title="Orientability">orientable</a></i> manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable. </p><p>Some illustrative examples of non-orientable manifolds include: (1) the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a>, which is a manifold with boundary, (2) the <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a>, which must intersect itself in its 3-space representation, and (3) the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>, which arises naturally in geometry. </p> <div class="mw-heading mw-heading4"><h4 id="Möbius_strip"><span id="M.C3.B6bius_strip"></span>Möbius strip</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=37" title="Edit section: Möbius strip"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Moebius_strip.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/140px-Moebius_strip.svg.png" decoding="async" width="140" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/210px-Moebius_strip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/280px-Moebius_strip.svg.png 2x" data-file-width="712" data-file-height="640" /></a><figcaption>Möbius strip</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a></div> <p>Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a <i>single</i> side. Similarly to the Klein Bottle below, this two dimensional surface would need to intersect itself in two dimensions, but can easily be constructed in three or more dimensions. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quasitoric_manifold" title="Quasitoric manifold">Quasitoric manifold</a></div> <div class="mw-heading mw-heading4"><h4 id="Klein_bottle">Klein bottle</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=38" title="Edit section: Klein bottle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a></div> <p>Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into <a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">cross-caps</a>. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. In three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space. </p> <div class="mw-heading mw-heading4"><h4 id="Real_projective_plane">Real projective plane</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=39" title="Edit section: Real projective plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Real_projective_space" title="Real projective space">Real projective space</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:BoysSurfaceTopView.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/BoysSurfaceTopView.PNG/220px-BoysSurfaceTopView.PNG" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2d/BoysSurfaceTopView.PNG 1.5x" data-file-width="289" data-file-height="282" /></a><figcaption>The <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a> is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as <a href="/wiki/Boy%27s_surface" title="Boy's surface">Boy's surface</a>.</figcaption></figure> <p>Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called <i>antipodes</i>. Although there is no way to do so physically, it is possible (by considering a <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a>) to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane". </p> <div class="mw-heading mw-heading3"><h3 id="Genus_and_the_Euler_characteristic">Genus and the Euler characteristic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=40" title="Edit section: Genus and the Euler characteristic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For two dimensional manifolds a key invariant property is the <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a>, or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a>, and more generally <a href="/wiki/Betti_number" title="Betti number">Betti numbers</a> and <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> and <a href="/wiki/Cohomology" title="Cohomology">cohomology</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Maps_of_manifolds">Maps of manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=41" title="Edit section: Maps of manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:MorinSurfaceAsSphere%27sInsideVersusOutside.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/MorinSurfaceAsSphere%27sInsideVersusOutside.PNG/220px-MorinSurfaceAsSphere%27sInsideVersusOutside.PNG" decoding="async" width="220" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/MorinSurfaceAsSphere%27sInsideVersusOutside.PNG/330px-MorinSurfaceAsSphere%27sInsideVersusOutside.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/7/73/MorinSurfaceAsSphere%27sInsideVersusOutside.PNG 2x" data-file-width="338" data-file-height="280" /></a><figcaption>A <a href="/wiki/Morin_surface" title="Morin surface">Morin surface</a>, an <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersion</a> used in <a href="/wiki/Sphere_eversion" title="Sphere eversion">sphere eversion</a></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Maps_of_manifolds" title="Maps of manifolds">Maps of manifolds</a></div> <p>Just as there are various types of manifolds, there are various types of <a href="/wiki/Maps_of_manifolds" title="Maps of manifolds">maps of manifolds</a>. In addition to continuous functions and smooth functions generally, there are maps with special properties. In <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a> a basic type are <a href="/wiki/Embedding" title="Embedding">embeddings</a>, of which <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a> is a central example, and generalizations such as <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersions</a>, <a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">submersions</a>, <a href="/wiki/Covering_space" title="Covering space">covering spaces</a>, and <a href="/wiki/Ramified_covering_space" class="mw-redirect" title="Ramified covering space">ramified covering spaces</a>. Basic results include the <a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding theorem</a> and <a href="/wiki/Whitney_immersion_theorem" title="Whitney immersion theorem">Whitney immersion theorem</a>. </p><p>In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of <a href="/wiki/Isometric_embedding" class="mw-redirect" title="Isometric embedding">isometric embeddings</a>, <a href="/wiki/Isometric_immersion" class="mw-redirect" title="Isometric immersion">isometric immersions</a>, and <a href="/wiki/Riemannian_submersion" title="Riemannian submersion">Riemannian submersions</a>; a basic result is the <a href="/wiki/Nash_embedding_theorem" class="mw-redirect" title="Nash embedding theorem">Nash embedding theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Scalar-valued_functions">Scalar-valued functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=42" title="Edit section: Scalar-valued functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spherical_harmonics.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Spherical_harmonics.png/260px-Spherical_harmonics.png" decoding="async" width="260" height="584" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Spherical_harmonics.png/390px-Spherical_harmonics.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Spherical_harmonics.png/520px-Spherical_harmonics.png 2x" data-file-width="1243" data-file-height="2791" /></a><figcaption>3D color plot of the <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a> of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb41e9a10a8fd7179b9170149a8d70949ba5d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}"></span></figcaption></figure> <p>A basic example of maps between manifolds are scalar-valued functions on a manifold, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon M\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b28c87bb68bdeda0931c6fa7210a19a7da47528a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.047ex; height:2.509ex;" alt="{\displaystyle f\colon M\to \mathbb {R} }"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon M\to \mathbb {C} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d982a5659e47ce1bfa1fa7874a7b9b7dbad1925b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.694ex; height:2.509ex;" alt="{\displaystyle f\colon M\to \mathbb {C} ,}"></span> </p><p>sometimes called <a href="/wiki/Regular_function" class="mw-redirect" title="Regular function">regular functions</a> or <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functionals</a>, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold. </p><p>In geometric topology, most commonly studied are <a href="/wiki/Morse_function" class="mw-redirect" title="Morse function">Morse functions</a>, which yield <a href="/wiki/Handlebody" title="Handlebody">handlebody</a> decompositions, while in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, one often studies solution to <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, an important example of which is <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>, where one studies <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic functions</a>: the kernel of the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a>. This leads to such functions as the <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>, and to <a href="/wiki/Heat_kernel" title="Heat kernel">heat kernel</a> methods of studying manifolds, such as <a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">hearing the shape of a drum</a> and some proofs of the <a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations_of_manifolds">Generalizations of manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=43" title="Edit section: Generalizations of manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Infinite dimensional manifolds</dt> <dd>The definition of a manifold can be generalized by dropping the requirement of finite dimensionality. Thus an infinite dimensional manifold is a topological space locally homeomorphic to a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> over the reals. This omits the point-set axioms, allowing higher cardinalities and <a href="/wiki/Non-Hausdorff_manifold" title="Non-Hausdorff manifold">non-Hausdorff manifolds</a>; and it omits finite dimension, allowing structures such as <a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifolds</a> to be modeled on <a href="/wiki/Hilbert_spaces" class="mw-redirect" title="Hilbert spaces">Hilbert spaces</a>, <a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifolds</a> to be modeled on <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>, and <a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifolds</a> to be modeled on <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a>. Usually one relaxes one or the other condition: manifolds with the point-set axioms are studied in <a href="/wiki/General_topology" title="General topology">general topology</a>, while infinite-dimensional manifolds are studied in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.</dd> <dt>Orbifolds</dt> <dd>An <a href="/wiki/Orbifold" title="Orbifold">orbifold</a> is a generalization of manifold allowing for certain kinds of "<a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a>" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (<i>e.g.</i> Euclidean space) by the <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">actions</a> of various <a href="/wiki/Finite_group" title="Finite group">finite groups</a>. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense.</dd> <dt>Algebraic varieties and schemes</dt> <dd><a href="/wiki/Algebraic_curve#Singularities" title="Algebraic curve">Non-singular</a> algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space, an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">Schemes</a> are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> instead of atlases.</dd> <dd>Because of <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singular points</a>, a variety is in general not a manifold, though linguistically the French <i>variété</i>, German <i>Mannigfaltigkeit</i> and English <i>manifold</i> are largely <a href="/wiki/Synonymous" class="mw-redirect" title="Synonymous">synonymous</a>. In French an algebraic variety is called <i>une <a href="https://fr.wikipedia.org/wiki/vari%C3%A9t%C3%A9_alg%C3%A9brique" class="extiw" title="fr:variété algébrique">variété algébrique</a></i> (an <i>algebraic variety</i>), while a smooth manifold is called <i>une <a href="https://fr.wikipedia.org/wiki/vari%C3%A9t%C3%A9_diff%C3%A9rentielle" class="extiw" title="fr:variété différentielle">variété différentielle</a></i> (a <i>differential variety</i>).</dd> <dt>Stratified space</dt> <dd>A "stratified space" is a space that can be divided into pieces ("strata"), with each stratum a manifold, with the strata fitting together in prescribed ways (formally, a <a href="/wiki/Filtration_(mathematics)" title="Filtration (mathematics)">filtration</a> by closed subsets).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> There are various technical definitions, notably a Whitney stratified space (see <a href="/wiki/Whitney_conditions" title="Whitney conditions">Whitney conditions</a>) for smooth manifolds and a <a href="/wiki/Topologically_stratified_space" class="mw-redirect" title="Topologically stratified space">topologically stratified space</a> for topological manifolds. Basic examples include <a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">manifold with boundary</a> (top dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney stratified spaces are a broad class of spaces, including algebraic varieties, analytic varieties, <a href="/wiki/Semialgebraic_set" title="Semialgebraic set">semialgebraic sets</a>, and <a href="/wiki/Subanalytic_set" title="Subanalytic set">subanalytic sets</a>.</dd> <dt>CW-complexes</dt> <dd>A <a href="/wiki/CW_complex" title="CW complex">CW complex</a> is a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, hence not a manifold. However, they are of central interest in algebraic topology, especially in <a href="/wiki/Homotopy_theory" title="Homotopy theory">homotopy theory</a>.</dd> <dt>Homology manifolds</dt> <dd>A <a href="/wiki/Homology_manifold" title="Homology manifold">homology manifold</a> is a space that behaves like a manifold from the point of view of homology theory. These are not all manifolds, but (in high dimension) can be analyzed by surgery theory similarly to manifolds, and failure to be a manifold is a local obstruction, as in surgery theory.<sup id="cite_ref-bfmw_9-0" class="reference"><a href="#cite_note-bfmw-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></dd> <dt>Differential spaces</dt> <dd>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> be a nonempty set. Suppose that some family of real functions on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> was chosen. Denote it by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\subseteq \mathbb {R} ^{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>⊆</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\subseteq \mathbb {R} ^{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48526365ad9e39365ad9a3233f0eb7c7a87ef387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.502ex; height:2.843ex;" alt="{\displaystyle C\subseteq \mathbb {R} ^{M}}"></span>. It is an algebra with respect to the pointwise addition and multiplication. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> be equipped with the topology induced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. Suppose also that the following conditions hold. First: for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\in C^{\infty }\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\in C^{\infty }\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e929b9f5b0e3341415d9d245732b8e0957eef2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.67ex; height:2.843ex;" alt="{\displaystyle H\in C^{\infty }\left(\mathbb {R} ^{n}\right)}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>, and arbitrary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\dots ,f_{n}\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},\dots ,f_{n}\in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664260609db96974bb469c93142450d3a392d68f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.336ex; height:2.509ex;" alt="{\displaystyle f_{1},\dots ,f_{n}\in C}"></span>, the composition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\circ \left(f_{1},\dots ,f_{n}\right)\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>∘</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\circ \left(f_{1},\dots ,f_{n}\right)\in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87140b87ed10301ffdc249c04949b4f7d23ac703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.404ex; height:2.843ex;" alt="{\displaystyle H\circ \left(f_{1},\dots ,f_{n}\right)\in C}"></span>. Second: every function, which in every point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> locally coincides with some function from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, also belongs to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. A pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5503cc157affa1581b3ef93777cf5b1fe416aa8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.052ex; height:2.843ex;" alt="{\displaystyle (M,C)}"></span> for which the above conditions hold, is called a Sikorski differential space.<sup id="cite_ref-sacd_10-0" class="reference"><a href="#cite_note-sacd-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=44" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a> – Subset of a manifold that is a manifold itself; an injective immersion into a manifold</li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a> – Straight path on a curved surface or a Riemannian manifold</li> <li><a href="/wiki/Directional_statistics" title="Directional statistics">Directional statistics</a> – Subdiscipline of statistics: statistics on manifolds</li> <li><a href="/wiki/List_of_manifolds" title="List of manifolds">List of manifolds</a></li> <li><a href="/wiki/Timeline_of_manifolds" title="Timeline of manifolds">Timeline of manifolds</a> – Mathematics timeline</li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="By_dimension">By dimension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=45" title="Edit section: By dimension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/3-manifold" title="3-manifold">3-manifold</a> – Mathematical space</li> <li><a href="/wiki/4-manifold" title="4-manifold">4-manifold</a> – Mathematical space</li> <li><a href="/wiki/5-manifold" title="5-manifold">5-manifold</a> – Manifold of dimension five</li> <li><a href="/wiki/Convenient_vector_space#Application:_Manifolds_of_mappings_between_finite_dimensional_manifolds" title="Convenient vector space">Manifolds of mappings</a> – locally convex vector spaces satisfying a very mild completeness condition<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=46" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 40em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">E.g. see <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRiaza2008" class="citation cs2">Riaza, Ricardo (2008), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HoOWxqWru1cC&pg=PA110"><i>Differential-Algebraic Systems: Analytical Aspects and Circuit Applications</i></a>, World Scientific, p. 110, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789812791818" title="Special:BookSources/9789812791818"><bdi>9789812791818</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential-Algebraic+Systems%3A+Analytical+Aspects+and+Circuit+Applications&rft.pages=110&rft.pub=World+Scientific&rft.date=2008&rft.isbn=9789812791818&rft.aulast=Riaza&rft.aufirst=Ricardo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHoOWxqWru1cC%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AManifold" class="Z3988"></span>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGunning1990" class="citation cs2">Gunning, R. C. (1990), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dKYhlJB1iOgC&pg=PA73"><i>Introduction to Holomorphic Functions of Several Variables, Volume 2</i></a>, CRC Press, p. 73, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780534133092" title="Special:BookSources/9780534133092"><bdi>9780534133092</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Holomorphic+Functions+of+Several+Variables%2C+Volume+2&rft.pages=73&rft.pub=CRC+Press&rft.date=1990&rft.isbn=9780534133092&rft.aulast=Gunning&rft.aufirst=R.+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdKYhlJB1iOgC%26pg%3DPA73&rfr_id=info%3Asid%2Fen.wikipedia.org%3AManifold" class="Z3988"></span>.</span> </li> <li id="cite_note-Morita-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Morita_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShigeyuki_MoritaTeruko_NagaseKatsumi_Nomizu2001" class="citation book cs1">Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometryofdiffer00mori"><i>Geometry of Differential Forms</i></a></span>. American Mathematical Society Bookstore. p. <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofdiffer00mori/page/12">12</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-1045-6" title="Special:BookSources/0-8218-1045-6"><bdi>0-8218-1045-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Differential+Forms&rft.pages=12&rft.pub=American+Mathematical+Society+Bookstore&rft.date=2001&rft.isbn=0-8218-1045-6&rft.au=Shigeyuki+Morita&rft.au=Teruko+Nagase&rft.au=Katsumi+Nomizu&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofdiffer00mori&rfr_id=info%3Asid%2Fen.wikipedia.org%3AManifold" class="Z3988"></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged June 2021">dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">The notion of a map can formalized as a <a href="/w/index.php?title=Cell_decomposition&action=edit&redlink=1" class="new" title="Cell decomposition (page does not exist)">cell decomposition</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1895" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré, H.</a> (1895). 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Surveys 53 (1998), no. 1, 229–236</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitney1936" class="citation journal cs1"><a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Whitney, H.</a> (1936). "Differentiable Manifolds". <i>Annals of Mathematics</i>. 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(1996). "Topology of homology manifolds". <i>Annals of Mathematics</i>. 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(1967). <a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fcm-18-1-251-272">"Abstract covariant derivative"</a>. <i>Colloquium Mathematicum</i>. <b>18</b>: 251–272. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fcm-18-1-251-272">10.4064/cm-18-1-251-272</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Colloquium+Mathematicum&rft.atitle=Abstract+covariant+derivative&rft.volume=18&rft.pages=251-272&rft.date=1967&rft_id=info%3Adoi%2F10.4064%2Fcm-18-1-251-272&rft.aulast=Sikorski&rft.aufirst=R.&rft_id=https%3A%2F%2Fdoi.org%2F10.4064%252Fcm-18-1-251-272&rfr_id=info%3Asid%2Fen.wikipedia.org%3AManifold" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=47" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Michael_Freedman" title="Michael Freedman">Freedman, Michael H.</a>, and Quinn, Frank (1990) <i>Topology of 4-Manifolds</i>. Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08577-3" title="Special:BookSources/0-691-08577-3">0-691-08577-3</a>.</li> <li><a href="/wiki/Victor_Guillemin" title="Victor Guillemin">Guillemin, Victor</a> and Pollack, Alan (1974) <i>Differential Topology</i>. Prentice-Hall. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-212605-2" title="Special:BookSources/0-13-212605-2">0-13-212605-2</a>. Advanced undergraduate / first-year graduate text inspired by Milnor.</li> <li>Hempel, John (1976) <i>3-Manifolds</i>. Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-3695-1" title="Special:BookSources/0-8218-3695-1">0-8218-3695-1</a>.</li> <li><a href="/wiki/Morris_Hirsch" title="Morris Hirsch">Hirsch, Morris</a>, (1997) <i>Differential Topology</i>. Springer Verlag. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90148-5" title="Special:BookSources/0-387-90148-5">0-387-90148-5</a>. The most complete account, with historical insights and excellent, but difficult, problems. The standard reference for those wishing to have a deep understanding of the subject.</li> <li><a href="/wiki/Robion_Kirby" title="Robion Kirby">Kirby, Robion C.</a> and Siebenmann, Laurence C. (1977) <i>Foundational Essays on Topological Manifolds. Smoothings, and Triangulations</i>. Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08190-5" title="Special:BookSources/0-691-08190-5">0-691-08190-5</a>. A detailed study of the <a href="/wiki/Category_theory" title="Category theory">category</a> of topological manifolds.</li> <li>Lee, John M. (2000) <i>Introduction to Topological Manifolds</i>. Springer-Verlag. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98759-2" title="Special:BookSources/0-387-98759-2">0-387-98759-2</a>. Detailed and comprehensive first-year graduate text.</li> <li>Lee, John M. (2003) <i><a rel="nofollow" class="external text" href="https://archive.org/details/GraduateTextsInMathematics218LeeJ.M.IntroductionToSmoothManifoldsSpringer2012">Introduction to Smooth Manifolds</a></i>. Springer-Verlag. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95495-3" title="Special:BookSources/0-387-95495-3">0-387-95495-3</a>. Detailed and comprehensive first-year graduate text; sequel to <i>Introduction to Topological Manifolds</i>.</li> <li><a href="/wiki/William_S._Massey" title="William S. Massey">Massey, William S.</a> (1977) <i>Algebraic Topology: An Introduction</i>. Springer-Verlag. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90271-6" title="Special:BookSources/0-387-90271-6">0-387-90271-6</a>.</li> <li><a href="/wiki/John_Milnor" title="John Milnor">Milnor, John</a> (1997) <i>Topology from the Differentiable Viewpoint</i>. Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-04833-9" title="Special:BookSources/0-691-04833-9">0-691-04833-9</a>. Classic brief introduction to differential topology.</li> <li><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (1991) <i><a rel="nofollow" class="external text" href="https://archive.org/details/MunkresJ.R.AnalysisOnManifolds">Analysis on Manifolds</a></i>. Addison-Wesley (reprinted by Westview Press) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-51035-9" title="Special:BookSources/0-201-51035-9">0-201-51035-9</a>. Undergraduate text treating manifolds in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>.</li> <li><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (2000) <i>Topology</i>. Prentice Hall. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-181629-2" title="Special:BookSources/0-13-181629-2">0-13-181629-2</a>.</li> <li>Neuwirth, L. P., ed. (1975) <i>Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox</i>. Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08170-0" title="Special:BookSources/978-0-691-08170-0">978-0-691-08170-0</a>.</li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann, Bernhard</a>, <i>Gesammelte mathematische Werke und wissenschaftlicher Nachlass</i>, Sändig Reprint. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-253-03059-8" title="Special:BookSources/3-253-03059-8">3-253-03059-8</a>. <ul><li><i><a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/">Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse.</a></i> The 1851 doctoral thesis in which "manifold" (<i>Mannigfaltigkeit</i>) first appears.</li> <li><i><a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/">Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.</a></i> The 1854 Göttingen inaugural lecture (<i>Habilitationsschrift</i>).</li></ul></li> <li><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1965) <i><a rel="nofollow" class="external text" href="https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint">Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus</a></i>. W.A. Benjamin Inc. (reprinted by Addison-Wesley and Westview Press). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-9021-9" title="Special:BookSources/0-8053-9021-9">0-8053-9021-9</a>. <a href="/wiki/Calculus_on_Manifolds_(book)" title="Calculus on Manifolds (book)">Famously terse</a> advanced undergraduate / first-year graduate text.</li> <li><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1999) <i>A Comprehensive Introduction to Differential Geometry</i> (3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTu2011" class="citation book cs1">Tu, Loring W. (2011). <a rel="nofollow" class="external text" href="https://www.springer.com/gb/book/9781441973993"><i>An Introduction to Manifolds</i></a> (2nd ed.). New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7399-3" title="Special:BookSources/978-1-4419-7399-3"><bdi>978-1-4419-7399-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Manifolds&rft.place=New+York&rft.edition=2nd&rft.pub=Springer&rft.date=2011&rft.isbn=978-1-4419-7399-3&rft.aulast=Tu&rft.aufirst=Loring+W.&rft_id=https%3A%2F%2Fwww.springer.com%2Fgb%2Fbook%2F9781441973993&rfr_id=info%3Asid%2Fen.wikipedia.org%3AManifold" class="Z3988"></span>. Concise first-year graduate text.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Manifold&action=edit&section=48" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Manifold">"Manifold"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Manifold&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DManifold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AManifold" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.dimensions-math.org/Dim_E.htm">Dimensions-math.org</a> (A film explaining and visualizing manifolds up to fourth dimension.)</li> <li>The <a rel="nofollow" class="external text" href="http://www.map.mpim-bonn.mpg.de">manifold atlas</a> project of the <a rel="nofollow" class="external text" href="http://www.mpim-bonn.mpg.de">Max Planck Institute for Mathematics in Bonn</a></li> <li>MIT Open Courseware: <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/18-965-geometry-of-manifolds-fall-2004/pages/lecture-notes/">Geometry of Manifolds</a>.</li></ul> <div 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.navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tensors" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tensors" title="Template:Tensors"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tensors" title="Template talk:Tensors"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tensors" title="Special:EditPage/Template:Tensors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tensors" style="font-size:114%;margin:0 4em"><a href="/wiki/Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor<br />definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a class="mw-selflink selflink">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" 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