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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atlases"> <div class="vector-toc-text"> 2.1 Atlases </div> </a> <ul id="toc-Atlases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Manifolds"> <div class="vector-toc-text"> 2.2 Manifolds </div> </a> <ul id="toc-Manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Patching_together_Euclidean_pieces_to_form_a_manifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Patching_together_Euclidean_pieces_to_form_a_manifold"> <div class="vector-toc-text"> 2.3 Patching together Euclidean pieces to form a manifold </div> </a> <ul id="toc-Patching_together_Euclidean_pieces_to_form_a_manifold-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Differentiable_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Differentiable_functions"> <div class="vector-toc-text"> 3 Differentiable functions </div> </a> <button aria-controls="toc-Differentiable_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Differentiable functions subsection </button> <ul id="toc-Differentiable_functions-sublist" class="vector-toc-list"> <li id="toc-Differentiation_of_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differentiation_of_functions"> <div class="vector-toc-text"> 3.1 Differentiation of functions </div> </a> <ul id="toc-Differentiation_of_functions-sublist" class="vector-toc-list"> <li id="toc-Directional_differentiation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Directional_differentiation"> <div class="vector-toc-text"> 3.1.1 Directional differentiation </div> </a> <ul id="toc-Directional_differentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangent_vector_and_the_differential" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tangent_vector_and_the_differential"> <div class="vector-toc-text"> 3.1.2 Tangent vector and the differential </div> </a> <ul id="toc-Tangent_vector_and_the_differential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_of_tangent_space_and_differentiation_in_local_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definition_of_tangent_space_and_differentiation_in_local_coordinates"> <div class="vector-toc-text"> 3.1.3 Definition of tangent space and differentiation in local coordinates </div> </a> <ul id="toc-Definition_of_tangent_space_and_differentiation_in_local_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Partitions_of_unity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partitions_of_unity"> <div class="vector-toc-text"> 3.2 Partitions of unity </div> </a> <ul id="toc-Partitions_of_unity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differentiability_of_mappings_between_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differentiability_of_mappings_between_manifolds"> <div class="vector-toc-text"> 3.3 Differentiability of mappings between manifolds </div> </a> <ul id="toc-Differentiability_of_mappings_between_manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bundles" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bundles"> <div class="vector-toc-text"> 4 Bundles </div> </a> <button aria-controls="toc-Bundles-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Bundles subsection </button> <ul id="toc-Bundles-sublist" class="vector-toc-list"> <li id="toc-Tangent_bundle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangent_bundle"> <div class="vector-toc-text"> 4.1 Tangent bundle </div> </a> <ul id="toc-Tangent_bundle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cotangent_bundle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cotangent_bundle"> <div class="vector-toc-text"> 4.2 Cotangent bundle </div> </a> <ul id="toc-Cotangent_bundle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_bundle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_bundle"> <div class="vector-toc-text"> 4.3 Tensor bundle </div> </a> <ul id="toc-Tensor_bundle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frame_bundle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frame_bundle"> <div class="vector-toc-text"> 4.4 Frame bundle </div> </a> <ul id="toc-Frame_bundle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jet_bundles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jet_bundles"> <div class="vector-toc-text"> 4.5 Jet bundles </div> </a> <ul id="toc-Jet_bundles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculus_on_manifolds" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculus_on_manifolds"> <div class="vector-toc-text"> 5 Calculus on manifolds </div> </a> <button aria-controls="toc-Calculus_on_manifolds-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Calculus on manifolds subsection </button> <ul id="toc-Calculus_on_manifolds-sublist" class="vector-toc-list"> <li id="toc-Differential_calculus_of_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_calculus_of_functions"> <div class="vector-toc-text"> 5.1 Differential calculus of functions </div> </a> <ul id="toc-Differential_calculus_of_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_derivative"> <div class="vector-toc-text"> 5.2 Lie derivative </div> </a> <ul id="toc-Lie_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exterior_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exterior_calculus"> <div class="vector-toc-text"> 5.3 Exterior calculus </div> </a> <ul id="toc-Exterior_calculus-sublist" class="vector-toc-list"> <li id="toc-Exterior_derivative" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Exterior_derivative"> <div class="vector-toc-text"> 5.3.1 Exterior derivative </div> </a> <ul id="toc-Exterior_derivative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Topology_of_differentiable_manifolds" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Topology_of_differentiable_manifolds"> <div class="vector-toc-text"> 6 Topology of differentiable manifolds </div> </a> <button aria-controls="toc-Topology_of_differentiable_manifolds-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Topology of differentiable manifolds subsection </button> <ul id="toc-Topology_of_differentiable_manifolds-sublist" class="vector-toc-list"> <li id="toc-Relationship_with_topological_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_with_topological_manifolds"> <div class="vector-toc-text"> 6.1 Relationship with topological manifolds </div> </a> <ul id="toc-Relationship_with_topological_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification"> <div class="vector-toc-text"> 6.2 Classification </div> </a> <ul id="toc-Classification-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Structures_on_smooth_manifolds" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Structures_on_smooth_manifolds"> <div class="vector-toc-text"> 7 Structures on smooth manifolds </div> </a> <button aria-controls="toc-Structures_on_smooth_manifolds-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Structures on smooth manifolds subsection </button> <ul id="toc-Structures_on_smooth_manifolds-sublist" class="vector-toc-list"> <li id="toc-(Pseudo-)Riemannian_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#(Pseudo-)Riemannian_manifolds"> <div class="vector-toc-text"> 7.1 (Pseudo-)Riemannian manifolds </div> </a> <ul id="toc-(Pseudo-)Riemannian_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symplectic_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symplectic_manifolds"> <div class="vector-toc-text"> 7.2 Symplectic manifolds </div> </a> <ul id="toc-Symplectic_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"> 7.3 Lie groups </div> </a> <ul id="toc-Lie_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Alternative_definitions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Alternative_definitions"> <div class="vector-toc-text"> 8 Alternative definitions </div> </a> <button aria-controls="toc-Alternative_definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Alternative definitions subsection </button> <ul id="toc-Alternative_definitions-sublist" class="vector-toc-list"> <li id="toc-Pseudogroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudogroups"> <div class="vector-toc-text"> 8.1 Pseudogroups </div> </a> <ul id="toc-Pseudogroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structure_sheaf" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structure_sheaf"> <div class="vector-toc-text"> 8.2 Structure sheaf </div> </a> <ul id="toc-Structure_sheaf-sublist" class="vector-toc-list"> <li id="toc-Sheaves_of_local_rings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sheaves_of_local_rings"> <div class="vector-toc-text"> 8.2.1 Sheaves of local rings </div> </a> <ul id="toc-Sheaves_of_local_rings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 9 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Non-commutative_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-commutative_geometry"> <div class="vector-toc-text"> 9.1 Non-commutative geometry </div> </a> <ul id="toc-Non-commutative_geometry-sublist" class="vector-toc-list"> </ul> </li> </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"> 10 See also </div> </a> <ul id="toc-See_also-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"> 11 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 12 Bibliography </div> </a> <ul id="toc-Bibliography-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 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Mannigfaltigkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diferentseeruv_muutkond" title="Diferentseeruv muutkond – Estonian" lang="et" hreflang="et" data-title="Diferentseeruv muutkond" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variedad_diferenciable" title="Variedad diferenciable – Spanish" lang="es" hreflang="es" data-title="Variedad diferenciable" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</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_%D8%AF%DB%8C%D9%81%D8%B1%D8%A7%D9%86%D8%B3%DB%8C%D9%84%E2%80%8C%D9%BE%D8%B0%DB%8C%D8%B1" title="منیفلد دیفرانسیل‌پذیر – Persian" lang="fa" hreflang="fa" data-title="منیفلد دیفرانسیل‌پذیر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_diff%C3%A9rentielle" title="Variété différentielle – French" lang="fr" hreflang="fr" data-title="Variété différentielle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Variedade_diferenciable" title="Variedade diferenciable – Galician" lang="gl" hreflang="gl" data-title="Variedade diferenciable" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lipatan_terdiferensialkan" title="Lipatan terdiferensialkan – Indonesian" lang="id" hreflang="id" data-title="Lipatan terdiferensialkan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variet%C3%A0_differenziabile" title="Varietà differenziabile – Italian" lang="it" hreflang="it" data-title="Varietà differenziabile" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</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_%D7%97%D7%9C%D7%A7%D7%94" title="יריעה חלקה – Hebrew" lang="he" hreflang="he" data-title="יריעה חלקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Differenci%C3%A1lhat%C3%B3_sokas%C3%A1g" title="Differenciálható sokaság – Hungarian" lang="hu" hreflang="hu" data-title="Differenciálható sokaság" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Differentieerbare_vari%C3%ABteit" title="Differentieerbare variëteit – Dutch" lang="nl" hreflang="nl" data-title="Differentieerbare variëteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%AF%E5%BE%AE%E5%88%86%E5%A4%9A%E6%A7%98%E4%BD%93" title="可微分多様体 – Japanese" lang="ja" hreflang="ja" data-title="可微分多様体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozmaito%C5%9B%C4%87_r%C3%B3%C5%BCniczkowa" title="Rozmaitość różniczkowa – Polish" lang="pl" hreflang="pl" data-title="Rozmaitość różniczkowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-tl badge-Q70893996 mw-list-item" title=""><a href="https://tl.wikipedia.org/wiki/Manipoldong_diperensiyable" title="Manipoldong diperensiyable – Tagalog" lang="tl" hreflang="tl" data-title="Manipoldong diperensiyable" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BE%D0%B2%D0%BD%D0%B8%D0%B9_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%B2%D0%B8%D0%B4" title="Диференційовний многовид – Ukrainian" lang="uk" hreflang="uk" data-title="Диференційовний многовид" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%B5%81%E5%BD%A2" title="微分流形 – Cantonese" lang="yue" hreflang="yue" data-title="微分流形" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%B5%81%E5%BD%A2" title="微分流形 – Chinese" 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data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Smooth_manifold&redirect=no" class="mw-redirect" title="Smooth manifold">Smooth manifold</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Manifold upon which it is possible to perform calculus</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Nondifferentiable_atlas.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9a/Nondifferentiable_atlas.png" decoding="async" width="328" height="167" class="mw-file-element" data-file-width="328" data-file-height="167" /></a><figcaption>A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the <a href="/wiki/Tropic_of_Cancer" title="Tropic of Cancer">Tropic of Cancer</a> is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.</figcaption></figure> In mathematics, a differentiable manifold (also differential manifold) is a type of <a href="/wiki/Manifold" title="Manifold">manifold</a> that is locally similar enough to a <a href="/wiki/Vector_space" title="Vector space">vector space</a> to allow one to apply <a href="/wiki/Calculus" title="Calculus">calculus</a>. Any <a href="/wiki/Manifold" title="Manifold">manifold</a> can be described by a collection of charts (<a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a>). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a>), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a> with a globally defined <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a>. Any topological manifold can be given a differential structure locally by using the <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a> in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their <a href="/wiki/Function_composition" title="Function composition">compositions</a> on chart intersections in the atlas must be differentiable functions on the corresponding vector space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called <a href="/wiki/Transition_map" class="mw-redirect" title="Transition map">transition maps</a>. The ability to define such a local differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A locally differential structure allows one to define the globally differentiable <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a>, differentiable functions, and differentiable <a href="/wiki/Tensor_field" title="Tensor field">tensor</a> and <a href="/wiki/Vector_field" title="Vector field">vector</a> fields. Differentiable manifolds are very important in <a href="/wiki/Physics" title="Physics">physics</a>. Special kinds of differentiable manifolds form the basis for physical theories such as <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, and <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior calculus.</a> The study of calculus on differentiable manifolds is known as <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. "Differentiability" of a manifold has been given several meanings, including: <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a>, k-times differentiable, <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> (which itself has many meanings), and <a href="/wiki/Analytic_function" title="Analytic function">analytic</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=1" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History of manifolds and varieties</a></div> The emergence of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> as a distinct discipline is generally credited to <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> and <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>. Riemann first described manifolds in his famous <a href="/wiki/Habilitation" title="Habilitation">habilitation</a> lecture before the faculty at <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">Göttingen</a>.<a href="#cite_note-1">[1]</a> He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments: <dl><dd>Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ... – B. Riemann</dd></dl> The works of physicists such as <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>,<a href="#cite_note-2">[2]</a> and mathematicians <a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a> and <a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a><a href="#cite_note-3">[3]</a> led to the development of <a href="/wiki/Tensor_analysis" class="mw-redirect" title="Tensor analysis">tensor analysis</a> and the notion of <a href="/wiki/General_covariance" title="General covariance">covariance</a>, which identifies an intrinsic geometric property as one that is invariant with respect to <a href="/wiki/Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">coordinate transformations</a>. These ideas found a key application in <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> and its underlying <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a>. A modern definition of a 2-dimensional manifold was given by <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> in his 1913 book on <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>.<a href="#cite_note-4">[4]</a> The widely accepted general definition of a manifold in terms of an <a href="/wiki/Atlas_(mathematics)" class="mw-redirect" title="Atlas (mathematics)">atlas</a> is due to <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a>.<a href="#cite_note-5">[5]</a> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Atlases">Atlases</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=3" title="Edit section: Atlases">edit</a>]</div> Let M be a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. A chart (U, φ) on M consists of an open subset U of M, and a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> φ from U to an open subset of some <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> Rn. Somewhat informally, one may refer to a chart φ : U → Rn, meaning that the image of φ is an open subset of Rn, and that φ is a homeomorphism onto its image; in the usage of some authors, this may instead mean that φ : U → Rn is itself a homeomorphism. The presence of a chart suggests the possibility of doing <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> on M; for instance, if given a function u : M → R and a chart (U, φ) on M, one could consider the composition u ∘ φ−1, which is a real-valued function whose domain is an open subset of a Euclidean space; as such, if it happens to be differentiable, one could consider its <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a>. This situation is not fully satisfactory for the following reason. Consider a second chart (V, ψ) on M, and suppose that U and V contain some points in common. The two corresponding functions u ∘ φ−1 and u ∘ ψ−1 are linked in the sense that they can be reparametrized into one another: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\circ \varphi ^{-1}={\big (}u\circ \psi ^{-1}{\big )}\circ {\big (}\psi \circ \varphi ^{-1}{\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∘</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>u</mi> <mo>∘</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ψ</mi> <mo>∘</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\circ \varphi ^{-1}={\big (}u\circ \psi ^{-1}{\big )}\circ {\big (}\psi \circ \varphi ^{-1}{\big )},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8bdad9dff9be58fb7f0d66953aaa00fbe73f8e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.508ex; height:3.343ex;" alt="{\displaystyle u\circ \varphi ^{-1}={\big (}u\circ \psi ^{-1}{\big )}\circ {\big (}\psi \circ \varphi ^{-1}{\big )},}"> the natural domain of the right-hand side being φ(U ∩ V). Since φ and ψ are homeomorphisms, it follows that ψ ∘ φ−1 is a homeomorphism from φ(U ∩ V) to ψ(U ∩ V). Consequently it's just a bicontinuous function, thus even if both functions u ∘ φ−1 and u ∘ ψ−1 are differentiable, their differential properties will not necessarily be strongly linked to one another, as ψ ∘ φ−1 is not guaranteed to be sufficiently differentiable for being able to compute the partial derivatives of the LHS applying the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> to the RHS. The same problem is found if one considers instead functions c : R → M; one is led to the reparametrization formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ c={\big (}\varphi \circ \psi ^{-1}{\big )}\circ {\big (}\psi \circ c{\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∘</mo> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>φ</mi> <mo>∘</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ψ</mi> <mo>∘</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \circ c={\big (}\varphi \circ \psi ^{-1}{\big )}\circ {\big (}\psi \circ c{\big )},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6481344801df7c25d874aa85a627911234fe3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.197ex; height:3.343ex;" alt="{\displaystyle \varphi \circ c={\big (}\varphi \circ \psi ^{-1}{\big )}\circ {\big (}\psi \circ c{\big )},}"> at which point one can make the same observation as before. This is resolved by the introduction of a "differentiable atlas" of charts, which specifies a collection of charts on M for which the <a href="/wiki/Transition_map" class="mw-redirect" title="Transition map">transition maps</a> ψ ∘ φ−1 are all differentiable. This makes the situation quite clean: if u ∘ φ−1 is differentiable, then due to the first reparametrization formula listed above, the map u ∘ ψ−1 is also differentiable on the region ψ(U ∩ V), and vice versa. Moreover, the derivatives of these two maps are linked to one another by the chain rule. Relative to the given atlas, this facilitates a notion of differentiable mappings whose domain or range is M, as well as a notion of the derivative of such maps. Formally, the word "differentiable" is somewhat ambiguous, as it is taken to mean different things by different authors; sometimes it means the existence of first derivatives, sometimes the existence of continuous first derivatives, and sometimes the existence of infinitely many derivatives. The following gives a formal definition of various (nonambiguous) meanings of "differentiable atlas". Generally, "differentiable" will be used as a catch-all term including all of these possibilities, provided k ≥ 1. <table class="wikitable"> <tbody><tr> <th colspan="5" style="text-align: left;">Given a topological space M... </th></tr> <tr> <td>a Ck atlas </td> <td rowspan="4">is a collection of charts </td> <td>{φα : Uα → Rn}α∈A </td> <td rowspan="4">such that {Uα}α∈A covers M, and such that for all α and β in A, the <a href="/wiki/Transition_map" class="mw-redirect" title="Transition map">transition map</a> φα ∘ φ−1 β is </td> <td>a <a href="/wiki/Smoothness" title="Smoothness">Ck</a> map </td></tr> <tr> <td>a smooth or C ∞ atlas </td> <td>{φα : Uα → Rn}α∈A </td> <td>a <a href="/wiki/Smoothness" title="Smoothness">smooth</a> map </td></tr> <tr> <td>an analytic or C ω atlas </td> <td>{φα : Uα → Rn}α∈A </td> <td>a <a href="/wiki/Analytic_function" title="Analytic function">real-analytic</a> map </td></tr> <tr> <td>a holomorphic atlas </td> <td>{φα : Uα → Cn}α∈A </td> <td>a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> map </td></tr></tbody></table> <div class="noresize thumb tright" style=";"> <div class="thumbinner" style="overflow:hidden;width:252px;"> <div class="thumbimage" style="overflow:hidden; position:relative; background-color:white;"> <div style=";left:0px; top:0px; width:250px; position:absolute;"> <a href="/wiki/File:Two_coordinate_charts_on_a_manifold.svg" class="mw-file-description" title="The transition map of two charts. '"`UNIQ--postMath-00000003-QINU`"' denotes '"`UNIQ--postMath-00000004-QINU`"' and '"`UNIQ--postMath-00000005-QINU`"' denotes '"`UNIQ--postMath-00000006-QINU`"'."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Two_coordinate_charts_on_a_manifold.svg/250px-Two_coordinate_charts_on_a_manifold.svg.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Two_coordinate_charts_on_a_manifold.svg/375px-Two_coordinate_charts_on_a_manifold.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Two_coordinate_charts_on_a_manifold.svg/500px-Two_coordinate_charts_on_a_manifold.svg.png 2x" data-file-width="350" data-file-height="280" /></a></div> <div style="text-align:left; background-color:transparent; line-height:110%;"> <div id="annotation_45x70" style="position:absolute; left:45px; top:70px; line-height:110%;"><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><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}"></div> <div id="annotation_70x54" style="position:absolute; left:70px; top:54px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c507812c8cdaf4cea8d2e7e1705b495a3010a352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.872ex; height:2.509ex;" alt="{\displaystyle U_{\alpha }}"></div> <div id="annotation_187x66" style="position:absolute; left:187px; top:66px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d87b923092fa558b2132b6a0e066f5f3bab6681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.762ex; height:2.843ex;" alt="{\displaystyle U_{\beta }}"></div> <div id="annotation_42x100" style="position:absolute; left:42px; top:100px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408558572f5616a304c4f8778f69e4c15d59cc0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.67ex; height:2.509ex;" alt="{\displaystyle \phi _{\alpha }}"></div> <div id="annotation_183x117" style="position:absolute; left:183px; top:117px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8beae3af4f52166fa996066900cc52ddbba48157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.56ex; height:2.843ex;" alt="{\displaystyle \phi _{\beta }}"></div> <div id="annotation_80x112" style="position:absolute; left:80px; top:112px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\beta \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\beta \alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af98f656e372b5fbc8a715ca4119b06922bf9b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.611ex; height:2.843ex;" alt="{\displaystyle \phi _{\beta \alpha }}"></div> <div id="annotation_90x145" style="position:absolute; left:90px; top:145px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe2eba5c573574045804a9c5a915495e78ee3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.611ex; height:2.843ex;" alt="{\displaystyle \phi _{\alpha \beta }}"></div> <div id="annotation_55x183" style="position:absolute; left:55px; top:183px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}"></div> <div id="annotation_145x183" style="position:absolute; left:145px; top:183px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}"></div> </div> <div style="visibility:hidden"><a href="/wiki/File:Two_coordinate_charts_on_a_manifold.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Two_coordinate_charts_on_a_manifold.svg/250px-Two_coordinate_charts_on_a_manifold.svg.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Two_coordinate_charts_on_a_manifold.svg/375px-Two_coordinate_charts_on_a_manifold.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Two_coordinate_charts_on_a_manifold.svg/500px-Two_coordinate_charts_on_a_manifold.svg.png 2x" data-file-width="350" data-file-height="280" /></a></div> </div> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Two_coordinate_charts_on_a_manifold.svg" title="File:Two coordinate charts on a manifold.svg"> </a></div>The transition map of two charts. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe2eba5c573574045804a9c5a915495e78ee3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.611ex; height:2.843ex;" alt="{\displaystyle \phi _{\alpha \beta }}"> denotes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha }\circ \phi _{\beta }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha }\circ \phi _{\beta }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d79753b9f0ed9ead8aa331305bd1597502407b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.583ex; height:3.676ex;" alt="{\displaystyle \phi _{\alpha }\circ \phi _{\beta }^{-1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\beta \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\beta \alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af98f656e372b5fbc8a715ca4119b06922bf9b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.611ex; height:2.843ex;" alt="{\displaystyle \phi _{\beta \alpha }}"> denotes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\beta }\circ \phi _{\alpha }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\beta }\circ \phi _{\alpha }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365626d594ef3f6c2110a580cd4fa75bf1da0b95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.472ex; height:3.343ex;" alt="{\displaystyle \phi _{\beta }\circ \phi _{\alpha }^{-1}}">.</div> </div></div> Since every real-analytic map is smooth, and every smooth map is Ck for any k, one can see that any analytic atlas can also be viewed as a smooth atlas, and every smooth atlas can be viewed as a Ck atlas. This chain can be extended to include holomorphic atlases, with the understanding that any holomorphic map between open subsets of Cn can be viewed as a real-analytic map between open subsets of R2n. Given a differentiable atlas on a topological space, one says that a chart is differentiably compatible with the atlas, or differentiable relative to the given atlas, if the inclusion of the chart into the collection of charts comprising the given differentiable atlas results in a differentiable atlas. A differentiable atlas determines a maximal differentiable atlas, consisting of all charts which are differentiably compatible with the given atlas. A maximal atlas is always very large. For instance, given any chart in a maximal atlas, its restriction to an arbitrary open subset of its domain will also be contained in the maximal atlas. A maximal smooth atlas is also known as a <a href="/wiki/Smooth_structure" title="Smooth structure">smooth structure</a>; a maximal holomorphic atlas is also known as a <a href="/wiki/Complex_manifold" title="Complex manifold">complex structure</a>. An alternative but equivalent definition, avoiding the direct use of maximal atlases, is to consider equivalence classes of differentiable atlases, in which two differentiable atlases are considered equivalent if every chart of one atlas is differentiably compatible with the other atlas. Informally, what this means is that in dealing with a smooth manifold, one can work with a single differentiable atlas, consisting of only a few charts, with the implicit understanding that many other charts and differentiable atlases are equally legitimate. According to the <a href="/wiki/Invariance_of_domain" title="Invariance of domain">invariance of domain</a>, each connected component of a topological space which has a differentiable atlas has a well-defined dimension n. This causes a small ambiguity in the case of a holomorphic atlas, since the corresponding dimension will be one-half of the value of its dimension when considered as an analytic, smooth, or Ck atlas. For this reason, one refers separately to the "real" and "complex" dimension of a topological space with a holomorphic atlas. <div class="mw-heading mw-heading3"><h3 id="Manifolds">Manifolds</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=4" title="Edit section: Manifolds">edit</a>]</div> A differentiable manifold is a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> and <a href="/wiki/Second_countable" class="mw-redirect" title="Second countable">second countable</a> topological space M, together with a maximal differentiable atlas on M. Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of <a href="/wiki/Bump_function" title="Bump function">bump functions</a> and <a href="/wiki/Partition_of_unity" title="Partition of unity">partitions of unity</a>, both of which are used ubiquitously. The notion of a C0 manifold is identical to that of a <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a>. However, there is a notable distinction to be made. Given a topological space, it is meaningful to ask whether or not it is a topological manifold. By contrast, it is not meaningful to ask whether or not a given topological space is (for instance) a smooth manifold, since the notion of a smooth manifold requires the specification of a smooth atlas, which is an additional structure. It could, however, be meaningful to say that a certain topological space cannot be given the structure of a smooth manifold. It is possible to reformulate the definitions so that this sort of imbalance is not present; one can start with a set M (rather than a topological space M), using the natural analogue of a smooth atlas in this setting to define the structure of a topological space on M. <div class="mw-heading mw-heading3"><h3 id="Patching_together_Euclidean_pieces_to_form_a_manifold">Patching together Euclidean pieces to form a manifold</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=5" title="Edit section: Patching together Euclidean pieces to form a manifold">edit</a>]</div> One can reverse-engineer the above definitions to obtain one perspective on the construction of manifolds. The idea is to start with the images of the charts and the transition maps, and to construct the manifold purely from this data. As in the above discussion, we use the "smooth" context but everything works just as well in other settings. Given an indexing set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582d40a9ff663187250948b07bb66456162c2042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle V_{\alpha }}"> be a collection of open subsets of <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><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}}"> and for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9a2e5019ee66d78e0032ef67d732a91a2b2aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.437ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta \in A}"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\alpha \beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c4e19bfbd9d037dfd97c0c0df8e67c25c0550d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.581ex; height:2.843ex;" alt="{\displaystyle V_{\alpha \beta }}"> be an open (possibly empty) subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66668ae9c071f388d25da00071eb3b1297c6ec60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.529ex; height:2.843ex;" alt="{\displaystyle V_{\beta }}"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \beta }:V_{\alpha \beta }\to V_{\beta \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \beta }:V_{\alpha \beta }\to V_{\beta \alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c75672aa39e6d8db48351a33f5ffc15675236b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.325ex; height:2.843ex;" alt="{\displaystyle \phi _{\alpha \beta }:V_{\alpha \beta }\to V_{\beta \alpha }}"> be a smooth map. Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c60e4200d97edd678e54eee8582620d8f12091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.721ex; height:2.509ex;" alt="{\displaystyle \phi _{\alpha \alpha }}"> is the identity map, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd42f3505be9b484a7a1ae2aa95b308616784b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.418ex; height:2.843ex;" alt="{\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \alpha }}"> is the identity map, and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \gamma }\circ \phi _{\gamma \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \gamma }\circ \phi _{\gamma \alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96dde5581d07b1faa77488f6fc6ed1ecf88e7c4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.015ex; height:2.843ex;" alt="{\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \gamma }\circ \phi _{\gamma \alpha }}"> is the identity map. Then define an equivalence relation on the disjoint union <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigsqcup _{\alpha \in A}V_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigsqcup _{\alpha \in A}V_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fadb5254d1968266857bbfa9216b89f0c61b69b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.575ex; height:3.009ex;" alt="{\textstyle \bigsqcup _{\alpha \in A}V_{\alpha }}"> by declaring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in V_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in V_{\alpha \beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8f69647c362a6c0117c31525258a8b310457b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:7.681ex; height:2.843ex;" alt="{\displaystyle p\in V_{\alpha \beta }}"> to be equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha \beta }(p)\in V_{\beta \alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha \beta }(p)\in V_{\beta \alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5910816e183dccec1c679d77c81d6a5609700a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.659ex; height:3.009ex;" alt="{\displaystyle \phi _{\alpha \beta }(p)\in V_{\beta \alpha }.}"> With some technical work, one can show that the set of equivalence classes can naturally be given a topological structure, and that the charts used in doing so form a smooth atlas. For the patching together the analytic structures(subset), see <a href="/wiki/Analytic_varieties" class="mw-redirect" title="Analytic varieties">analytic varieties</a>. <div class="mw-heading mw-heading2"><h2 id="Differentiable_functions">Differentiable functions</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=6" title="Edit section: Differentiable functions">edit</a>]</div> A real valued function f on an n-dimensional differentiable manifold M is called differentiable at a point p ∈ M if it is differentiable in any coordinate chart defined around p. In more precise terms, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (U,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be3491c45cdc56c113de78c13f5cb0b70d4c6d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.011ex; height:2.843ex;" alt="{\displaystyle (U,\phi )}"> is a differentiable chart where <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an open set in <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><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}"> containing p and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :U\to {\mathbf {R} }^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :U\to {\mathbf {R} }^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c115f4934bee5ec7e5244ab207d23291e292a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.941ex; height:2.676ex;" alt="{\displaystyle \phi :U\to {\mathbf {R} }^{n}}"> is the map defining the chart, then f is differentiable at p <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \phi ^{-1}\colon \phi (U)\subset {\mathbf {R} }^{n}\to {\mathbf {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ \phi ^{-1}\colon \phi (U)\subset {\mathbf {R} }^{n}\to {\mathbf {R} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ab364a0d4eaccd70a09f38df3a9572523bc4e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.14ex; height:3.176ex;" alt="{\displaystyle f\circ \phi ^{-1}\colon \phi (U)\subset {\mathbf {R} }^{n}\to {\mathbf {R} }}"> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772d7fe1c5a3439845a87f55d45af038956b3f12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.364ex; height:2.843ex;" alt="{\displaystyle \phi (p)}">, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \phi ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ \phi ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa227c8abb67f2401820a4c4f4931173448dee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.192ex; height:3.009ex;" alt="{\displaystyle f\circ \phi ^{-1}}"> is a differentiable function from the open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d0afd506728742ba3cdc66abc637c8ab349d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle \phi (U)}">, considered as a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {R} }^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {R} }^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96fef157cdfd6a7bb01b79ea8467bcbb6c3d78b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle {\mathbf {R} }^{n}}">, to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }">. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at p. It follows from the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> applied to the transition functions between one chart and another that if f is differentiable in any particular chart at p, then it is differentiable in all charts at p. Analogous considerations apply to defining Ck functions, smooth functions, and analytic functions. <div class="mw-heading mw-heading3"><h3 id="Differentiation_of_functions">Differentiation of functions</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=7" title="Edit section: Differentiation of functions">edit</a>]</div> There are various ways to define the <a href="/wiki/Derivative" title="Derivative">derivative</a> of a function on a differentiable manifold, the most fundamental of which is the <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivative</a>. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable <a href="/wiki/Affine_space" title="Affine space">affine</a> structure with which to define <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a>. Therefore, the directional derivative looks at curves in the manifold instead of vectors. <div class="mw-heading mw-heading4"><h4 id="Directional_differentiation">Directional differentiation</h4>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=8" title="Edit section: Directional differentiation">edit</a>]</div> Given a real valued function f on an n dimensional differentiable manifold M, the directional derivative of f at a point p in M is defined as follows. Suppose that γ(t) is a curve in M with γ(0) = p, which is differentiable in the sense that its composition with any chart is a <a href="/wiki/Differentiable_curve" title="Differentiable curve">differentiable curve</a> in Rn. Then the directional derivative of f at p along γ is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd6a9ded54a2cbd15a13c1f406426888fe1ad7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.111ex; height:5.843ex;" alt="{\displaystyle \left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}"> If γ1 and γ2 are two curves such that γ1(0) = γ2(0) = p, and in any coordinate chart <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }">, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>∘</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>∘</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa18b65975fc76e637ee2c95721a563607b16839" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.003ex; height:5.843ex;" alt="{\displaystyle \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}"> then, by the chain rule, f has the same directional derivative at p along γ1 as along γ2. This means that the directional derivative depends only on the <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> of the curve at p. Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space. <div class="mw-heading mw-heading4"><h4 id="Tangent_vector_and_the_differential">Tangent vector and the differential</h4>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=9" title="Edit section: Tangent vector and the differential">edit</a>]</div> A tangent vector at p ∈ M is an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of differentiable curves γ with γ(0) = p, modulo the equivalence relation of first-order <a href="/wiki/Contact_(mathematics)" title="Contact (mathematics)">contact</a> between the curves. Therefore, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}\equiv \gamma _{2}\iff \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≡</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>∘</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>∘</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}\equiv \gamma _{2}\iff \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3b94a6c925b08000317ef416843c9bd6f155f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.516ex; height:5.843ex;" alt="{\displaystyle \gamma _{1}\equiv \gamma _{2}\iff \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}"> in every coordinate chart <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }">. Therefore, the equivalence classes are curves through p with a prescribed <a href="/wiki/Velocity_vector" class="mw-redirect" title="Velocity vector">velocity vector</a> at p. The collection of all tangent vectors at p forms a <a href="/wiki/Vector_space" title="Vector space">vector space</a>: the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> to M at p, denoted TpM. If X is a tangent vector at p and f a differentiable function defined near p, then differentiating f along any curve in the equivalence class defining X gives a well-defined directional derivative along X: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Xf(p):=\left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Xf(p):=\left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5343a1dad5b17723429b1b98adc91d3bb5d3066" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.094ex; height:5.843ex;" alt="{\displaystyle Xf(p):=\left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}"> Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative. If the function f is fixed, then the mapping <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto Xf(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <mi>X</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto Xf(p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01fa2c7b44e242158a5b5847b79621ac55853839" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.831ex; height:2.843ex;" alt="{\displaystyle X\mapsto Xf(p)}"> is a <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on the tangent space. This linear functional is often denoted by df(p) and is called the differential of f at p: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df(p)\colon T_{p}M\to {\mathbf {R} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df(p)\colon T_{p}M\to {\mathbf {R} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91704cc64a1016499ffbdd262275c773a1840e62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.63ex; height:3.009ex;" alt="{\displaystyle df(p)\colon T_{p}M\to {\mathbf {R} }.}"> <div class="mw-heading mw-heading4"><h4 id="Definition_of_tangent_space_and_differentiation_in_local_coordinates">Definition of tangent space and differentiation in local coordinates</h4>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=10" title="Edit section: Definition of tangent space and differentiation in local coordinates">edit</a>]</div> Let <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><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}"> be a topological <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><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}">-manifold with a smooth atlas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b37d6859a1d93e54910e827321b46b5d9d57c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.969ex; height:2.843ex;" alt="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}.}"> Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ea46c20042fba4142a87ecd1f7c29776a6ce46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.802ex; height:2.843ex;" alt="{\displaystyle A_{p}}"> denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\alpha \in A:p\in U_{\alpha }\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mo>∈</mo> <mi>A</mi> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\alpha \in A:p\in U_{\alpha }\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d5257fc2cc5aa4e8adc62a806782ba5eb137df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.862ex; height:2.843ex;" alt="{\displaystyle \{\alpha \in A:p\in U_{\alpha }\}.}"> A "tangent vector at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}">" is a mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:A_{p}\to \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>:</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <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>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v:A_{p}\to \mathbb {R} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcaec81b0616fc57c6c835ba36a04238e99d67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.025ex; height:3.009ex;" alt="{\displaystyle v:A_{p}\to \mathbb {R} ^{n},}"> here denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \mapsto v_{\alpha },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">↦</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \mapsto v_{\alpha },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6229961b799359e60cd76e2085a90107bdd9ea9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.16ex; height:2.176ex;" alt="{\displaystyle \alpha \mapsto v_{\alpha },}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\alpha }=D{\Big |}_{\phi _{\beta }(p)}(\phi _{\alpha }\circ \phi _{\beta }^{-1})(v_{\beta })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mi>D</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\alpha }=D{\Big |}_{\phi _{\beta }(p)}(\phi _{\alpha }\circ \phi _{\beta }^{-1})(v_{\beta })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bd48809fca20f546306f3c43cc473a074fd6c9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.831ex; height:5.009ex;" alt="{\displaystyle v_{\alpha }=D{\Big |}_{\phi _{\beta }(p)}(\phi _{\alpha }\circ \phi _{\beta }^{-1})(v_{\beta })}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in A_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in A_{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afd8534e348c71f1d2dd6b3eaf2412461ac74ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.143ex; height:2.843ex;" alt="{\displaystyle \alpha ,\beta \in A_{p}.}"> Let the collection of tangent vectors at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> be denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b69cac5a8c2a370102468e4b47abc73ac692fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.506ex; height:2.843ex;" alt="{\displaystyle T_{p}M.}"> Given a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f: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:M\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7c129dfb3446a2251aabc1cb3eba0b13ee7b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.95ex; height:2.509ex;" alt="{\displaystyle f:M\to \mathbb {R} }">, define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df_{p}:T_{p}M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <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 df_{p}:T_{p}M\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ea410da20ed7b211ef5d6d02fe16239173d763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.503ex; height:2.843ex;" alt="{\displaystyle df_{p}:T_{p}M\to \mathbb {R} }"> by sending a tangent vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:A_{p}\to \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>:</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v:A_{p}\to \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd82c857c98b76ac56e391ccc6b7c771bb6abba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.378ex; height:3.009ex;" alt="{\displaystyle v:A_{p}\to \mathbb {R} ^{n}}"> to the number given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D{\Big |}_{\phi _{\alpha }(p)}(f\circ \phi _{\alpha }^{-1})(v_{\alpha }),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D{\Big |}_{\phi _{\alpha }(p)}(f\circ \phi _{\alpha }^{-1})(v_{\alpha }),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5411556b8a993f135bc150db4cd8f5a2da5d16" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.776ex; height:4.676ex;" alt="{\displaystyle D{\Big |}_{\phi _{\alpha }(p)}(f\circ \phi _{\alpha }^{-1})(v_{\alpha }),}"> which due to the chain rule and the constraint in the definition of a tangent vector does not depend on the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in A_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in A_{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713ed3ea0fe3992cd40f38cdc6c89eb606124d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.777ex; height:2.843ex;" alt="{\displaystyle \alpha \in A_{p}.}"> One can check that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b20e0304a65ead2cafb33412a383b34fe527ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}"> naturally has the structure of a <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><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}">-dimensional real vector space, and that with this structure, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1142d069553f6a55274e87055720127c02d3ec02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.414ex; height:2.843ex;" alt="{\displaystyle df_{p}}"> is a linear map. The key observation is that, due to the constraint appearing in the definition of a tangent vector, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff50695dd0048523c0558c20e434114f2871719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.302ex; height:2.343ex;" alt="{\displaystyle v_{\beta }}"> for a single element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ea46c20042fba4142a87ecd1f7c29776a6ce46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.802ex; height:2.843ex;" alt="{\displaystyle A_{p}}"> automatically determines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e8acf871998209aa082e10fb1a5738cf721193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.009ex;" alt="{\displaystyle v_{\alpha }}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a833dd21d115a96c1c5b55a7136fadf84ce3a262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.718ex; height:2.176ex;" alt="{\displaystyle \alpha \in A.}"> The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{i}={\widetilde {v}}^{j}{\frac {\partial x^{i}}{\partial {\widetilde {x}}^{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{i}={\widetilde {v}}^{j}{\frac {\partial x^{i}}{\partial {\widetilde {x}}^{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d20542fe976ddcaf5b4904a52df086e099190a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.556ex; height:6.176ex;" alt="{\displaystyle v^{i}={\widetilde {v}}^{j}{\frac {\partial x^{i}}{\partial {\widetilde {x}}^{j}}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df_{p}(v)={\frac {\partial f}{\partial x^{i}}}v^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df_{p}(v)={\frac {\partial f}{\partial x^{i}}}v^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b0688761b02706ee8a6a5f93e5aec9330a46ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.307ex; height:5.843ex;" alt="{\displaystyle df_{p}(v)={\frac {\partial f}{\partial x^{i}}}v^{i}.}"></dd></dl> With the idea of the formal definitions understood, this shorthand notation is, for most purposes, much easier to work with. <div class="mw-heading mw-heading3"><h3 id="Partitions_of_unity">Partitions of unity</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=11" title="Edit section: Partitions of unity">edit</a>]</div> One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits <a href="/wiki/Partition_of_unity" title="Partition of unity">partitions of unity</a>. This distinguishes the differential structure on a manifold from stronger structures (such as analytic and holomorphic structures) that in general fail to have partitions of unity. Suppose that M is a manifold of class Ck, where 0 ≤ k ≤ ∞. Let {Uα} be an open covering of M. Then a partition of unity subordinate to the cover {Uα} is a collection of real-valued Ck functions φi on M satisfying the following conditions: <ul><li>The <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">supports</a> of the φi are <a href="/wiki/Compact_space" title="Compact space">compact</a> and <a href="/wiki/Locally_finite_collection" title="Locally finite collection">locally finite</a>;</li> <li>The support of φi is completely contained in Uα for some α;</li> <li>The φi sum to one at each point of M: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\phi _{i}(x)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\phi _{i}(x)=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277f015657c1b5d837ae0e985b4be9fc52b1a170" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.974ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\phi _{i}(x)=1.}"></li></ul> (Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φi.) Every open covering of a Ck manifold M has a Ck partition of unity. This allows for certain constructions from the topology of Ck functions on Rn to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of Rn. Partitions of unity therefore allow for certain other kinds of <a href="/wiki/Function_space" title="Function space">function spaces</a> to be considered: for instance <a href="/wiki/Lp_space" title="Lp space">Lp spaces</a>, <a href="/wiki/Sobolev_spaces" class="mw-redirect" title="Sobolev spaces">Sobolev spaces</a>, and other kinds of spaces that require integration. <div class="mw-heading mw-heading3"><h3 id="Differentiability_of_mappings_between_manifolds">Differentiability of mappings between manifolds</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=12" title="Edit section: Differentiability of mappings between manifolds">edit</a>]</div> Suppose M and N are two differentiable manifolds with dimensions m and n, respectively, and f is a function from M to N. Since differentiable manifolds are topological spaces we know what it means for f to be continuous. But what does "f is Ck(M, N)" mean for k ≥ 1? We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map that goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be Ck(Rm, Rn). We define "f is Ck(M, N)" to mean that all such compositions of f with charts are Ck(Rm, Rn). Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on M and N are selected. However, defining the derivative itself is more subtle. If M or N is itself already a Euclidean space, then we don't need a chart to map it to one. <div class="mw-heading mw-heading2"><h2 id="Bundles">Bundles</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=13" title="Edit section: Bundles">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Tangent_bundle">Tangent bundle</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=14" title="Edit section: Tangent bundle">edit</a>]</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/Tangent_bundle" title="Tangent bundle">tangent bundle</a></div> The <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> of a point consists of the possible directional derivatives at that point, and has the same <a href="/wiki/Dimension" title="Dimension">dimension</a> n as does the manifold. For a set of (non-singular) coordinates xk local to the point, the coordinate derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{k}={\frac {\partial }{\partial x_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{k}={\frac {\partial }{\partial x_{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6403bbe5dd2d0d034e9c9eb79880b7a061d56fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.994ex; height:5.843ex;" alt="{\displaystyle \partial _{k}={\frac {\partial }{\partial x_{k}}}}"> define a <a href="/wiki/Holonomic_basis" title="Holonomic basis">holonomic basis</a> of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a>, whose dimension is 2n. The tangent bundle is where <a href="/wiki/Vector_field" title="Vector field">tangent vectors</a> lie, and is itself a differentiable manifold. The <a href="/wiki/Lagrangian_system" title="Lagrangian system">Lagrangian</a> is a function on the tangent bundle. One can also define the tangent bundle as the bundle of 1-<a href="/wiki/Jet_(mathematics)" title="Jet (mathematics)">jets</a> from R (the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>) to M. One may construct an atlas for the tangent bundle consisting of charts based on Uα × Rn, where Uα denotes one of the charts in the atlas for M. Each of these new charts is the tangent bundle for the charts Uα. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class. <div class="mw-heading mw-heading3"><h3 id="Cotangent_bundle">Cotangent bundle</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=15" title="Edit section: Cotangent bundle">edit</a>]</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/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a></div> The <a href="/wiki/Dual_space" title="Dual space">dual space</a> of a vector space is the set of real valued linear functions on the vector space. The <a href="/wiki/Cotangent_space" title="Cotangent space">cotangent space</a> at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> is the collection of all cotangent vectors, along with the natural differentiable manifold structure. Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> is a scalar on the cotangent bundle. The <a href="/wiki/Total_space" class="mw-redirect" title="Total space">total space</a> of a cotangent bundle has the structure of a <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a>. Cotangent vectors are sometimes called <a href="/wiki/Covector" class="mw-redirect" title="Covector">covectors</a>. One can also define the cotangent bundle as the bundle of 1-<a href="/wiki/Jet_(mathematics)" title="Jet (mathematics)">jets</a> of functions from M to R. Elements of the cotangent space can be thought of as <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> displacements: if f is a differentiable function we can define at each point p a cotangent vector dfp, which sends a tangent vector Xp to the derivative of f associated with Xp. However, not every covector field can be expressed this way. Those that can are referred to as <a href="/wiki/Exact_differential" title="Exact differential">exact differentials</a>. For a given set of local coordinates xk, the differentials dxk p form a basis of the cotangent space at p. <div class="mw-heading mw-heading3"><h3 id="Tensor_bundle">Tensor bundle</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=16" title="Edit section: Tensor bundle">edit</a>]</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/Tensor_bundle" title="Tensor bundle">tensor bundle</a></div> The tensor bundle is the <a href="/wiki/Direct_sum_of_vector_bundles" class="mw-redirect" title="Direct sum of vector bundles">direct sum</a> of all <a href="/wiki/Tensor_product" title="Tensor product">tensor products</a> of the tangent bundle and the cotangent bundle. Each element of the bundle is a <a href="/wiki/Tensor_field" title="Tensor field">tensor field</a>, which can act as a <a href="/wiki/Multilinear_operator" class="mw-redirect" title="Multilinear operator">multilinear operator</a> on vector fields, or on other tensor fields. The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional. It is however an <a href="/wiki/Algebra_(ring_theory)" class="mw-redirect" title="Algebra (ring theory)">algebra</a> over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent factors it has. Sometimes these ranks are referred to as <a href="/wiki/Covariance" title="Covariance">covariant</a> and <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> ranks, signifying tangent and cotangent ranks, respectively. <div class="mw-heading mw-heading3"><h3 id="Frame_bundle">Frame bundle</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=17" title="Edit section: Frame bundle">edit</a>]</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/Frame_bundle" title="Frame bundle">frame bundle</a></div> A frame (or, in more precise terms, a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of Rn to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F(M), a <a href="/wiki/General_linear_group" title="General linear group">GL(n, R)</a> <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> made up of the set of all frames over M. The frame bundle is useful because tensor fields on M can be regarded as <a href="/wiki/Equivariant" class="mw-redirect" title="Equivariant">equivariant</a> vector-valued functions on F(M). <div class="mw-heading mw-heading3"><h3 id="Jet_bundles">Jet bundles</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=18" title="Edit section: Jet bundles">edit</a>]</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/Jet_bundle" title="Jet bundle">jet bundle</a></div> On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order <a href="/wiki/Contact_(mathematics)" title="Contact (mathematics)">contact</a>. By analogy, the k-th order tangent bundle is the collection of curves modulo the relation of k-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the k-jet bundle is the bundle of their k-jets. These and other examples of the general idea of jet bundles play a significant role in the study of <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a> on manifolds. The notion of a frame also generalizes to the case of higher-order jets. Define a k-th order frame to be the k-jet of a <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> from Rn to M.<a href="#cite_note-6">[6]</a> The collection of all k-th order frames, Fk(M), is a principal Gk bundle over M, where Gk is the <a href="/wiki/Jet_group" title="Jet group">group of k-jets</a>; i.e., the group made up of <a href="/wiki/Jet_(mathematics)" title="Jet (mathematics)">k-jets</a> of diffeomorphisms of Rn that fix the origin. Note that GL(n, R) is naturally isomorphic to G1, and a subgroup of every Gk, k ≥ 2. In particular, a section of F2(M) gives the frame components of a <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connection</a> on M. Thus, the quotient bundle F2(M) / GL(n, R) is the bundle of symmetric linear connections over M. <div class="mw-heading mw-heading2"><h2 id="Calculus_on_manifolds">Calculus on manifolds</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=19" title="Edit section: Calculus on manifolds">edit</a>]</div> Many of the techniques from <a href="/wiki/Multivariate_calculus" class="mw-redirect" title="Multivariate calculus">multivariate calculus</a> also apply, <a href="/wiki/Mutatis_mutandis" title="Mutatis mutandis">mutatis mutandis</a>, to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the <a href="/wiki/Total_derivative" title="Total derivative">total derivative</a> of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least <a href="/wiki/Local_property" title="Local property">locally</a>. For example, there are versions of the <a href="/wiki/Implicit_function" title="Implicit function">implicit</a> and <a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">inverse function theorems</a> for such functions. There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are: <ul><li>The <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a>, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.</li> <li>An <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a>, which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.</li></ul> Ideas from <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a> also carry over to differential manifolds. These are naturally expressed in the language of <a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">exterior calculus</a> and <a href="/wiki/Differential_form" title="Differential form">differential forms</a>. The fundamental theorems of integral calculus in several variables—namely <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a>, the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, and <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a>—generalize to a theorem (also called Stokes' theorem) relating the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> and integration over <a href="/wiki/Submanifold" title="Submanifold">submanifolds</a>. <div class="mw-heading mw-heading3"><h3 id="Differential_calculus_of_functions">Differential calculus of functions</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=20" title="Edit section: Differential calculus of functions">edit</a>]</div> Differentiable functions between two manifolds are needed in order to formulate suitable notions of <a href="/wiki/Submanifold" title="Submanifold">submanifolds</a>, and other related concepts. If f : M → N is a differentiable function from a differentiable manifold M of dimension m to another differentiable manifold N of dimension n, then the <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">differential</a> of f is a mapping df : TM → TN. It is also denoted by Tf and called the tangent map. At each point of M, this is a linear transformation from one tangent space to another: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df(p)\colon T_{p}M\to T_{f(p)}N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df(p)\colon T_{p}M\to T_{f(p)}N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901667ae950847c9878fef1fcf30acfe5c5ec250" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.291ex; height:3.176ex;" alt="{\displaystyle df(p)\colon T_{p}M\to T_{f(p)}N.}"> The rank of f at p is the <a href="/wiki/Rank_of_a_matrix" class="mw-redirect" title="Rank of a matrix">rank</a> of this linear transformation. Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given by <a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's theorem</a>. Functions of maximal rank at a point are called <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersions</a> and <a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">submersions</a>: <ul><li>If m ≤ n, and f : M → N has rank m at p ∈ M, then f is called an immersion at p. If f is an immersion at all points of M and is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> onto its image, then f is an <a href="/wiki/Embedding" title="Embedding">embedding</a>. Embeddings formalize the notion of M being a <a href="/wiki/Submanifold" title="Submanifold">submanifold</a> of N. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.</li> <li>If m ≥ n, and f : M → N has rank n at p ∈ M, then f is called a submersion at p. The implicit function theorem states that if f is a submersion at p, then M is locally a product of N and Rm−n near p. In formal terms, there exist coordinates (y1, ..., yn) in a neighborhood of f(p) in N, and m − n functions x1, ..., xm−n defined in a neighborhood of p in M such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{1}\circ f,\dotsc ,y_{n}\circ f,x_{1},\dotsc ,x_{m-n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y_{1}\circ f,\dotsc ,y_{n}\circ f,x_{1},\dotsc ,x_{m-n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c6c6765ea72310463f45e7ae6735d2f556fdcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.351ex; height:2.843ex;" alt="{\displaystyle (y_{1}\circ f,\dotsc ,y_{n}\circ f,x_{1},\dotsc ,x_{m-n})}"> is a system of local coordinates of M in a neighborhood of p. Submersions form the foundation of the theory of <a href="/wiki/Fibration" title="Fibration">fibrations</a> and <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">fibre bundles</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Lie_derivative">Lie derivative</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=21" title="Edit section: Lie derivative">edit</a>]</div> A <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a>, named after <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a>, is a <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a> on the <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a> of <a href="/wiki/Tensor_field" title="Tensor field">tensor fields</a> over a <a href="/wiki/Manifold" title="Manifold">manifold</a> M. The <a href="/wiki/Vector_space" title="Vector space">vector space</a> of all Lie derivatives on M forms an infinite dimensional <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> with respect to the <a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a> defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A,B]:={\mathcal {L}}_{A}B=-{\mathcal {L}}_{B}A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>B</mi> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A,B]:={\mathcal {L}}_{A}B=-{\mathcal {L}}_{B}A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52a85bba0a0a65aec00893b1af31acff9b2b32e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.792ex; height:2.843ex;" alt="{\displaystyle [A,B]:={\mathcal {L}}_{A}B=-{\mathcal {L}}_{B}A.}"> The Lie derivatives are represented by <a href="/wiki/Vector_field" title="Vector field">vector fields</a>, as <a href="/wiki/Lie_group#The_Lie_algebra_associated_to_a_Lie_group" title="Lie group">infinitesimal generators</a> of flows (<a href="/wiki/Active_transformation" class="mw-redirect" title="Active transformation">active</a> <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>) on M. Looking at it the other way around, the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of diffeomorphisms of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the <a href="/wiki/Lie_group" title="Lie group">Lie group</a> theory. <div class="mw-heading mw-heading3"><h3 id="Exterior_calculus">Exterior calculus</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=22" title="Edit section: Exterior calculus">edit</a>]</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/Differential_form" title="Differential form">differential form</a></div> The exterior calculus allows for a generalization of the <a href="/wiki/Gradient" title="Gradient">gradient</a>, <a href="/wiki/Divergence" title="Divergence">divergence</a> and <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> operators. The bundle of <a href="/wiki/Differential_form" title="Differential form">differential forms</a>, at each point, consists of all totally <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric</a> <a href="/wiki/Multilinear_map" title="Multilinear map">multilinear</a> maps on the tangent space at that point. It is naturally divided into n-forms for each n at most equal to the dimension of the manifold; an n-form is an n-variable form, also called a form of degree n. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, an n-form is a tensor with cotangent rank n and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric. <div class="mw-heading mw-heading4"><h4 id="Exterior_derivative">Exterior derivative</h4>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=23" title="Edit section: Exterior derivative">edit</a>]</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/Exterior_derivative" title="Exterior derivative">exterior derivative</a></div> The exterior derivative is a linear operator on the <a href="/wiki/Graded_vector_space" title="Graded vector space">graded vector space</a> of all smooth differential forms on a smooth manifold <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><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}">. It is usually denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">. More precisely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\dim(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\dim(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f129dc2cc6749381d8d5a34d490651436968fde2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.62ex; height:2.843ex;" alt="{\displaystyle n=\dim(M)}">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq k\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq k\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b429c3c44b3dc10332285272eee6f754dbf985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0\leq k\leq n}"> the operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> maps the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{k}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e527515b7c9de608cccd789454806924363647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.018ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k}(M)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-forms on <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><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}"> into the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{k+1}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{k+1}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92e1a9eb6d032924f3099d4c4072fa0f3793b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.119ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k+1}(M)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9f13644a6be482d7ddb19a6e0c706564773085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}">-forms (if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e81682bf174c978e9008ffb557ba4da2cf7478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k>n}"> there are no non-zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-forms on <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><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}"> so the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> is identically zero on <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><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}">-forms). For example, the exterior differential of a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is given in local coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}"> <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> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02a5fbf8bc31d443c91025339f9fd1de1065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}">, with associated local co-frame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx_{1},\ldots ,dx_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx_{1},\ldots ,dx_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0b10be2d458f01860db3b84cdcc0f7da7c2b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.542ex; height:2.509ex;" alt="{\displaystyle dx_{1},\ldots ,dx_{n}}"> by the formula : <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}dx_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}dx_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8d245f1aa1f450c7586dfad6475fe5d48eeb09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.611ex; height:6.843ex;" alt="{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}dx_{i}.}"> The exterior differential satisfies the following identity, similar to a <a href="/wiki/Product_rule" title="Product rule">product rule</a> with respect to the wedge product of forms: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{\deg \omega }\omega \wedge d\eta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>∧</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>ω</mi> <mo>∧</mo> <mi>η</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>deg</mi> <mo>⁡</mo> <mi>ω</mi> </mrow> </msup> <mi>ω</mi> <mo>∧</mo> <mi>d</mi> <mi>η</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{\deg \omega }\omega \wedge d\eta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9071a74dd65ed0e7fd39102a89378d62719467" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.524ex; height:3.176ex;" alt="{\displaystyle d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{\deg \omega }\omega \wedge d\eta .}"> The exterior derivative also satisfies the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\circ d=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>∘</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\circ d=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b28b25c1eefce2c0374e96c1638eae720af6041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.887ex; height:2.176ex;" alt="{\displaystyle d\circ d=0}">. That is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-form then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k+2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8af963731a6f9664ee81b251d48c96d92f5582a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+2)}">-form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(df)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(df)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c17343e94c583a288b4cd430df8256e0c6b24c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle d(df)}"> is identically vanishing. A form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\omega =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\omega =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36675fed06f4f9cab60ae8e9479020cf17f3244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.923ex; height:2.176ex;" alt="{\displaystyle d\omega =0}"> is called closed, while a form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =d\eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>d</mi> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =d\eta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/335b4226d5a601d08618fd580221cefd1861f6ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.93ex; height:2.676ex;" alt="{\displaystyle \omega =d\eta }"> for some other form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"> is called exact. Another formulation of the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\circ d=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>∘</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\circ d=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b28b25c1eefce2c0374e96c1638eae720af6041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.887ex; height:2.176ex;" alt="{\displaystyle d\circ d=0}"> is that an exact form is closed. This allows one to define <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> of the manifold <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><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}">, where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">th cohomology group is the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> of the closed forms on <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><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}"> by the exact forms on <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><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}">. <div class="mw-heading mw-heading2"><h2 id="Topology_of_differentiable_manifolds">Topology of differentiable manifolds</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=24" title="Edit section: Topology of differentiable manifolds">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Relationship_with_topological_manifolds">Relationship with topological manifolds</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=25" title="Edit section: Relationship with topological manifolds">edit</a>]</div> Suppose that <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><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}"> is a topological <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><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}">-manifold. If given any smooth atlas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f39f90fa6b69ece1e5a2c8bbcace4842442fa5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.322ex; height:2.843ex;" alt="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}}">, it is easy to find a smooth atlas which defines a different smooth manifold structure on <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> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775a47d0e02fa0a7a5a7a9687176fd3907cbe2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M;}"> consider a homeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi :M\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi :M\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6962920b8756d2e8aee1bb074fd596e1a77366b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.114ex; height:2.176ex;" alt="{\displaystyle \Phi :M\to M}"> which is not smooth relative to the given atlas; for instance, one can modify the identity map localized non-smooth bump. Then consider the new atlas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(\Phi ^{-1}(U_{\alpha }),\phi _{\alpha }\circ \Phi )\}_{\alpha \in A},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(\Phi ^{-1}(U_{\alpha }),\phi _{\alpha }\circ \Phi )\}_{\alpha \in A},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22d4ad00ab0cbd70763bce088e3e36c51d912b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.662ex; height:3.176ex;" alt="{\displaystyle \{(\Phi ^{-1}(U_{\alpha }),\phi _{\alpha }\circ \Phi )\}_{\alpha \in A},}"> which is easily verified as a smooth atlas. However, the charts in the new atlas are not smoothly compatible with the charts in the old atlas, since this would require that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha }\circ \Phi \circ \phi _{\beta }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> <mo>∘</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha }\circ \Phi \circ \phi _{\beta }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856a7b6ea24c9637aa695c6087ac5cc80f8970f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.455ex; height:3.676ex;" alt="{\displaystyle \phi _{\alpha }\circ \Phi \circ \phi _{\beta }^{-1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\alpha }\circ \Phi ^{-1}\circ \phi _{\beta }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∘</mo> <msubsup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\alpha }\circ \Phi ^{-1}\circ \phi _{\beta }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d73ebb682c36cacc9b788d5ed790bc0428f11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.788ex; height:3.676ex;" alt="{\displaystyle \phi _{\alpha }\circ \Phi ^{-1}\circ \phi _{\beta }^{-1}}"> are smooth for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ab677d974cccb0132cac08bd67fc8ac765627e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.979ex; height:2.509ex;" alt="{\displaystyle \beta ,}"> with these conditions being exactly the definition that both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab22c7cf7f1a54d85993e0257a93f28eae546df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.011ex; height:2.676ex;" alt="{\displaystyle \Phi ^{-1}}"> are smooth, in contradiction to how <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> was selected. With this observation as motivation, one can define an equivalence relation on the space of smooth atlases on <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><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}"> by declaring that smooth atlases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f39f90fa6b69ece1e5a2c8bbcace4842442fa5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.322ex; height:2.843ex;" alt="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(V_{\beta },\psi _{\beta })\}_{\beta \in B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>∈</mo> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(V_{\beta },\psi _{\beta })\}_{\beta \in B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/006cf33791ed84b806c232b9a99c3b05bf0f011a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.903ex; height:3.009ex;" alt="{\displaystyle \{(V_{\beta },\psi _{\beta })\}_{\beta \in B}}"> are equivalent if there is a homeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi :M\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi :M\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6962920b8756d2e8aee1bb074fd596e1a77366b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.114ex; height:2.176ex;" alt="{\displaystyle \Phi :M\to M}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(\Phi ^{-1}(U_{\alpha }),\phi _{\alpha }\circ \Phi )\}_{\alpha \in A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(\Phi ^{-1}(U_{\alpha }),\phi _{\alpha }\circ \Phi )\}_{\alpha \in A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf83b92e8b35af77e28445a73e8136bc525b9927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.015ex; height:3.176ex;" alt="{\displaystyle \{(\Phi ^{-1}(U_{\alpha }),\phi _{\alpha }\circ \Phi )\}_{\alpha \in A}}"> is smoothly compatible with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(V_{\beta },\psi _{\beta })\}_{\beta \in B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>∈</mo> <mi>B</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(V_{\beta },\psi _{\beta })\}_{\beta \in B},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8275d25ea2c3fff6a54997b004267368ff44c162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.549ex; height:3.009ex;" alt="{\displaystyle \{(V_{\beta },\psi _{\beta })\}_{\beta \in B},}"> and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(\Phi (V_{\beta }),\psi _{\beta }\circ \Phi ^{-1})\}_{\beta \in B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>∘</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>∈</mo> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(\Phi (V_{\beta }),\psi _{\beta }\circ \Phi ^{-1})\}_{\beta \in B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f2db10dd99abc650399354e9a6f36373b42401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.595ex; height:3.343ex;" alt="{\displaystyle \{(\Phi (V_{\beta }),\psi _{\beta }\circ \Phi ^{-1})\}_{\beta \in B}}"> is smoothly compatible with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b37d6859a1d93e54910e827321b46b5d9d57c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.969ex; height:2.843ex;" alt="{\displaystyle \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in A}.}"> More briefly, one could say that two smooth atlases are equivalent if there exists a diffeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\to M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\to M,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76674b85f00a3e422f62743faa3b0325f1723060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.145ex; height:2.509ex;" alt="{\displaystyle M\to M,}"> in which one smooth atlas is taken for the domain and the other smooth atlas is taken for the range. Note that this equivalence relation is a refinement of the equivalence relation which defines a smooth manifold structure, as any two smoothly compatible atlases are also compatible in the present sense; one can take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> to be the identity map. If the dimension of <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><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}"> is 1, 2, or 3, then there exists a smooth structure on <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><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}">, and all distinct smooth structures are equivalent in the above sense. The situation is more complicated in higher dimensions, although it isn't fully understood. <ul><li>Some topological manifolds admit no smooth structures, as was originally shown with a <a href="/wiki/Kervaire_manifold" title="Kervaire manifold">ten-dimensional example</a> by <a href="#CITEREFKervaire1960">Kervaire (1960)</a>. A <a href="/wiki/Donaldson%27s_theorem" title="Donaldson's theorem">major application</a> of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> in differential geometry due to <a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Simon Donaldson</a>, in combination with results of <a href="/wiki/Michael_Freedman" title="Michael Freedman">Michael Freedman</a>, shows that many simply-connected compact topological 4-manifolds do not admit smooth structures. A well-known particular example is the <a href="/wiki/E8_manifold" title="E8 manifold">E8 manifold</a>.</li> <li>Some topological manifolds admit many smooth structures which are not equivalent in the sense given above. This was originally discovered by <a href="/wiki/John_Milnor" title="John Milnor">John Milnor</a> in the form of the <a href="/wiki/Exotic_sphere" title="Exotic sphere">exotic 7-spheres</a>.<a href="#cite_note-7">[7]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Classification">Classification</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=26" title="Edit section: Classification">edit</a>]</div> Every one-dimensional connected smooth manifold is diffeomorphic to either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8d5f879619008f9a2851aa290f438c32e9232a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.223ex; height:3.009ex;" alt="{\displaystyle S^{1},}"> each with their standard smooth structures. For a classification of smooth 2-manifolds, see <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a>. A particular result is that every two-dimensional connected compact smooth manifold is diffeomorphic to one of the following: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2790d16fc095fefd35493399f92b5557ced3b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.223ex; height:3.009ex;" alt="{\displaystyle S^{2},}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S^{1}\times S^{1})\sharp \cdots \sharp (S^{1}\times S^{1}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi mathvariant="normal">♯</mi> <mo>⋯</mo> <mi mathvariant="normal">♯</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S^{1}\times S^{1})\sharp \cdots \sharp (S^{1}\times S^{1}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0028114e9f26e4f8a7261491455039a6ef60d1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.556ex; height:3.176ex;" alt="{\displaystyle (S^{1}\times S^{1})\sharp \cdots \sharp (S^{1}\times S^{1}),}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {RP} ^{2}\sharp \cdots \sharp \mathbb {RP} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">♯</mi> <mo>⋯</mo> <mi mathvariant="normal">♯</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {RP} ^{2}\sharp \cdots \sharp \mathbb {RP} ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a20a495114ffa87d5c55a3f724ab14764e7dec80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.259ex; height:3.176ex;" alt="{\displaystyle \mathbb {RP} ^{2}\sharp \cdots \sharp \mathbb {RP} ^{2}.}"> The situation is <a href="/wiki/Teichm%C3%BCller_space" title="Teichmüller space">more nontrivial</a> if one considers complex-differentiable structure instead of smooth structure. The situation in three dimensions is quite a bit more complicated, and known results are more indirect. A remarkable result, proved in 2002 by methods of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, is the <a href="/wiki/Thurston%27s_geometrization_conjecture" class="mw-redirect" title="Thurston's geometrization conjecture">geometrization conjecture</a>, stating loosely that any compact smooth 3-manifold can be split up into different parts, each of which admits Riemannian metrics which possess many symmetries. There are also various "recognition results" for geometrizable 3-manifolds, such as <a href="/wiki/Mostow_rigidity" class="mw-redirect" title="Mostow rigidity">Mostow rigidity</a> and Sela's algorithm for the isomorphism problem for hyperbolic groups.<a href="#cite_note-8">[8]</a> The classification of n-manifolds for n greater than three is known to be impossible, even up to <a href="/wiki/Homotopy_equivalence" class="mw-redirect" title="Homotopy equivalence">homotopy equivalence</a>. Given any finitely <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presented</a> group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to <a href="/wiki/Decision_problem" title="Decision problem">decide</a> the isomorphism problem for finitely presented groups, there is no algorithm to decide whether two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is <a href="/wiki/Decision_problem" title="Decision problem">undecidable</a>. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide whether a manifold has trivial fundamental group, i.e. is <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>. Simply connected <a href="/wiki/4-manifold" title="4-manifold">4-manifolds</a> have been classified up to homeomorphism by <a href="/wiki/Michael_Freedman" title="Michael Freedman">Freedman</a> using the <a href="/wiki/Intersection_theory" title="Intersection theory">intersection form</a> and <a href="/wiki/Kirby%E2%80%93Siebenmann_invariant" class="mw-redirect" title="Kirby–Siebenmann invariant">Kirby–Siebenmann invariant</a>. Smooth 4-manifold theory is known to be much more complicated, as the <a href="/wiki/Exotic_R4" title="Exotic R4">exotic smooth structures</a> on R4 demonstrate. However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the <a href="/wiki/H-cobordism_theorem" class="mw-redirect" title="H-cobordism theorem">h-cobordism theorem</a> can be used to reduce the classification to a classification up to homotopy equivalence, and <a href="/wiki/Surgery_theory" title="Surgery theory">surgery theory</a> can be applied.<a href="#cite_note-9">[9]</a> This has been carried out to provide an explicit classification of simply connected <a href="/wiki/5-manifold" title="5-manifold">5-manifolds</a> by Dennis Barden. <div class="mw-heading mw-heading2"><h2 id="Structures_on_smooth_manifolds">Structures on smooth manifolds</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=27" title="Edit section: Structures on smooth manifolds">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="(Pseudo-)Riemannian_manifolds">(Pseudo-)Riemannian manifolds</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=28" title="Edit section: (Pseudo-)Riemannian manifolds">edit</a>]</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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></div> A <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> consists of a smooth manifold together with a positive-definite <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a> on each of the individual tangent spaces. This collection of inner products is called the <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a>, and is naturally a symmetric 2-tensor field. This "metric" identifies a natural vector space isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M\to T_{p}^{\ast }M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}M\to T_{p}^{\ast }M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1ed6f8241397485e8f050862002438d6a011d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.689ex; height:2.843ex;" alt="{\displaystyle T_{p}M\to T_{p}^{\ast }M}"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c41366ebf5222955ca9977172cd6dd4e2d8aed3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.189ex; height:2.509ex;" alt="{\displaystyle p\in M.}"> On a Riemannian manifold one can define notions of length, volume, and angle. Any smooth manifold can be given many different Riemannian metrics. A <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> (also called a semi-Riemannian manifold) is a generalization of the notion of <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> where the inner products are allowed to have an <a href="/wiki/Metric_signature" title="Metric signature">indefinite signature</a>, as opposed to being <a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">positive-definite</a>; they are still required to be non-degenerate. Every smooth pseudo-Riemannian and Riemmannian manifold defines a number of associated tensor fields, such as the <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>. <a href="/wiki/Lorentzian_manifolds" class="mw-redirect" title="Lorentzian manifolds">Lorentzian manifolds</a> are pseudo-Riemannian manifolds of signature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158f4d6958e3923de0db80edec6c5f90a97d3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.403ex; height:2.843ex;" alt="{\displaystyle (n-1,1)}">; the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d928ec15aeef83aade867992ee473933adb6139d" 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=4}"> is fundamental in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. Not every smooth manifold can be given a non-Riemannian pseudo-Riemannian structure; there are topological restrictions on doing so. A <a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler manifold</a> is a different generalization of a Riemannian manifold, in which the inner product is replaced with a <a href="/wiki/Vector_norm" class="mw-redirect" title="Vector norm">vector norm</a>; as such, this allows the definition of length, but not angle. <div class="mw-heading mw-heading3"><h3 id="Symplectic_manifolds">Symplectic manifolds</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=29" title="Edit section: Symplectic manifolds">edit</a>]</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/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a></div> A <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> is a manifold equipped with a <a href="/wiki/Closed_form_(calculus)" class="mw-redirect" title="Closed form (calculus)">closed</a>, <a href="/wiki/Nondegenerate_form" class="mw-redirect" title="Nondegenerate form">nondegenerate</a> <a href="/wiki/2-form" class="mw-redirect" title="2-form">2-form</a>. This condition forces symplectic manifolds to be even-dimensional, due to the fact that skew-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n+1)\times (2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n+1)\times (2n+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc0b190d5d6bc6f60fad9cdb429ef500675b50b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.579ex; height:2.843ex;" alt="{\displaystyle (2n+1)\times (2n+1)}"> matrices all have zero determinant. There are two basic examples: <ul><li>Cotangent bundles, which arise as phase spaces in <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, are a motivating example, since they admit a <a href="/wiki/Tautological_one-form" title="Tautological one-form">natural symplectic form</a>.</li> <li>All oriented two-dimensional Riemannian manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e27d2e539fd0c3a9a7efab6257abd17de7fc57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}"> are, in a natural way, symplectic, by defining the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (u,v)=g(u,J(v))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (u,v)=g(u,J(v))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ba3a170160f183308eb81aa2d77f0f3f78d6cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.542ex; height:2.843ex;" alt="{\displaystyle \omega (u,v)=g(u,J(v))}"> where, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in T_{p}M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in T_{p}M,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9fc3cc87a756a6d74df405cecfb6745c130189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.474ex; height:2.843ex;" alt="{\displaystyle v\in T_{p}M,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(v)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f70a4a8467e7ea53af84425527b86d98605f81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.408ex; height:2.843ex;" alt="{\displaystyle J(v)}"> denotes the vector such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,J(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,J(v)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be8b6344d4a4c03d04cc9e55f69da363272fdb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.57ex; height:2.843ex;" alt="{\displaystyle v,J(v)}"> is an oriented <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8699118cdd0dd2bded6fe4d529a6af245d7a83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.168ex; height:2.343ex;" alt="{\displaystyle g_{p}}">-orthonormal basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b69cac5a8c2a370102468e4b47abc73ac692fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.506ex; height:2.843ex;" alt="{\displaystyle T_{p}M.}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Lie_groups">Lie groups</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=30" title="Edit section: Lie groups">edit</a>]</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> A <a href="/wiki/Lie_group" title="Lie group">Lie group</a> consists of a C∞ manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> together with a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> such that the product and inversion maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m:G\times G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>:</mo> <mi>G</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m:G\times G\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513b3d99d2a660a1552a8ae4bd19bd82e78ad871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.912ex; height:2.176ex;" alt="{\displaystyle m:G\times G\to G}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:G\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413e0936c524dbc4d440b73ac947d32ebd0215b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.007ex; height:2.176ex;" alt="{\displaystyle i:G\to G}"> are smooth as maps of manifolds. These objects often arise naturally in describing (continuous) symmetries, and they form an important source of examples of smooth manifolds. Many otherwise familiar examples of smooth manifolds, however, cannot be given a Lie group structure, since given a Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> and any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be73903416a0dd94b8cbc2268ce480810c0e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}">, one could consider the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(g,\cdot ):G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(g,\cdot ):G\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689244b229cb62b97de38ed8f0e153e8edad516f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.851ex; height:2.843ex;" alt="{\displaystyle m(g,\cdot ):G\to G}"> which sends the identity element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> and hence, by considering the differential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{e}G\to T_{g}G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>G</mi> <mo stretchy="false">→</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mi>G</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{e}G\to T_{g}G,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facc6176eae2b6184437ca53cb57b059cb2c5cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.649ex; height:2.843ex;" alt="{\displaystyle T_{e}G\to T_{g}G,}"> gives a natural identification between any two tangent spaces of a Lie group. In particular, by considering an arbitrary nonzero vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{e}G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>G</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{e}G,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729522289c49c40b6bf4ad1908c79e795f5db20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.83ex; height:2.509ex;" alt="{\displaystyle T_{e}G,}"> one can use these identifications to give a smooth non-vanishing vector field on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}"> This shows, for instance, that no <a href="/wiki/Hairy_ball_theorem" title="Hairy ball theorem">even-dimensional sphere</a> can support a Lie group structure. The same argument shows, more generally, that every Lie group must be <a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">parallelizable</a>. <div class="mw-heading mw-heading2"><h2 id="Alternative_definitions">Alternative definitions</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=31" title="Edit section: Alternative definitions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Pseudogroups">Pseudogroups</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=32" title="Edit section: Pseudogroups">edit</a>]</div> The notion of a <a href="/wiki/Pseudogroup" title="Pseudogroup">pseudogroup</a><a href="#cite_note-10">[10]</a> provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. A pseudogroup consists of a topological space S and a collection Γ consisting of homeomorphisms from open subsets of S to other open subsets of S such that <ol><li>If f ∈ Γ, and U is an open subset of the domain of f, then the restriction f|U is also in Γ.</li> <li>If f is a homeomorphism from a union of open subsets of S, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup _{i}\,U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup _{i}\,U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cc0b4be858829765a46a821b21068d98d3ad37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.124ex; height:2.509ex;" alt="{\displaystyle \cup _{i}\,U_{i}}">, to an open subset of S, then f ∈ Γ provided <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{U_{i}}\in \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{U_{i}}\in \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36d29126f5183147095dc135e57721dbe87bf4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.199ex; height:3.176ex;" alt="{\displaystyle f|_{U_{i}}\in \Gamma }"> for every i.</li> <li>For every open U ⊂ S, the identity transformation of U is in Γ.</li> <li>If f ∈ Γ, then f−1 ∈ Γ.</li> <li>The composition of two elements of Γ is in Γ.</li></ol> These last three conditions are analogous to the definition of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. Note that Γ need not be a group, however, since the functions are not globally defined on S. For example, the collection of all local Ck <a href="/wiki/Diffeomorphisms" class="mw-redirect" title="Diffeomorphisms">diffeomorphisms</a> on Rn form a pseudogroup. All <a href="/wiki/Biholomorphism" title="Biholomorphism">biholomorphisms</a> between open sets in Cn form a pseudogroup. More examples include: orientation preserving maps of Rn, <a href="/wiki/Symplectomorphism" title="Symplectomorphism">symplectomorphisms</a>, <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>, <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>, and so on. Thus, a wide variety of function classes determine pseudogroups. An atlas (Ui, φi) of homeomorphisms φi from Ui ⊂ M to open subsets of a topological space S is said to be compatible with a pseudogroup Γ provided that the transition functions φj ∘ φi−1 : φi(Ui ∩ Uj) → φj(Ui ∩ Uj) are all in Γ. A differentiable manifold is then an atlas compatible with the pseudogroup of Ck functions on Rn. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in Cn. And so forth. Thus, pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology. <div class="mw-heading mw-heading3"><h3 id="Structure_sheaf">Structure sheaf</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=33" title="Edit section: Structure sheaf">edit</a>]</div> Sometimes, it can be useful to use an alternative approach to endow a manifold with a Ck-structure. Here k = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">structure sheaf</a> of M, denoted Ck, is a sort of <a href="/wiki/Functor" title="Functor">functor</a> that defines, for each open set U ⊂ M, an algebra Ck(U) of continuous functions U → R. A structure sheaf Ck is said to give M the structure of a Ck manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x1, ..., xn ∈ Ck(U) such that the map f = (x1, ..., xn) : U → Rn is a homeomorphism onto an open set in Rn, and such that Ck|U is the <a href="/wiki/Pullback" title="Pullback">pullback</a> of the sheaf of k-times continuously differentiable functions on Rn.<a href="#cite_note-11">[11]</a> In particular, this latter condition means that any function h in Ck(V), for V, can be written uniquely as h(x) = H(x1(x), ..., xn(x)), where H is a k-times differentiable function on f(V) (an open set in Rn). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on Rn, and <a href="/wiki/A_fortiori_argument" class="mw-redirect" title="A fortiori argument">a fortiori</a> this is sufficient to characterize the differential structure on the manifold. <div class="mw-heading mw-heading4"><h4 id="Sheaves_of_local_rings">Sheaves of local rings</h4>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=34" title="Edit section: Sheaves of local rings">edit</a>]</div> A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a <a href="/wiki/Ringed_space" title="Ringed space">ringed space</a>. This approach is strongly influenced by the theory of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, but uses <a href="/wiki/Local_ring" title="Local ring">local rings</a> of the <a href="/wiki/Germ_(mathematics)" title="Germ (mathematics)">germs</a> of differentiable functions. It is especially popular in the context of complex manifolds. We begin by describing the basic structure sheaf on Rn. If U is an open set in Rn, let <dl><dd>O(U) = Ck(U, R)</dd></dl> consist of all real-valued k-times continuously differentiable functions on U. As U varies, this determines a sheaf of rings on Rn. The stalk Op for p ∈ Rn consists of <a href="/wiki/Germ_(mathematics)" title="Germ (mathematics)">germs</a> of functions near p, and is an algebra over R. In particular, this is a <a href="/wiki/Local_ring" title="Local ring">local ring</a> whose unique <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> consists of those functions that vanish at p. The pair (Rn, O) is an example of a <a href="/wiki/Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed space</a>: it is a topological space equipped with a sheaf whose stalks are each local rings. A differentiable manifold (of class Ck) consists of a pair (M, OM) where M 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>, and OM is a sheaf of local R-algebras defined on M, such that the locally ringed space (M, OM) is locally isomorphic to (Rn, O). In this way, differentiable manifolds can be thought of as <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> modeled on Rn. This means that <a href="#cite_note-12">[12]</a> for each point p ∈ M, there is a neighborhood U of p, and a pair of functions (f, f#), where <ol><li>f : U → f(U) ⊂ Rn is a homeomorphism onto an open set in Rn.</li> <li>f#: O|f(U) → f∗ (OM|U) is an isomorphism of sheaves.</li> <li>The localization of f# is an isomorphism of local rings</li></ol> <dl><dd><dl><dd>f#f(p) : Of(p) → OM,p.</dd></dl></dd></dl> There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be Rn. For example, (in particular in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>), one could take this to be the space of complex numbers Cn equipped with the sheaf of <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> (thus arriving at the spaces of <a href="/wiki/Complex_analytic_geometry" class="mw-redirect" title="Complex analytic geometry">complex analytic geometry</a>), or the sheaf of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> (thus arriving at the spaces of interest in complex algebraic geometry). In broader terms, this concept can be adapted for any suitable notion of a scheme (see <a href="/wiki/Topos" title="Topos">topos theory</a>). Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair (f, f#), but these merely quantify the idea of local isomorphism rather than being central to the discussion (as in the case of charts and atlases). Third, the sheaf OM is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals). Hence, it is a more primitive definition of the structure (see <a href="/wiki/Synthetic_differential_geometry" title="Synthetic differential geometry">synthetic differential geometry</a>). A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology. <ul><li>The <a href="/wiki/Cotangent_space" title="Cotangent space">cotangent space</a> at a point is Ip/Ip2, where Ip is the maximal ideal of the stalk OM,p.</li> <li>In general, the entire <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> can be obtained by a related technique (see <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> for details).</li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> (and <a href="/wiki/Jet_(mathematics)" title="Jet (mathematics)">jets</a>) can be approached in a coordinate-independent manner using the <a href="/wiki/Completion_(algebra)#Krull_topology" class="mw-redirect" title="Completion (algebra)">Ip-adic filtration</a> on OM,p.</li> <li>The <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> (or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of OM into the ring of <a href="/wiki/Dual_numbers" class="mw-redirect" title="Dual numbers">dual numbers</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=35" title="Edit section: Generalizations">edit</a>]</div> The <a href="/wiki/Category_theory" title="Category theory">category</a> of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. <a href="/wiki/Diffeological_space" class="mw-redirect" title="Diffeological space">Diffeological spaces</a> use a different notion of chart known as a "plot". <a href="/wiki/Fr%C3%B6licher_space" title="Frölicher space">Frölicher spaces</a> and <a href="/wiki/Orbifold" title="Orbifold">orbifolds</a> are other attempts. A <a href="/wiki/Rectifiable_set" title="Rectifiable set">rectifiable set</a> generalizes the idea of a piece-wise 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. <a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifolds</a> and <a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifolds</a>, in particular <a href="/wiki/Convenient_vector_space#Application:_Manifolds_of_mappings_between_finite_dimensional_manifolds" title="Convenient vector space">manifolds of mappings</a> are infinite dimensional differentiable manifolds. <div class="mw-heading mw-heading3"><h3 id="Non-commutative_geometry">Non-commutative geometry</h3>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=36" title="Edit section: Non-commutative geometry">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></td><td class="mbox-text"><div class="mbox-text-span">This section needs expansion. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Differentiable_manifold&action=edit&section=">adding to it</a>. (June 2008)</div></td></tr></tbody></table> For a Ck manifold M, the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of real-valued Ck functions on the manifold forms an <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a> under pointwise addition and multiplication, called the algebra of scalar fields or simply the algebra of scalars. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of <a href="/wiki/Regular_function" class="mw-redirect" title="Regular function">regular functions</a> in algebraic geometry. It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the <a href="/wiki/Banach%E2%80%93Stone_theorem" title="Banach–Stone theorem">Banach–Stone theorem</a>, and is more formally known as the <a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">spectrum of a C*-algebra</a>. First, there is a one-to-one correspondence between the points of M and the algebra homomorphisms φ: Ck(M) → R, as such a homomorphism φ corresponds to a codimension one ideal in Ck(M) (namely the kernel of φ), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec (the Max Spec) of Ck(M) recovers M as a point set, though in fact it recovers M as a topological space. One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry (interpreting rings geometrically) and <a href="/wiki/Operator_theory" title="Operator theory">operator theory</a> (interpreting Banach spaces geometrically). For example, the tangent bundle to M can be defined as the derivations of the algebra of smooth functions on M. This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of a <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> – a commutative C*-algebra being precisely the ring of scalars of a manifold, by Banach–Stone, and allows one to consider noncommutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of <a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=37" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 34em;"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas (topology)</a></li> <li><a href="/wiki/Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a></li> <li><a href="/wiki/Introduction_to_the_mathematics_of_general_relativity" title="Introduction to the mathematics of general relativity">Introduction to the mathematics of general relativity</a></li> <li><a href="/wiki/List_of_formulas_in_Riemannian_geometry" title="List of formulas in Riemannian geometry">List of formulas in Riemannian geometry</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Space_(mathematics)" title="Space (mathematics)">Space (mathematics)</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=38" title="Edit section: References">edit</a>]</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: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> B. Riemann (1867). </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Maxwell himself worked with <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> rather than tensors, but his equations for electromagnetism were used as an early example of the tensor formalism; 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="CITEREFDimitrienko2002" class="citation cs2">Dimitrienko, Yuriy I. (2002), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7UMYToTiYDsC&pg=PR11">Tensor Analysis and Nonlinear Tensor Functions</a>, Springer, p. xi, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781402010156" title="Special:BookSources/9781402010156"><bdi>9781402010156</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensor+Analysis+and+Nonlinear+Tensor+Functions&rft.pages=xi&rft.pub=Springer&rft.date=2002&rft.isbn=9781402010156&rft.aulast=Dimitrienko&rft.aufirst=Yuriy+I.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7UMYToTiYDsC%26pg%3DPR11&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988">. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> See G. Ricci (1888), G. Ricci and T. Levi-Civita (1901), T. Levi-Civita (1927). </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> See H. Weyl (1955). </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> H. Whitney (1936). </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> See S. Kobayashi (1972). </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> J. Milnor (1956). </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Z. Sela (1995). However, 3-manifolds are only classified in the sense that there is an (impractical) algorithm for generating a non-redundant list of all compact 3-manifolds. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> See A. Ranicki (2002). </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Kobayashi and Nomizu (1963), Volume 1. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> This definition can be found in MacLane and Moerdijk (1992). For an equivalent, ad hoc definition, see Sternberg (1964) Chapter II. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Hartshorne (1997) </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Differentiable_manifold&action=edit&section=39" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonaldson,_Simon1983" class="citation journal cs1"><a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Donaldson, Simon</a> (1983). <a rel="nofollow" class="external text" href="https://projecteuclid.org/download/pdf_1/euclid.jdg/1214437665">"An application of gauge theory to four-dimensional topology"</a>. <a href="/wiki/Journal_of_Differential_Geometry" title="Journal of Differential Geometry">Journal of Differential Geometry</a>. 18 (2): 279–315. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214437665">10.4310/jdg/1214437665</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Differential+Geometry&rft.atitle=An+application+of+gauge+theory+to+four-dimensional+topology&rft.volume=18&rft.issue=2&rft.pages=279-315&rft.date=1983&rft_id=info%3Adoi%2F10.4310%2Fjdg%2F1214437665&rft.au=Donaldson%2C+Simon&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fdownload%2Fpdf_1%2Feuclid.jdg%2F1214437665&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne1977" class="citation book cs1"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977). Algebraic Geometry. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90244-9" title="Special:BookSources/0-387-90244-9"><bdi>0-387-90244-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=0-387-90244-9&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988"></li> <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=Differentiable_manifold">"Differentiable manifold"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <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=Differentiable+manifold&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDifferentiable_manifold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKervaire1960" class="citation journal cs1"><a href="/wiki/Michel_Kervaire" title="Michel Kervaire">Kervaire, Michel A.</a> (1960). "A manifold which does not admit any differentiable structure". <a href="/wiki/Commentarii_Mathematici_Helvetici" title="Commentarii Mathematici Helvetici">Commentarii Mathematici Helvetici</a>. 34 (1): 257–270. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02565940">10.1007/BF02565940</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120977898">120977898</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Commentarii+Mathematici+Helvetici&rft.atitle=A+manifold+which+does+not+admit+any+differentiable+structure&rft.volume=34&rft.issue=1&rft.pages=257-270&rft.date=1960&rft_id=info%3Adoi%2F10.1007%2FBF02565940&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120977898%23id-name%3DS2CID&rft.aulast=Kervaire&rft.aufirst=Michel+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKobayashi1972" class="citation book cs1"><a href="/wiki/Shoshichi_Kobayashi" title="Shoshichi Kobayashi">Kobayashi, Shoshichi</a> (1972). Transformation groups in differential geometry. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transformation+groups+in+differential+geometry&rft.pub=Springer&rft.date=1972&rft.aulast=Kobayashi&rft.aufirst=Shoshichi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2009" class="citation cs2">Lee, Jeffrey M. 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Prentice-Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Differential+Geometry&rft.pub=Prentice-Hall&rft.date=1964&rft.aulast=Sternberg&rft.aufirst=Shlomo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flecturesondiffer0000ster&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferentiable+manifold" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/SmoothManifold.html">"Smooth Manifold"</a>. 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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 class="mw-selflink selflink">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 <a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</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/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 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> 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