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class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition_and_first_consequences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_and_first_consequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition and first consequences</span> </div> </a> <button aria-controls="toc-Definition_and_first_consequences-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition and first consequences subsection</span> </button> <ul id="toc-Definition_and_first_consequences-sublist" class="vector-toc-list"> <li id="toc-Transition_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transition_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Transition functions</span> </div> </a> <ul id="toc-Transition_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subbundles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subbundles"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Subbundles</span> </div> </a> <ul id="toc-Subbundles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vector_bundle_morphisms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vector_bundle_morphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vector bundle morphisms</span> </div> </a> <ul id="toc-Vector_bundle_morphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sections_and_locally_free_sheaves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sections_and_locally_free_sheaves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Sections and locally free sheaves</span> </div> </a> <ul id="toc-Sections_and_locally_free_sheaves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operations_on_vector_bundles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operations_on_vector_bundles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Operations on vector bundles</span> </div> </a> <ul id="toc-Operations_on_vector_bundles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_structures_and_generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Additional_structures_and_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Additional structures and generalizations</span> </div> </a> <ul id="toc-Additional_structures_and_generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Smooth_vector_bundles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Smooth_vector_bundles"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Smooth vector bundles</span> </div> </a> <ul id="toc-Smooth_vector_bundles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-K-theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#K-theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>K-theory</span> </div> </a> <ul id="toc-K-theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-General_notions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_notions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>General notions</span> </div> </a> <ul id="toc-General_notions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology_and_differential_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology_and_differential_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Topology and differential geometry</span> </div> </a> <ul id="toc-Topology_and_differential_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_and_analytic_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_and_analytic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Algebraic and analytic geometry</span> </div> </a> <ul id="toc-Algebraic_and_analytic_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div 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class="mw-page-title-main">Vector bundle</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 21 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">21 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B2%D9%85%D8%A9_%D8%B4%D8%B9%D8%A7%D8%B9%D9%8A%D8%A9" title="حزمة شعاعية – Arabic" lang="ar" hreflang="ar" data-title="حزمة شعاعية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Fibrat_vectorial" title="Fibrat vectorial – Catalan" lang="ca" hreflang="ca" data-title="Fibrat vectorial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vektorb%C3%BCndel" title="Vektorbündel – German" lang="de" hreflang="de" data-title="Vektorbündel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fibrado_vectorial" title="Fibrado vectorial – Spanish" lang="es" hreflang="es" data-title="Fibrado vectorial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektora_fasko" title="Vektora fasko – Esperanto" lang="eo" hreflang="eo" data-title="Vektora fasko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D9%84%D8%A7%D9%81_%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1%DB%8C" title="کلاف برداری – Persian" lang="fa" hreflang="fa" data-title="کلاف برداری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fibr%C3%A9_vectoriel" title="Fibré vectoriel – French" lang="fr" hreflang="fr" data-title="Fibré vectoriel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A1%ED%84%B0_%EB%8B%A4%EB%B0%9C" title="벡터 다발 – Korean" lang="ko" hreflang="ko" data-title="벡터 다발" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Fibrato_vettoriale" title="Fibrato vettoriale – Italian" lang="it" hreflang="it" data-title="Fibrato vettoriale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%92%D7%93_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="אגד וקטורי – Hebrew" lang="he" hreflang="he" data-title="אגד וקטורי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vectorbundel" title="Vectorbundel – Dutch" lang="nl" hreflang="nl" data-title="Vectorbundel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E6%9D%9F" title="ベクトル束 – Japanese" lang="ja" hreflang="ja" data-title="ベクトル束" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wi%C4%85zka_wektorowa" title="Wiązka wektorowa – Polish" lang="pl" hreflang="pl" data-title="Wiązka wektorowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fibrado_vetorial" title="Fibrado vetorial – Portuguese" lang="pt" hreflang="pt" data-title="Fibrado vetorial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D1%81%D0%BB%D0%BE%D0%B5%D0%BD%D0%B8%D0%B5" title="Векторное расслоение – Russian" lang="ru" hreflang="ru" data-title="Векторное расслоение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vektorikimppu" title="Vektorikimppu – Finnish" lang="fi" hreflang="fi" data-title="Vektorikimppu" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Vektorknippe" title="Vektorknippe – Swedish" lang="sv" hreflang="sv" data-title="Vektorknippe" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%B5_%D1%80%D0%BE%D0%B7%D1%88%D0%B0%D1%80%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Векторне розшарування – Ukrainian" lang="uk" hreflang="uk" data-title="Векторне розшарування" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_th%E1%BB%9B_v%C3%A9ct%C6%A1" title="Phân thớ véctơ – Vietnamese" lang="vi" hreflang="vi" data-title="Phân thớ véctơ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E5%8F%A2" title="向量叢 – Cantonese" lang="yue" hreflang="yue" data-title="向量叢" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E4%B8%9B" title="向量丛 – Chinese" lang="zh" hreflang="zh" data-title="向量丛" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical parametrization of vector spaces by another space</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mobius_strip_illus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Mobius_strip_illus.svg/250px-Mobius_strip_illus.svg.png" decoding="async" width="250" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Mobius_strip_illus.svg/375px-Mobius_strip_illus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Mobius_strip_illus.svg/500px-Mobius_strip_illus.svg.png 2x" data-file-width="300" data-file-height="280" /></a><figcaption>The (infinitely extended) <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> is a <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a> over the <a href="/wiki/N-sphere" title="N-sphere">1-sphere</a> <b>S</b><sup>1</sup>. Locally around every point in <b>S</b><sup>1</sup>, it <a href="/wiki/Homeomorphism" title="Homeomorphism">looks like</a> <i>U</i> × <b>R</b> (where <i>U</i> is an open <a href="/wiki/Arc_(topology)" class="mw-redirect" title="Arc (topology)">arc</a> including the point), but the total bundle is different from <b>S</b><sup>1</sup> × <b>R</b> (which is a <a href="/wiki/Cartesian_product" title="Cartesian product">cylinder</a> instead).</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>vector bundle</b> is a <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> construction that makes precise the idea of a <a href="/wiki/Family_of_sets" title="Family of sets">family</a> of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> parameterized by another <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (for example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> could be a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, a <a href="/wiki/Manifold" title="Manifold">manifold</a>, or an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>): to every point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> we associate (or "attach") a vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab3e825c2bf9c80d11d12e070a4626d48e03c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}"></span> in such a way that these vector spaces fit together to form another space of the same kind as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (e.g. a topological space, manifold, or algebraic variety), which is then called a <b>vector bundle over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span></b>. </p><p>The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)=V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)=V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c3b02ea27a0711748cda32ad3ddb07ecfe7883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.812ex; height:2.843ex;" alt="{\displaystyle V(x)=V}"></span> <a href="/wiki/For_all" class="mw-redirect" title="For all">for all</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>: in this case there is a copy of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and these copies fit together to form the vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7debf43b00ac69f7b4dbaacb835837ff4e63d358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.176ex;" alt="{\displaystyle X\times V}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Such vector bundles are said to be <a href="/wiki/Fiber_bundle#Trivial_bundle" title="Fiber bundle"><i>trivial</i></a>. A more complicated (and prototypical) class of examples are the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundles</a> of <a href="/wiki/Manifold" title="Manifold">smooth (or differentiable) manifolds</a>: to every point of such a manifold we attach the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the <a href="/wiki/Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>. In general, a manifold is said to be <a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">parallelizable</a> if, and only if, its tangent bundle is trivial. </p><p>Vector bundles are almost always required to be <i>locally trivial</i>, which means they are examples of <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundles</a>. Also, the vector spaces are usually required to be over the <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, in which case the vector bundle is said to be a real or complex vector bundle (respectively). <a href="/wiki/Complex_vector_bundle" title="Complex vector bundle">Complex vector bundles</a> can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the <a href="/wiki/Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_first_consequences">Definition and first consequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=1" title="Edit section: Definition and first consequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vector_bundle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Vector_bundle.png/300px-Vector_bundle.png" decoding="async" width="300" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Vector_bundle.png/450px-Vector_bundle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Vector_bundle.png/600px-Vector_bundle.png 2x" data-file-width="1912" data-file-height="1583" /></a><figcaption>A vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> over a base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. A point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(=X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mo>=</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(=X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f203dafa6a7caaafae1afa7ebe78d16a084b1be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.685ex; height:2.843ex;" alt="{\displaystyle M(=X)}"></span> corresponds to the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> in a fibre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfd6e415c3ac8cee5e5b9a79e8303bc8af844fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.222ex; height:2.843ex;" alt="{\displaystyle E_{m_{1}}}"></span> of the vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, and this fibre is mapped down to the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span> by the <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">projection</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6639e46e22a131d7cf81be9460c78e68e673fa51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.101ex; height:2.176ex;" alt="{\displaystyle \pi :E\to M}"></span>.</figcaption></figure> <p>A <b>real vector bundle</b> consists of: </p> <ol><li>topological spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (<i>base space</i>) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> (<i>total space</i>)</li> <li>a <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Surjection" class="mw-redirect" title="Surjection">surjection</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf8de3c25d0905f50abaf4f9374daa0a4bd796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}"></span> (<i>bundle projection</i>)</li> <li>for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, the structure of a <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">finite-dimensional</a> <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Vector_space" title="Vector space">vector space</a> on the <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(\{x\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(\{x\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1324bc8d82b0ad4271b2ea25dcd89f6fcf936a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(\{x\})}"></span></li></ol> <p>where the following compatibility condition is satisfied: for every point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, there is an <a href="/wiki/Open_neighborhood" class="mw-redirect" title="Open neighborhood">open neighborhood</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>, a <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>, and a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>U</mi> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78126d4f5262fa66874845674c8fe06cf75ee243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.817ex; height:3.176ex;" alt="{\displaystyle \varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)}"></span></dd></dl> <p>such that for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span>, </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi \circ \varphi )(x,v)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>π</mi> <mo>∘</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi \circ \varphi )(x,v)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9941764a8a38d21116503cfdf3226dc1d87884d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.585ex; height:2.843ex;" alt="{\displaystyle (\pi \circ \varphi )(x,v)=x}"></span> for all <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{k}}"> <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>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}"></span>, and</li> <li>the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\mapsto \varphi (x,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">↦</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\mapsto \varphi (x,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c91df1d568d242f410e1136d954b578668790221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.562ex; height:2.843ex;" alt="{\displaystyle v\mapsto \varphi (x,v)}"></span> is a <a href="/wiki/Linear_map" title="Linear map">linear</a> <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> between the vector spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{k}}"> <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>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(\{x\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(\{x\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1324bc8d82b0ad4271b2ea25dcd89f6fcf936a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(\{x\})}"></span>.</li></ul> <p>The open neighborhood <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> together with the homeomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is called a <b><a href="/wiki/Local_trivialization" class="mw-redirect" title="Local trivialization">local trivialization</a></b> of the vector bundle. The local trivialization shows that <i>locally</i> the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> "looks like" the projection of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\times \mathbb {R} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\times \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b80843ab74234e43f73e372586de584aade0bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.39ex; height:2.676ex;" alt="{\displaystyle U\times \mathbb {R} ^{k}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span>. </p><p>Every fiber <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(\{x\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(\{x\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1324bc8d82b0ad4271b2ea25dcd89f6fcf936a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(\{x\})}"></span> is a finite-dimensional real vector space and hence has a <a href="/wiki/Dimension" title="Dimension">dimension</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbcc5a960cbbd063fb012b087a8fdef03066d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}"></span>. The local trivializations show that the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to k_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to k_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44202a2dfa93a48f09d8026cf6a8c79d1de7701c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.327ex; height:2.509ex;" alt="{\displaystyle x\to k_{x}}"></span> is <a href="/wiki/Locally_constant" class="mw-redirect" title="Locally constant">locally constant</a>, and is therefore constant on each <a href="/wiki/Locally_connected_space" title="Locally connected space">connected component</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbcc5a960cbbd063fb012b087a8fdef03066d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}"></span> is equal to a constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> on all of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is called the <b>rank</b> of the vector bundle, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is said to be a <b>vector bundle of rank <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span></b>. Often the definition of a vector bundle includes that the rank is well defined, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbcc5a960cbbd063fb012b087a8fdef03066d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}"></span> is constant. Vector bundles of rank 1 are called <a href="/wiki/Line_bundle" title="Line bundle">line bundles</a>, while those of rank 2 are less commonly called plane bundles. </p><p>The <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb393f0441bbee93188d64d08ec6710993bd2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.587ex; height:2.676ex;" alt="{\displaystyle X\times \mathbb {R} ^{k}}"></span>, equipped with the projection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} ^{k}\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {R} ^{k}\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c022dc9321d0c05ef498784bf67de967698cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.181ex; height:2.676ex;" alt="{\displaystyle X\times \mathbb {R} ^{k}\to X}"></span>, is called the <b>trivial bundle</b> of rank <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Transition_functions">Transition functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=2" title="Edit section: Transition functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Transition_functions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Transition_functions.png/300px-Transition_functions.png" decoding="async" width="300" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Transition_functions.png/450px-Transition_functions.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Transition_functions.png/600px-Transition_functions.png 2x" data-file-width="2066" data-file-height="1979" /></a><figcaption>Two trivial vector bundles over <a href="/wiki/Open_set" title="Open set">open sets</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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 }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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 }}"></span> may be <a href="/wiki/Gluing_(topology)" class="mw-redirect" title="Gluing (topology)">glued</a> over the intersection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\alpha \beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d8973640ae2416841ae4f25a1784c266292c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.814ex; height:2.843ex;" alt="{\displaystyle U_{\alpha \beta }}"></span> by transition functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\alpha \beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8eaf30ff3ec1f116c0cacac01122f01243072fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.335ex; height:2.343ex;" alt="{\displaystyle g_{\alpha \beta }}"></span> which serve to stick the shaded grey regions together after applying a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> to the fibres (note the transformation of the blue <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> under the effect of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\alpha \beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8eaf30ff3ec1f116c0cacac01122f01243072fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.335ex; height:2.343ex;" alt="{\displaystyle g_{\alpha \beta }}"></span>). Different choices of transition functions may result in different vector bundles which are non-trivial after gluing is complete.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mobius_transition_functions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Mobius_transition_functions.png/300px-Mobius_transition_functions.png" decoding="async" width="300" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Mobius_transition_functions.png/450px-Mobius_transition_functions.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Mobius_transition_functions.png/600px-Mobius_transition_functions.png 2x" data-file-width="3375" data-file-height="1098" /></a><figcaption>The <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> can be constructed by a non-trivial gluing of two trivial bundles on open <a href="/wiki/Subset" title="Subset">subsets</a> <i>U</i> and <i>V</i> of the <a href="/wiki/1-sphere" class="mw-redirect" title="1-sphere">circle <i>S<sup>1</sup></i></a>. When glued trivially (with <i>g<sub>UV</sub>=1</i>) one obtains the trivial bundle, but with the non-trivial gluing of <i>g<sub>UV</sub>=1</i> on one overlap and <i>g<sub>UV</sub>=-1</i> on the second overlap, one obtains the non-trivial bundle <i>E</i>, the Möbius strip. This can be visualised as a "twisting" of one of the local <a href="/wiki/Chart_(topology)" class="mw-redirect" title="Chart (topology)">charts</a>.</figcaption></figure> <p>Given a vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e385963fc9a729e22866cd3c05d5002c361bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.37ex; height:2.176ex;" alt="{\displaystyle E\to X}"></span> of rank <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>, and a pair of neighborhoods <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> over which the bundle trivializes via </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>:</mo> <mi>U</mi> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-REL"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mo>≅</mo> </mpadded> </mover> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>:</mo> <mi>V</mi> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-REL"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mo>≅</mo> </mpadded> </mover> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d6925f8dbabbb8fa8d3b0c26d951c4c61aaf40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-top: -0.339ex; width:23.994ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}}"></span></dd></dl> <p>the <a href="/wiki/Composite_Function" class="mw-redirect" title="Composite Function">composite function</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>∘</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>:</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo>∩</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo>∩</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4194d4ebcfe431f84d3377d9705429b467c63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.85ex; height:3.343ex;" alt="{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}}"></span></dd></dl> <p>is well-defined on the overlap, and satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>∘</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeaf146c9756b97ddb6867046214233491a7127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.768ex; height:3.343ex;" alt="{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)}"></span></dd></dl> <p>for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}"></span>-valued function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>V</mi> </mrow> </msub> <mo>:</mo> <mi>U</mi> <mo>∩</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb009b693db02865b497fc99a228e5a832ed1e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.611ex; height:2.843ex;" alt="{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (k).}"></span></dd></dl> <p>These are called the <b><a href="/wiki/Transition_map" class="mw-redirect" title="Transition map">transition functions</a></b> (or the <b>coordinate transformations</b>) of the vector bundle. </p><p>The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of transition functions forms a <a href="/wiki/%C4%8Cech_cocycle" class="mw-redirect" title="Čech cocycle">Čech cocycle</a> in the sense that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo>,</mo> <mspace width="1em" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mi>W</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9354e02750aa109d0a9476cdfd75636134050902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.831ex; height:2.843ex;" alt="{\displaystyle g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}"></span></dd></dl> <p>for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V,W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U,V,W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4d31f98b67bd24af02cbb34ade648ad59dd201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.073ex; height:2.509ex;" alt="{\displaystyle U,V,W}"></span> over which the bundle trivializes satisfying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap V\cap W\neq \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∩</mo> <mi>V</mi> <mo>∩</mo> <mi>W</mi> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cap V\cap W\neq \emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcf743f4101ee66a1bbd8a44e762e3d95d52d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.431ex; height:2.843ex;" alt="{\displaystyle U\cap V\cap W\neq \emptyset }"></span>. Thus the data <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>π</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d522e4b89a83e991a5f3c4d3d9068b0488280ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.766ex; height:3.176ex;" alt="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}"></span> defines a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a>; the additional data of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{UV}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{UV}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb934498b382f3276942bdd431e1d435a9cdb76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.866ex; height:2.009ex;" alt="{\displaystyle g_{UV}}"></span> specifies a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}"></span> structure group in which the <a href="/wiki/Group_action" title="Group action">action</a> on the fiber is the standard action of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}"></span>. </p><p>Conversely, given a fiber bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>π</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d522e4b89a83e991a5f3c4d3d9068b0488280ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.766ex; height:3.176ex;" alt="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}"></span> with a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}"></span> cocycle acting in the standard way on the fiber <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{k}}"> <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>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}"></span>, there is <a href="/wiki/Associated_bundle" title="Associated bundle">associated</a> a vector bundle. This is an example of the <a href="/wiki/Fibre_bundle_construction_theorem" class="mw-redirect" title="Fibre bundle construction theorem">fibre bundle construction theorem</a> for vector bundles, and can be taken as an alternative definition of a vector bundle. </p> <div class="mw-heading mw-heading3"><h3 id="Subbundles">Subbundles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=3" title="Edit section: Subbundles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Subbundle" title="Subbundle">Subbundle</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Subbundle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Subbundle.png/300px-Subbundle.png" decoding="async" width="300" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Subbundle.png/450px-Subbundle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Subbundle.png/600px-Subbundle.png 2x" data-file-width="2132" data-file-height="1284" /></a><figcaption>A line subbundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> of a trivial rank 2 vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> over a one-dimensional manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>.</figcaption></figure> <p>One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf8de3c25d0905f50abaf4f9374daa0a4bd796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}"></span> over a topological space, a subbundle is simply a <a href="/wiki/Linear_subspace" title="Linear subspace">subspace</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc790c91db34007686074e392dfed5bb12f482e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:2.176ex;" alt="{\displaystyle F\subset E}"></span> for which the <a href="/wiki/Restriction_of_a_map" class="mw-redirect" title="Restriction of a map">restriction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.\pi \right|_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mi>π</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.\pi \right|_{F}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d446dfad36ee9b74b7494280da783f9dfb2791b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.442ex; height:2.843ex;" alt="{\displaystyle \left.\pi \right|_{F}}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.\pi \right|_{F}:F\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mi>π</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>:</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.\pi \right|_{F}:F\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04af20074643e5c0c4cad037cf69a20f8c76de72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.714ex; height:2.843ex;" alt="{\displaystyle \left.\pi \right|_{F}:F\to X}"></span> the structure of a vector bundle also. In this case the fibre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}\subset E_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊂</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{x}\subset E_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00bce5646d7964a1281ed33a9c75a413183a278f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.653ex; height:2.509ex;" alt="{\displaystyle F_{x}\subset E_{x}}"></span> is a vector subspace for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>. </p><p>A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the <a href="/wiki/M%C3%B6bius_band" class="mw-redirect" title="Möbius band">Möbius band</a>, a non-trivial <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a> over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle. </p> <div class="mw-heading mw-heading2"><h2 id="Vector_bundle_morphisms">Vector bundle morphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=4" title="Edit section: Vector bundle morphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b><a href="/wiki/Morphism" title="Morphism">morphism</a></b> from the vector bundle <span class="texhtml mvar" style="font-style:italic;">π</span><sub>1</sub>: <i>E</i><sub>1</sub> → <i>X</i><sub>1</sub> to the vector bundle <span class="texhtml mvar" style="font-style:italic;">π</span><sub>2</sub>: <i>E</i><sub>2</sub> → <i>X</i><sub>2</sub> is given by a pair of continuous maps <i>f</i>: <i>E</i><sub>1</sub> → <i>E</i><sub>2</sub> and <i>g</i>: <i>X</i><sub>1</sub> → <i>X</i><sub>2</sub> such that </p> <dl><dd><i>g</i> ∘ <span class="texhtml mvar" style="font-style:italic;">π</span><sub>1</sub> = <span class="texhtml mvar" style="font-style:italic;">π</span><sub>2</sub> ∘ <i>f</i> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:BundleMorphism-01.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d3/BundleMorphism-01.png" decoding="async" width="137" height="136" class="mw-file-element" data-file-width="137" data-file-height="136" /></a></span></dd></dl></dd> <dd>for every <i>x</i> in <i>X</i><sub>1</sub>, the map <span class="texhtml mvar" style="font-style:italic;">π</span><sub>1</sub><sup>−1</sup>({<i>x</i>}) → <span class="texhtml mvar" style="font-style:italic;">π</span><sub>2</sub><sup>−1</sup>({<i>g</i>(<i>x</i>)}) <a href="/wiki/Induced_map" class="mw-redirect" title="Induced map">induced</a> by <i>f</i> is a <a href="/wiki/Linear_map" title="Linear map">linear map</a> between vector spaces.</dd></dl> <p>Note that <i>g</i> is determined by <i>f</i> (because <span class="texhtml mvar" style="font-style:italic;">π</span><sub>1</sub> is surjective), and <i>f</i> is then said to <b>cover <i>g</i></b>. </p><p>The class of all vector bundles together with bundle morphisms forms a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a <a href="/wiki/Bundle_map" title="Bundle map">bundle map</a> between <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundles</a>, and are sometimes called <b>(vector) bundle homomorphisms</b>. </p><p>A bundle homomorphism from <i>E</i><sub>1</sub> to <i>E</i><sub>2</sub> with an <a href="/wiki/Inverse_element" title="Inverse element">inverse</a> which is also a bundle homomorphism (from <i>E</i><sub>2</sub> to <i>E</i><sub>1</sub>) is called a <b>(vector) bundle isomorphism</b>, and then <i>E</i><sub>1</sub> and <i>E</i><sub>2</sub> are said to be <b>isomorphic</b> vector bundles. An isomorphism of a (rank <i>k</i>) vector bundle <i>E</i> over <i>X</i> with the trivial bundle (of rank <i>k</i> over <i>X</i>) is called a <b>trivialization</b> of <i>E</i>, and <i>E</i> is then said to be <b>trivial</b> (or <b>trivializable</b>). The definition of a vector bundle shows that any vector bundle is <b>locally trivial</b>. </p><p>We can also consider the category of all vector bundles over a fixed base space <i>X</i>. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the <a href="/wiki/Identity_function" title="Identity function">identity map</a> on <i>X</i>. That is, bundle morphisms for which the following diagram <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutes</a>: </p> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:BundleMorphism-02.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2e/BundleMorphism-02.png" decoding="async" width="113" height="120" class="mw-file-element" data-file-width="113" data-file-height="120" /></a></span></dd></dl> <p>(Note that this category is <i>not</i> <a href="/wiki/Abelian_category" title="Abelian category">abelian</a>; the <a href="/wiki/Kernel_(category_theory)" title="Kernel (category theory)">kernel</a> of a morphism of vector bundles is in general not a vector bundle in any natural way.) </p><p>A vector bundle morphism between vector bundles <span class="texhtml mvar" style="font-style:italic;">π</span><sub>1</sub>: <i>E</i><sub>1</sub> → <i>X</i><sub>1</sub> and <span class="texhtml mvar" style="font-style:italic;">π</span><sub>2</sub>: <i>E</i><sub>2</sub> → <i>X</i><sub>2</sub> covering a map <i>g</i> from <i>X</i><sub>1</sub> to <i>X</i><sub>2</sub> can also be viewed as a vector bundle morphism over <i>X</i><sub>1</sub> from <i>E</i><sub>1</sub> to the <a href="/wiki/Pullback_bundle" title="Pullback bundle">pullback bundle</a> <i>g</i>*<i>E</i><sub>2</sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Sections_and_locally_free_sheaves">Sections and locally free sheaves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=5" title="Edit section: Sections and locally free sheaves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vector_bundle_with_section.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Vector_bundle_with_section.png/300px-Vector_bundle_with_section.png" decoding="async" width="300" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Vector_bundle_with_section.png/450px-Vector_bundle_with_section.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Vector_bundle_with_section.png/600px-Vector_bundle_with_section.png 2x" data-file-width="1899" data-file-height="1595" /></a><figcaption>A vector bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> over a base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> with section <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Surface_normals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Surface_normals.svg/300px-Surface_normals.svg.png" decoding="async" width="300" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Surface_normals.svg/450px-Surface_normals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Surface_normals.svg/600px-Surface_normals.svg.png 2x" data-file-width="575" data-file-height="449" /></a><figcaption>The map associating a <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal</a> to each point on a <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a> can be thought of as a section. The surface is the space <i>X</i>, and at each point <i>x</i> there is a vector in the vector space attached at <i>x</i>.</figcaption></figure> <p>Given a vector bundle <span class="texhtml mvar" style="font-style:italic;">π</span>: <i>E</i> → <i>X</i> and an open subset <i>U</i> of <i>X</i>, we can consider <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)"><b>sections</b></a> of <span class="texhtml mvar" style="font-style:italic;">π</span> on <i>U</i>, i.e. continuous functions <i>s</i>: <i>U</i> → <i>E</i> where the composite <span class="texhtml mvar" style="font-style:italic;">π</span> ∘ <i>s</i> is such that <span class="nowrap">(<span class="texhtml mvar" style="font-style:italic;">π</span> ∘ <i>s</i>)(<i>u</i>) = <i>u</i></span> for all <i>u</i> in <i>U</i>. Essentially, a section assigns to every point of <i>U</i> a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but <a href="/wiki/Vector_field" title="Vector field">vector fields</a> on that manifold. </p><p>Let <i>F</i>(<i>U</i>) be the set of all sections on <i>U</i>. <i>F</i>(<i>U</i>) always contains at least one element, namely the <b>zero section</b>: the function <i>s</i> that maps every element <i>x</i> of <i>U</i> to the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero element of the vector space</a> <span class="texhtml mvar" style="font-style:italic;">π</span><sup>−1</sup>({<i>x</i>}). With the <a href="/wiki/Pointwise" title="Pointwise">pointwise</a> addition and <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> of sections, <i>F</i>(<i>U</i>) becomes itself a real vector space. The collection of these vector spaces is a <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of vector spaces on <i>X</i>. </p><p>If <i>s</i> is an element of <i>F</i>(<i>U</i>) and α: <i>U</i> → <b>R</b> is a continuous map, then α<i>s</i> (pointwise scalar multiplication) is in <i>F</i>(<i>U</i>). We see that <i>F</i>(<i>U</i>) is a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of continuous <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued functions</a> on <i>U</i>. Furthermore, if O<sub><i>X</i></sub> denotes the structure sheaf of continuous real-valued functions on <i>X</i>, then <i>F</i> becomes a sheaf of O<sub><i>X</i></sub>-modules. </p><p>Not every sheaf of O<sub><i>X</i></sub>-modules arises in this fashion from a vector bundle: only the <a href="/wiki/Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free</a> ones do. (The reason: locally we are looking for sections of a projection <i>U</i> × <b>R</b><sup><i>k</i></sup> → <i>U</i>; these are precisely the continuous functions <i>U</i> → <b>R</b><sup><i>k</i></sup>, and such a function is a <i>k</i>-<a href="/wiki/Tuple" title="Tuple">tuple</a> of continuous functions <i>U</i> → <b>R</b>.) </p><p>Even more: the category of real vector bundles on <i>X</i> is <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to the category of locally free and <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated</a> sheaves of O<sub><i>X</i></sub>-modules. </p><p>So we can think of the category of real vector bundles on <i>X</i> as sitting inside the category of <a href="/wiki/Sheaf_of_modules" title="Sheaf of modules">sheaves of O<sub><i>X</i></sub>-modules</a>; this latter category is abelian, so this is where we can compute <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernels</a> and <a href="/wiki/Cokernel" title="Cokernel">cokernels</a> of morphisms of vector bundles. </p><p>A rank <i>n</i> vector bundle is trivial <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it has <i>n</i> <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> global sections. </p> <div class="mw-heading mw-heading2"><h2 id="Operations_on_vector_bundles">Operations on vector bundles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=6" title="Edit section: Operations on vector bundles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> on vector spaces can be extended to vector bundles by performing the vector space operation <i>fiberwise</i>. </p><p>For example, if <i>E</i> is a vector bundle over <i>X</i>, then there is a bundle <i>E*</i> over <i>X</i>, called the <b><a href="/wiki/Dual_bundle" title="Dual bundle">dual bundle</a></b>, whose fiber at <i>x</i> ∈ <i>X</i> is the <a href="/wiki/Dual_vector_space" class="mw-redirect" title="Dual vector space">dual vector space</a> (<i>E<sub>x</sub></i>)*. Formally <i>E*</i> can be defined as the set of pairs (<i>x</i>, φ), where <i>x</i> ∈ <i>X</i> and φ ∈ (<i>E</i><sub><i>x</i></sub>)*. The dual bundle is locally trivial because the <a href="/wiki/Transpose" title="Transpose">dual space</a> of the inverse of a local trivialization of <i>E</i> is a local trivialization of <i>E*</i>: the key point here is that the operation of taking the dual vector space is <a href="/wiki/Functorial" class="mw-redirect" title="Functorial">functorial</a>. </p><p>There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles <i>E</i>, <i>F</i> on <i>X</i> (over the given field). A few examples follow. </p> <ul><li>The <b>Whitney sum</b> (named for <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a>) or <b>direct sum bundle</b> of <i>E</i> and <i>F</i> is a vector bundle <i>E</i> ⊕ <i>F</i> over <i>X</i> whose fiber over <i>x</i> is the <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> <i>E<sub>x</sub></i> ⊕ <i>F<sub>x</sub></i> of the vector spaces <i>E<sub>x</sub></i> and <i>F<sub>x</sub></i>.</li> <li>The <b><a href="/wiki/Tensor_product_bundle" title="Tensor product bundle">tensor product bundle</a></b> <i>E</i> ⊗ <i>F</i> is defined in a similar way, using fiberwise <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of vector spaces.</li> <li>The <b>Hom-bundle</b> Hom(<i>E</i>, <i>F</i>) is a vector bundle whose fiber at <i>x</i> is the space of linear maps from <i>E<sub>x</sub></i> to <i>F<sub>x</sub></i> (which is often denoted Hom(<i>E</i><sub><i>x</i></sub>, <i>F<sub>x</sub></i>) or <i>L</i>(<i>E</i><sub><i>x</i></sub>, <i>F</i><sub><i>x</i></sub>)). The Hom-bundle is so-called (and useful) because there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> between vector bundle homomorphisms from <i>E</i> to <i>F</i> over <i>X</i> and sections of Hom(<i>E</i>, <i>F</i>) over <i>X</i>.</li> <li>Building on the previous example, given a section <i>s</i> of an <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> bundle Hom(<i>E</i>, <i>E</i>) and a function <i>f</i>: <i>X</i> → <b>R</b>, one can construct an <b>eigenbundle</b> by taking the fiber over a point <i>x</i> ∈ <i>X</i> to be the <i>f</i>(<i>x</i>)-<a href="/wiki/Eigenvector#Eigenspace_and_spectrum" class="mw-redirect" title="Eigenvector">eigenspace</a> of the linear map <i>s</i>(<i>x</i>): <i>E</i><sub><i>x</i></sub> → <i>E</i><sub><i>x</i></sub>. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of <i>s</i> being the zero section and <i>f</i> having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in <i>E</i>, while everywhere else the fiber is the trivial 0-dimensional vector space.</li> <li>The <a href="/wiki/Dual_bundle" title="Dual bundle">dual vector bundle</a> <i>E*</i> is the Hom bundle Hom(<i>E</i>, <b>R</b> × <i>X</i>) of bundle homomorphisms of <i>E</i> and the trivial bundle <b>R</b> × <i>X</i>. There is a canonical vector bundle isomorphism Hom(<i>E</i>, <i>F</i>) = <i>E*</i> ⊗ <i>F</i>.</li></ul> <p>Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the <a href="/wiki/Category_of_vector_spaces" class="mw-redirect" title="Category of vector spaces">category of vector spaces</a> can also be performed on the category of vector bundles in a <a href="/wiki/Functor" title="Functor">functorial</a> manner. This is made precise in the language of <a href="/wiki/Smooth_functor" title="Smooth functor">smooth functors</a>. An operation of a different nature is the <b><a href="/wiki/Pullback_bundle" title="Pullback bundle">pullback bundle</a></b> construction. Given a vector bundle <i>E</i> → <i>Y</i> and a continuous map <i>f</i>: <i>X</i> → <i>Y</i> one can "pull back" <i>E</i> to a vector bundle <i>f*E</i> over <i>X</i>. The fiber over a point <i>x</i> ∈ <i>X</i> is essentially just the fiber over <i>f</i>(<i>x</i>) ∈ <i>Y</i>. Hence, Whitney summing <i>E</i> ⊕ <i>F</i> can be defined as the pullback bundle of the diagonal map from <i>X</i> to <i>X</i> × <i>X</i> where the bundle over <i>X</i> × <i>X</i> is <i>E</i> × <i>F</i>. </p><p><b>Remark</b>: Let <i>X</i> be a <a href="/wiki/Compact_space" title="Compact space">compact space</a>. Any vector bundle <i>E</i> over <i>X</i> is a direct summand of a trivial bundle; i.e., there exists a bundle <i>E</i><span class="nowrap" style="padding-left:0.1em;">'</span> such that <i>E</i> ⊕ <i>E</i><span class="nowrap" style="padding-left:0.1em;">'</span> is trivial. This fails if <i>X</i> is not compact: for example, the <a href="/wiki/Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">tautological line bundle</a> over the infinite real projective space does not have this property.<sup id="cite_ref-FOOTNOTEHatcher2003Example_3.6_1-0" class="reference"><a href="#cite_note-FOOTNOTEHatcher2003Example_3.6-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Additional_structures_and_generalizations">Additional structures and generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=7" title="Edit section: Additional structures and generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vector bundles are often given more structure. For instance, vector bundles may be equipped with a <a href="/wiki/Metric_(vector_bundle)" class="mw-redirect" title="Metric (vector bundle)">vector bundle metric</a>. Usually this metric is required to be <a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">positive definite</a>, in which case each fibre of <i>E</i> becomes a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. A vector bundle with a <a href="/wiki/Linear_complex_structure" title="Linear complex structure">complex structure</a> corresponds to a <a href="/wiki/Complex_vector_bundle" title="Complex vector bundle">complex vector bundle</a>, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be <a href="/wiki/Complex_linear_structure" class="mw-redirect" title="Complex linear structure">complex-linear</a> in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting <a href="/wiki/Reduction_of_the_structure_group_of_a_bundle" class="mw-redirect" title="Reduction of the structure group of a bundle">reduction of the structure group of a bundle</a>. Vector bundles over more general <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological fields</a> may also be used. </p><p>If instead of a finite-dimensional vector space, if the fiber <i>F</i> is taken to be a <a href="/wiki/Banach_space" title="Banach space">Banach space</a> then a <b><a href="/wiki/Banach_bundle" title="Banach bundle">Banach bundle</a></b> is obtained.<sup id="cite_ref-FOOTNOTELang1995_2-0" class="reference"><a href="#cite_note-FOOTNOTELang1995-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>V</mi> </mrow> </msub> <mo>:</mo> <mi>U</mi> <mo>∩</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ee22af2ae10ffc1714836c5042461aa3f68b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.493ex; height:2.843ex;" alt="{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (F)}"></span></dd></dl> <p>are continuous mappings of <a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifolds</a>. In the corresponding theory for C<sup><i>p</i></sup> bundles, all mappings are required to be C<sup><i>p</i></sup>. </p><p>Vector bundles are special <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundles</a>, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example <a href="/wiki/Sphere_bundle" title="Sphere bundle">sphere bundles</a> are fibered by spheres. </p> <div class="mw-heading mw-heading2"><h2 id="Smooth_vector_bundles">Smooth vector bundles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=8" title="Edit section: Smooth vector bundles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Smooth_vs_non-smooth_vector_bundle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Smooth_vs_non-smooth_vector_bundle.png/300px-Smooth_vs_non-smooth_vector_bundle.png" decoding="async" width="300" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Smooth_vs_non-smooth_vector_bundle.png/450px-Smooth_vs_non-smooth_vector_bundle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Smooth_vs_non-smooth_vector_bundle.png/600px-Smooth_vs_non-smooth_vector_bundle.png 2x" data-file-width="2439" data-file-height="1651" /></a><figcaption>The regularity of transition functions describing a vector bundle determines the type of the vector bundle. If the continuous transition functions <i>g<sub>UV</sub></i> are used, the resulting vector bundle <i>E</i> is only continuous but not smooth. If the smooth transition functions <i>h<sub>UV</sub></i> are used, then the resulting vector bundle <i>F</i> is a smooth vector bundle.</figcaption></figure> <p>A vector bundle (<i>E</i>, <i>p</i>, <i>M</i>) is <b>smooth</b>, if <i>E</i> and <i>M</i> are <a href="/wiki/Manifold" title="Manifold">smooth manifolds</a>, p: <i>E</i> → <i>M</i> is a smooth map, and the local trivializations are <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>. Depending on the required degree of <a href="/wiki/Smoothness" title="Smoothness">smoothness</a>, there are different corresponding notions of <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable"><i>C<sup>p</sup></i></a> bundles, <a href="/wiki/Infinitely_differentiable" class="mw-redirect" title="Infinitely differentiable">infinitely differentiable</a> <i>C</i><sup>∞</sup>-bundles and <a href="/wiki/Real_analytic" class="mw-redirect" title="Real analytic">real analytic</a> <i>C</i><sup>ω</sup>-bundles. In this section we will concentrate on <a href="/wiki/C-infinity" class="mw-redirect" title="C-infinity"><i>C</i><sup>∞</sup></a>-bundles. The most important example of a <i>C</i><sup>∞</sup>-vector bundle is the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> (<i>TM</i>, <span class="texhtml mvar" style="font-style:italic;">π</span><sub><i>TM</i></sub>, <i>M</i>) of a <i>C</i><sup>∞</sup>-manifold <i>M</i>. </p><p>A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are <i>smooth</i> functions on overlaps of trivializing charts <i>U</i> and <i>V</i>. That is, a vector bundle <i>E</i> is smooth if it admits a covering by trivializing open sets such that for any two such sets <i>U</i> and <i>V</i>, the transition function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{UV}:U\cap V\to \operatorname {GL} (k,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>V</mi> </mrow> </msub> <mo>:</mo> <mi>U</mi> <mo>∩</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{UV}:U\cap V\to \operatorname {GL} (k,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc244558b7b8dd9aaf8f066745b64e74418e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.579ex; height:2.843ex;" alt="{\displaystyle g_{UV}:U\cap V\to \operatorname {GL} (k,\mathbb {R} )}"></span></dd></dl> <p>is a smooth function into the <a href="/wiki/Matrix_group" class="mw-redirect" title="Matrix group">matrix group</a> GL(k,<b>R</b>), which is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. </p><p>Similarly, if the transition functions are: </p> <ul><li><i>C<sup>r</sup></i> then the vector bundle is a <b><i>C<sup>r</sup></i> vector bundle</b>,</li> <li><i>real <a href="/wiki/Analytic_function" title="Analytic function">analytic</a></i> then the vector bundle is a <b>real analytic vector bundle</b> (this requires the matrix group to have a real analytic structure),</li> <li><i>holomorphic</i> then the vector bundle is a <b><a href="/wiki/Holomorphic_vector_bundle" title="Holomorphic vector bundle">holomorphic vector bundle</a></b> (this requires the matrix group to be a <a href="/wiki/Complex_Lie_group" title="Complex Lie group">complex Lie group</a>),</li> <li><i>algebraic functions</i> then the vector bundle is an <b><a href="/wiki/Algebraic_vector_bundle" class="mw-redirect" title="Algebraic vector bundle">algebraic vector bundle</a></b> (this requires the matrix group to be an <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic group</a>).</li></ul> <p>The <i>C</i><sup>∞</sup>-vector bundles (<i>E</i>, <i>p</i>, <i>M</i>) have a very important property not shared by more general <i>C</i><sup>∞</sup>-fibre bundles. Namely, the tangent space <i>T<sub>v</sub></i>(<i>E</i><sub><i>x</i></sub>) at any <i>v</i> ∈ <i>E</i><sub><i>x</i></sub> can be naturally identified with the fibre <i>E</i><sub><i>x</i></sub> itself. This identification is obtained through the <i>vertical lift</i> <i>vl</i><sub><i>v</i></sub>: <i>E<sub>x</sub></i> → <i>T</i><sub><i>v</i></sub>(<i>E</i><sub><i>x</i></sub>), defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {vl} _{v}w[f]:=\left.{\frac {d}{dt}}\right|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>vl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>:=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>t</mi> <mi>w</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {vl} _{v}w[f]:=\left.{\frac {d}{dt}}\right|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd374a28264c91b99a13a8fb0313d29e97537e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.79ex; height:5.843ex;" alt="{\displaystyle \operatorname {vl} _{v}w[f]:=\left.{\frac {d}{dt}}\right|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).}"></span></dd></dl> <p>The vertical lift can also be seen as a natural <i>C</i><sup>∞</sup>-vector bundle isomorphism <i>p*E</i> → <i>VE</i>, where (<i>p*E</i>, <i>p*p</i>, <i>E</i>) is the pull-back bundle of (<i>E</i>, <i>p</i>, <i>M</i>) over <i>E</i> through <i>p</i>: <i>E</i> → <i>M</i>, and <i>VE</i> := Ker(<i>p</i><sub>*</sub>) ⊂ <i>TE</i> is the <i>vertical tangent bundle</i>, a natural vector subbundle of the tangent bundle (<i>TE</i>, <span class="texhtml mvar" style="font-style:italic;">π</span><sub><i>TE</i></sub>, <i>E</i>) of the total space <i>E</i>. </p><p>The total space <i>E</i> of any smooth vector bundle carries a natural vector field <i>V</i><sub><i>v</i></sub> := vl<sub><i>v</i></sub><i>v</i>, known as the <i>canonical vector field</i>. More formally, <i>V</i> is a smooth section of (<i>TE</i>, <span class="texhtml mvar" style="font-style:italic;">π</span><sub><i>TE</i></sub>, <i>E</i>), and it can also be defined as the infinitesimal generator of the <a href="/wiki/Lie_group_action" title="Lie group action">Lie-group action</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,v)\mapsto e^{tv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>v</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,v)\mapsto e^{tv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2270520be288b87abfe11ad21428a258ca82b48d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.131ex; height:3.009ex;" alt="{\displaystyle (t,v)\mapsto e^{tv}}"></span> given by the fibrewise scalar multiplication. The canonical vector field <i>V</i> characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when <i>X</i> is a smooth vector field on a smooth manifold <i>M</i> and <i>x</i> ∈ <i>M</i> such that <i>X</i><sub><i>x</i></sub> = 0, the linear mapping </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}(X):T_{x}M\to T_{x}M;\quad C_{x}(X)Y=(\nabla _{Y}X)_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>M</mi> <mo stretchy="false">→</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>M</mi> <mo>;</mo> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>Y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}(X):T_{x}M\to T_{x}M;\quad C_{x}(X)Y=(\nabla _{Y}X)_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9244f44cdd9afd7ec85b328a8fe77e58367dd987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.355ex; height:2.843ex;" alt="{\displaystyle C_{x}(X):T_{x}M\to T_{x}M;\quad C_{x}(X)Y=(\nabla _{Y}X)_{x}}"></span></dd></dl> <p>does not depend on the choice of the linear <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a> ∇ on <i>M</i>. The canonical vector field <i>V</i> on <i>E</i> satisfies the axioms </p> <ol><li>The flow (<i>t</i>, <i>v</i>) → Φ<sup><i>t</i></sup><sub><i>V</i></sub>(<i>v</i>) of <i>V</i> is globally defined.</li> <li>For each <i>v</i> ∈ <i>V</i> there is a unique lim<sub>t→∞</sub> Φ<sup><i>t</i></sup><sub><i>V</i></sub>(<i>v</i>) ∈ <i>V</i>.</li> <li><i>C</i><sub>v</sub>(<i>V</i>)∘<i>C</i><sub>v</sub>(<i>V</i>) = <i>C</i><sub>v</sub>(<i>V</i>) whenever <i>V</i><sub><i>v</i></sub> = 0.</li> <li>The <a href="/wiki/Zero_set" class="mw-redirect" title="Zero set">zero set</a> of <i>V</i> is a smooth <a href="/wiki/Submanifold" title="Submanifold">submanifold</a> of <i>E</i> whose <a href="/wiki/Codimension" title="Codimension">codimension</a> is equal to the rank of <i>C</i><sub>v</sub>(<i>V</i>).</li></ol> <p>Conversely, if <i>E</i> is any smooth manifold and <i>V</i> is a smooth vector field on <i>E</i> satisfying 1–4, then there is a unique vector bundle structure on <i>E</i> whose canonical vector field is <i>V</i>. </p><p>For any smooth vector bundle (<i>E</i>, <i>p</i>, <i>M</i>) the total space <i>TE</i> of its tangent bundle (<i>TE</i>, <span class="texhtml mvar" style="font-style:italic;">π</span><sub><i>TE</i></sub>, <i>E</i>) has a natural <a href="/wiki/Secondary_vector_bundle_structure" title="Secondary vector bundle structure">secondary vector bundle structure</a> (<i>TE</i>, <i>p</i><sub>*</sub>, <i>TM</i>), where <i>p</i><sub>*</sub> is the <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">push-forward</a> of the canonical projection <i>p</i>: <i>E</i> → <i>M</i>. The vector bundle operations in this secondary vector bundle structure are the push-forwards +<sub>*</sub>: <i>T</i>(<i>E</i> × <i>E</i>) → <i>TE</i> and λ<sub>*</sub>: <i>TE</i> → <i>TE</i> of the original addition +: <i>E</i> × <i>E</i> → <i>E</i> and scalar multiplication λ: <i>E</i> → <i>E</i>. </p> <div class="mw-heading mw-heading2"><h2 id="K-theory">K-theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=9" title="Edit section: K-theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The K-theory group, <span class="texhtml"><i>K</i>(<i>X</i>)</span>, of a compact <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> topological space is defined as the <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> generated by <a href="/wiki/Isomorphism_class" class="mw-redirect" title="Isomorphism class">isomorphism classes</a> <span class="texhtml">[<i>E</i>]</span> of <a href="/wiki/Complex_vector_bundle" title="Complex vector bundle">complex vector bundles</a> modulo the <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a> that, whenever we have an <a href="/wiki/Exact_sequence" title="Exact sequence">exact sequence</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to A\to B\to C\to 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to A\to B\to C\to 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ec13ea2c5cbd567e406ce3f794e092bfe5dd3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.702ex; height:2.509ex;" alt="{\displaystyle 0\to A\to B\to C\to 0,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [B]=[A]+[C]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>C</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [B]=[A]+[C]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581e4086c717d2c2649e5cd20354e243be7c7bfa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.093ex; height:2.843ex;" alt="{\displaystyle [B]=[A]+[C]}"></span> in <a href="/wiki/Topological_K-theory" title="Topological K-theory">topological K-theory</a>. <a href="/wiki/KO-theory" class="mw-redirect" title="KO-theory">KO-theory</a> is a version of this construction which considers real vector bundles. K-theory with <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact supports</a> can also be defined, as well as higher K-theory groups. </p><p>The famous <a href="/wiki/Bott_periodicity" class="mw-redirect" title="Bott periodicity">periodicity theorem</a> of <a href="/wiki/Raoul_Bott" title="Raoul Bott">Raoul Bott</a> asserts that the K-theory of any space <span class="texhtml"><i>X</i></span> is isomorphic to that of the <span class="texhtml"><i>S</i><sup>2</sup><i>X</i></span>, the double suspension of <span class="texhtml"><i>X</i></span>. </p><p>In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, one considers the K-theory groups consisting of <a href="/wiki/Coherent_sheaf" title="Coherent sheaf">coherent sheaves</a> on a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> <span class="texhtml"><i>X</i></span>, as well as the K-theory groups of vector bundles on the scheme with the above <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>. The two constructs are the same provided that the underlying scheme is <a href="/wiki/Smooth_morphism" title="Smooth morphism">smooth</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="General_notions">General notions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=11" title="Edit section: General notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Grassmannian" title="Grassmannian">Grassmannian</a>: <a href="/wiki/Classifying_space" title="Classifying space">classifying spaces</a> for vector bundle, among which <a href="/wiki/Projective_space" title="Projective space">projective spaces</a> for <a href="/wiki/Line_bundle" title="Line bundle">line bundles</a></li> <li><a href="/wiki/Characteristic_class" title="Characteristic class">Characteristic class</a></li> <li><a href="/wiki/Splitting_principle" title="Splitting principle">Splitting principle</a></li> <li><a href="/wiki/Stable_bundle" class="mw-redirect" title="Stable bundle">Stable bundle</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Topology_and_differential_geometry">Topology and differential geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=12" title="Edit section: Topology and differential geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Connection</a>: the notion needed to differentiate sections of vector bundles.</li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a>: the general study of connections on vector bundles and principal bundles and their relations to physics.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Algebraic_and_analytic_geometry">Algebraic and analytic geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=13" title="Edit section: Algebraic and analytic geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_vector_bundle" class="mw-redirect" title="Algebraic vector bundle">Algebraic vector bundle</a></li> <li><a href="/wiki/Picard_group" title="Picard group">Picard group</a></li> <li><a href="/wiki/Holomorphic_vector_bundle" title="Holomorphic vector bundle">Holomorphic vector bundle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEHatcher2003Example_3.6-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHatcher2003Example_3.6_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHatcher2003">Hatcher 2003</a>, Example 3.6.</span> </li> <li id="cite_note-FOOTNOTELang1995-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELang1995_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLang1995">Lang 1995</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=15" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 refbegin-columns references-column-width" style="column-width: 35em"> <ul><li><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="CITEREFAbrahamMarsden1978" class="citation cs2"><a href="/wiki/Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Abraham, Ralph H.</a>; <a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden, Jerrold E.</a> (1978), <i>Foundations of mechanics</i>, London: Benjamin-Cummings, see section 1.5, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-0102-1" title="Special:BookSources/978-0-8053-0102-1"><bdi>978-0-8053-0102-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+mechanics&rft.place=London&rft.pages=see+section+1.5&rft.pub=Benjamin-Cummings&rft.date=1978&rft.isbn=978-0-8053-0102-1&rft.aulast=Abraham&rft.aufirst=Ralph+H.&rft.au=Marsden%2C+Jerrold+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHatcher2003" class="citation cs2"><a href="/wiki/Allen_Hatcher" title="Allen Hatcher">Hatcher, Allen</a> (2003), <a rel="nofollow" class="external text" href="https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html"><i>Vector Bundles & K-Theory</i></a> (2.0 ed.)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Bundles+%26+K-Theory&rft.edition=2.0&rft.date=2003&rft.aulast=Hatcher&rft.aufirst=Allen&rft_id=https%3A%2F%2Fpi.math.cornell.edu%2F~hatcher%2FVBKT%2FVBpage.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJost2002" class="citation cs2">Jost, Jürgen (2002), <i>Riemannian Geometry and Geometric Analysis</i> (3rd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42627-1" title="Special:BookSources/978-3-540-42627-1"><bdi>978-3-540-42627-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemannian+Geometry+and+Geometric+Analysis&rft.place=Berlin%2C+New+York&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-3-540-42627-1&rft.aulast=Jost&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span>, see section 1.5.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1995" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1995), <i>Differential and Riemannian manifolds</i>, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94338-1" title="Special:BookSources/978-0-387-94338-1"><bdi>978-0-387-94338-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+and+Riemannian+manifolds&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-0-387-94338-1&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2009" class="citation cs2">Lee, Jeffrey M. (2009), <a rel="nofollow" class="external text" href="http://www.ams.org/bookstore-getitem/item=gsm-107"><i>Manifolds and Differential Geometry</i></a>, <a href="/wiki/Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 107, Providence: American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4815-9" title="Special:BookSources/978-0-8218-4815-9"><bdi>978-0-8218-4815-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manifolds+and+Differential+Geometry&rft.place=Providence&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4815-9&rft.aulast=Lee&rft.aufirst=Jeffrey+M.&rft_id=http%3A%2F%2Fwww.ams.org%2Fbookstore-getitem%2Fitem%3Dgsm-107&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2003" class="citation cs2">Lee, John M. (2003), <a rel="nofollow" class="external text" href="http://www.math.washington.edu/~lee/Books/smooth.html"><i>Introduction to Smooth Manifolds</i></a>, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95448-1" title="Special:BookSources/0-387-95448-1"><bdi>0-387-95448-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Smooth+Manifolds&rft.place=New+York&rft.pub=Springer&rft.date=2003&rft.isbn=0-387-95448-1&rft.aulast=Lee&rft.aufirst=John+M.&rft_id=http%3A%2F%2Fwww.math.washington.edu%2F~lee%2FBooks%2Fsmooth.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span> see Ch.5</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRubei2014" class="citation cs2">Rubei, Elena (2014), <i>Algebraic Geometry, a concise dictionary</i>, Berlin/Boston: Walter De Gruyter, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-031622-3" title="Special:BookSources/978-3-11-031622-3"><bdi>978-3-11-031622-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry%2C+a+concise+dictionary&rft.place=Berlin%2FBoston&rft.pub=Walter+De+Gruyter&rft.date=2014&rft.isbn=978-3-11-031622-3&rft.aulast=Rubei&rft.aufirst=Elena&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vector_bundle&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Vector_bundle">"Vector bundle"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Vector+bundle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DVector_bundle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+bundle" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/7836">Why is it useful to study vector bundles ?</a> on <a href="/wiki/MathOverflow" title="MathOverflow">MathOverflow</a></li> <li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/16240">Why is it useful to classify the vector bundles of a space ?</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output 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autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a class="mw-selflink selflink">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Vector_bundle&oldid=1259958684">https://en.wikipedia.org/w/index.php?title=Vector_bundle&oldid=1259958684</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Vector_bundles" title="Category:Vector bundles">Vector bundles</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks 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