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<span class="vector-toc-numb">2</span> <span>De Rham cohomology computed</span> </div> </a> <button aria-controls="toc-De_Rham_cohomology_computed-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 De Rham cohomology computed subsection</span> </button> <ul id="toc-De_Rham_cohomology_computed-sublist" class="vector-toc-list"> <li id="toc-The_n-sphere" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_n-sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>The <span><i>n</i></span>-sphere</span> </div> </a> <ul id="toc-The_n-sphere-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_n-torus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_n-torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>The <span><i>n</i></span>-torus</span> </div> </a> <ul id="toc-The_n-torus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Punctured_Euclidean_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Punctured_Euclidean_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Punctured Euclidean space</span> </div> </a> <ul id="toc-Punctured_Euclidean_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Möbius_strip" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Möbius_strip"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>The Möbius strip</span> </div> </a> <ul id="toc-The_Möbius_strip-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-De_Rham_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#De_Rham_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>De Rham theorem</span> </div> </a> <ul id="toc-De_Rham_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sheaf-theoretic_de_Rham_isomorphism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sheaf-theoretic_de_Rham_isomorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sheaf-theoretic de Rham isomorphism</span> </div> </a> <button aria-controls="toc-Sheaf-theoretic_de_Rham_isomorphism-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 Sheaf-theoretic de Rham isomorphism subsection</span> </button> <ul id="toc-Sheaf-theoretic_de_Rham_isomorphism-sublist" class="vector-toc-list"> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_ideas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_ideas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Related ideas</span> </div> </a> <button aria-controls="toc-Related_ideas-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 Related ideas subsection</span> </button> <ul id="toc-Related_ideas-sublist" class="vector-toc-list"> <li id="toc-Harmonic_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Harmonic forms</span> </div> </a> <ul id="toc-Harmonic_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hodge_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hodge_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Hodge decomposition</span> </div> </a> <ul id="toc-Hodge_decomposition-sublist" class="vector-toc-list"> </ul> </li> </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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">De Rham cohomology</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 13 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-13" 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">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cohomologia_de_De_Rham" title="Cohomologia de De Rham – Catalan" lang="ca" hreflang="ca" data-title="Cohomologia de De Rham" 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/De-Rham-Kohomologie" title="De-Rham-Kohomologie – German" lang="de" hreflang="de" data-title="De-Rham-Kohomologie" 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/Cohomolog%C3%ADa_de_De_Rham" title="Cohomología de De Rham – Spanish" lang="es" hreflang="es" data-title="Cohomología de De Rham" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cohomologie_de_De_Rham" title="Cohomologie de De Rham – French" lang="fr" hreflang="fr" data-title="Cohomologie de De Rham" 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%93%9C%EB%9E%8C_%EC%BD%94%ED%98%B8%EB%AA%B0%EB%A1%9C%EC%A7%80" title="드람 코호몰로지 – Korean" lang="ko" hreflang="ko" data-title="드람 코호몰로지" 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/Coomologia_di_De_Rham" title="Coomologia di De Rham – Italian" lang="it" hreflang="it" data-title="Coomologia di De Rham" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/De_Rham-cohomologie" title="De Rham-cohomologie – Dutch" lang="nl" hreflang="nl" data-title="De Rham-cohomologie" 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%89%E3%83%BB%E3%83%A9%E3%83%BC%E3%83%A0%E3%82%B3%E3%83%9B%E3%83%A2%E3%83%AD%E3%82%B8%E3%83%BC" title="ド・ラームコホモロジー – Japanese" lang="ja" hreflang="ja" data-title="ド・ラームコホモロジー" 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/Kompleks_de_Rhama" title="Kompleks de Rhama – Polish" lang="pl" hreflang="pl" data-title="Kompleks de Rhama" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B8_%D0%B4%D0%B5_%D0%A0%D0%B0%D0%BC%D0%B0" title="Когомологии де Рама – Russian" lang="ru" hreflang="ru" data-title="Когомологии де Рама" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/De_Rhamkohomologi" title="De Rhamkohomologi – Swedish" lang="sv" hreflang="sv" data-title="De Rhamkohomologi" 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%9A%D0%BE%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%8F_%D0%B4%D0%B5_%D0%A0%D0%B0%D0%BC%D0%B0" title="Когомологія де Рама – Ukrainian" lang="uk" hreflang="uk" data-title="Когомологія де Рама" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Cohomology with real coefficients computed using differential forms</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">For Grothendieck's de Rham cohomology of varieties, see <a href="/wiki/Algebraic_de_Rham_cohomology" class="mw-redirect" title="Algebraic de Rham cohomology">algebraic de Rham cohomology</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Irrotationalfield.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Irrotationalfield.svg/220px-Irrotationalfield.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Irrotationalfield.svg/330px-Irrotationalfield.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Irrotationalfield.svg/440px-Irrotationalfield.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Vector field corresponding to a differential form on the <a href="/wiki/Punctured_plane" class="mw-redirect" title="Punctured plane">punctured plane</a> that is closed but not exact, showing that the de Rham cohomology of this space is non-trivial.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>de Rham cohomology</b> (named after <a href="/wiki/Georges_de_Rham" title="Georges de Rham">Georges de Rham</a>) is a tool belonging both to <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> and to <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, capable of expressing basic topological information about <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a> in a form particularly adapted to computation and the concrete representation of <a href="/wiki/Cohomology_class" class="mw-redirect" title="Cohomology class">cohomology classes</a>. It is a <a href="/wiki/Cohomology_theory" class="mw-redirect" title="Cohomology theory">cohomology theory</a> based on the existence of <a href="/wiki/Differential_form" title="Differential form">differential forms</a> with prescribed properties. </p><p>On any smooth manifold, every <a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">exact form</a> is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of <a href="/wiki/Hole#In_mathematics" title="Hole">"holes"</a> in the manifold, and the <b>de Rham cohomology groups</b> comprise a set of <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariants</a> of smooth manifolds that precisely quantify this relationship.<sup id="cite_ref-FOOTNOTELee2013440_1-0" class="reference"><a href="#cite_note-FOOTNOTELee2013440-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <blockquote class="quote-frame pullquote" style="font-size: 95%; padding: 0.5em 2em; background-color: var( --background-color-neutral-subtle, #f8f9fa ); color: var( --color-base, black ); border: 1px solid #aaa; display:table; float:none;"><div style="padding: 0.6em 1em;">The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of <i>cohomology</i>, namely <i>de Rham cohomology</i>, which (roughly speaking) measures precisely the extent to which the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> fails in higher dimensions and on general manifolds.<br /><cite style="display: block; text-align: right;"> — <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a>, <i>Differential Forms and Integration</i><sup id="cite_ref-PCM_2-0" class="reference"><a href="#cite_note-PCM-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>de Rham complex</b> is the <a href="/wiki/Cochain_complex" class="mw-redirect" title="Cochain complex">cochain complex</a> of <a href="/wiki/Differential_form" title="Differential form">differential forms</a> on some <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> <span class="texhtml mvar" style="font-style:italic;">M</span>, with the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> as the differential: </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 0\to \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\to \cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\to \cdots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be4a0439c75ecdc48d1e98b608414dd6dbc7bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.538ex; height:4.176ex;" alt="{\displaystyle 0\to \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\to \cdots ,}" /></span></dd></dl> <p>where <span class="texhtml">Ω<sup>0</sup>(<i>M</i>)</span> is the space of <a href="/wiki/Smoothness" title="Smoothness">smooth functions</a> on <span class="texhtml mvar" style="font-style:italic;">M</span>, <span class="texhtml">Ω<sup>1</sup>(<i>M</i>)</span> is the space of <a href="/wiki/1-form" class="mw-redirect" title="1-form"><span class="texhtml">1</span>-forms</a>, and so forth. Forms that are the image of other forms under the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a>, plus the constant <span class="texhtml">0</span> function in <span class="texhtml">Ω<sup>0</sup>(<i>M</i>)</span>, are called <b>exact</b> and forms whose exterior derivative is <span class="texhtml">0</span> are called <b>closed</b> (see <i><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed and exact differential forms</a></i>); the relationship <span class="texhtml"><i>d</i><span style="padding-left:0.12em;"><sup>2</sup></span> = 0</span> then says that exact forms are closed. </p><p>In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the <span class="texhtml">1</span>-form corresponding to the derivative of angle from a reference point at its centre, typically written as <span class="texhtml"><i>dθ</i></span> (described at <i><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed and exact differential forms</a></i>). There is no function <span class="texhtml"><i>θ</i></span> defined on the whole circle such that <span class="texhtml"><i>dθ</i></span> is its derivative; the increase of <span class="texhtml">2<i>π</i></span> in going once around the circle in the positive direction implies a <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued function</a> <span class="texhtml"><i>θ</i></span>. Removing one point of the circle obviates this, at the same time changing the topology of the manifold. </p><p>One prominent example when all closed forms are exact is when the underlying space is <a href="/wiki/Contractible_space" title="Contractible space">contractible</a> to a point or, more generally, if it is <a href="/wiki/Simply_connected_space" title="Simply connected space">simply connected</a> (no-holes condition). In this case the exterior derivative <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> restricted to closed forms has a local inverse called a <a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">homotopy operator</a>.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Since it is also <a href="/wiki/Nilpotent" title="Nilpotent">nilpotent</a>,<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> it forms a dual <a href="/wiki/Chain_complex" title="Chain complex">chain complex</a> with the arrows reversed<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> compared to the de Rham complex. This is the situation described in the <a href="/wiki/Poincar%C3%A9_lemma" title="Poincaré lemma">Poincaré lemma</a>. </p><p>The idea behind de Rham cohomology is to define <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of closed forms on a manifold. One classifies two closed forms <span class="texhtml"><i>α</i>, <i>β</i> ∈ Ω<sup><i>k</i></sup>(<i>M</i>)</span> as <b>cohomologous</b> if they differ by an exact form, that is, if <span class="texhtml"><i>α</i> − <i>β</i></span> is exact. This classification induces an equivalence relation on the space of closed forms in <span class="texhtml">Ω<sup><i>k</i></sup>(<i>M</i>)</span>. One then defines the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <b>de Rham cohomology group</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathrm {dR} }^{k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{k}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb5996549a160b0bed09ef61e14b9e6da2c830c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.539ex; height:3.176ex;" alt="{\displaystyle H_{\mathrm {dR} }^{k}(M)}" /></span> to be the set of equivalence classes, that is, the set of closed forms in <span class="texhtml">Ω<sup><i>k</i></sup>(<i>M</i>)</span> modulo the exact forms. </p><p>Note that, for any manifold <span class="texhtml mvar" style="font-style:italic;">M</span> composed of <span class="texhtml"><i>m</i></span> disconnected components, each of which is <a href="/wiki/Connected_space" title="Connected space">connected</a>, we have 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 H_{\mathrm {dR} }^{0}(M)\cong \mathbb {R} ^{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</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>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{0}(M)\cong \mathbb {R} ^{m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f10c9be2f8e3c3c816279212971257bb9333acab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.637ex; height:3.176ex;" alt="{\displaystyle H_{\mathrm {dR} }^{0}(M)\cong \mathbb {R} ^{m}.}" /></span></dd></dl> <p>This follows from the fact that any smooth function on <span class="texhtml mvar" style="font-style:italic;">M</span> with zero derivative everywhere is separately constant on each of the connected components of <span class="texhtml mvar" style="font-style:italic;">M</span>. </p> <div class="mw-heading mw-heading2"><h2 id="De_Rham_cohomology_computed">De Rham cohomology computed</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=2" title="Edit section: De Rham cohomology computed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and a <a href="/wiki/Mayer%E2%80%93Vietoris_sequence" title="Mayer–Vietoris sequence">Mayer–Vietoris sequence</a>. Another useful fact is that the de Rham cohomology is a <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> objects: </p> <div class="mw-heading mw-heading3"><h3 id="The_n-sphere">The <span class="texhtml"><i>n</i></span>-sphere</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=3" title="Edit section: The n-sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the <a href="/wiki/N-sphere" title="N-sphere"><span class="texhtml mvar" style="font-style:italic;">n</span>-sphere</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 S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}" /></span>, and also when taken together with a product of open intervals, we have the following. Let <span class="texhtml"><i>n</i> > 0, <i>m</i> ≥ 0</span>, and <span class="texhtml mvar" style="font-style:italic;">I</span> be an open real interval. Then </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 H_{\mathrm {dR} }^{k}(S^{n}\times I^{m})\simeq {\begin{cases}\mathbb {R} &k=0{\text{ or }}k=n,\\0&k\neq 0{\text{ and }}k\neq n.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>×</mo> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> <mtd> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>k</mi> <mo>≠</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>k</mi> <mo>≠</mo> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{k}(S^{n}\times I^{m})\simeq {\begin{cases}\mathbb {R} &k=0{\text{ or }}k=n,\\0&k\neq 0{\text{ and }}k\neq n.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d67cfba6b57acc80af6321b50e9f818286fcfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.894ex; height:6.176ex;" alt="{\displaystyle H_{\mathrm {dR} }^{k}(S^{n}\times I^{m})\simeq {\begin{cases}\mathbb {R} &k=0{\text{ or }}k=n,\\0&k\neq 0{\text{ and }}k\neq n.\end{cases}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_n-torus">The <span class="texhtml"><i>n</i></span>-torus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=4" title="Edit section: The n-torus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>-torus is the Cartesian product: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6887b3f2a474e94ce6df60ec6b8414c605ea4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-right: -0.028ex; width:19.621ex; height:6.009ex;" alt="{\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}}" /></span>. Similarly, allowing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" /></span> here, we obtain </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 H_{\mathrm {dR} }^{k}(T^{n})\simeq \mathbb {R} ^{n \choose k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≃</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{k}(T^{n})\simeq \mathbb {R} ^{n \choose k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71b464131649048b2692e7e02720be2a5f787e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.454ex; height:4.509ex;" alt="{\displaystyle H_{\mathrm {dR} }^{k}(T^{n})\simeq \mathbb {R} ^{n \choose k}.}" /></span></dd></dl> <p>We can also find explicit generators for the de Rham cohomology of the torus directly using differential forms. Given a quotient 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 \pi :X\to X/G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :X\to X/G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d1aff81d6358ef440fec1e3ea66248f514d07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.832ex; height:2.843ex;" alt="{\displaystyle \pi :X\to X/G}" /></span> and a differential form <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 \omega \in \Omega ^{k}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega ^{k}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e393f6faddb22b719b0f2d64ed9d1c1d121b026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.843ex; height:3.176ex;" alt="{\displaystyle \omega \in \Omega ^{k}(X)}" /></span> we can say 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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /></span> is <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /></span><b>-invariant</b> if given any diffeomorphism induced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot g:X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot g:X\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad810c3f34fbf3475956d00ba682fb39a6886d9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.274ex; height:2.509ex;" alt="{\displaystyle \cdot g:X\to X}" /></span> we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot g)^{*}(\omega )=\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot g)^{*}(\omega )=\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921c0f39092cff19b5a0163a962d3275c6541240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.426ex; height:2.843ex;" alt="{\displaystyle (\cdot g)^{*}(\omega )=\omega }" /></span>. In particular, the pullback of any form 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 X/G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb103556c45e5f866285543c6b149b0f42f9121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.969ex; height:2.843ex;" alt="{\displaystyle X/G}" /></span> is <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /></span>-invariant. Also, the pullback is an injective morphism. In our case of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}/\mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}/\mathbb {Z} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508a95b410b6cc6a29d5659fbf855e1dbc229271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.828ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{n}/\mathbb {Z} ^{n}}" /></span> the differential forms <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 dx_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fcf9025bc3f34040616197a3819d50741aeb031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.345ex; height:2.509ex;" alt="{\displaystyle dx_{i}}" /></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}" /></span>-invariant since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x_{i}+k)=dx_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x_{i}+k)=dx_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ebc8453f797f45ce39eb3bf9120b78c5acb59a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.65ex; height:2.843ex;" alt="{\displaystyle d(x_{i}+k)=dx_{i}}" /></span>. But, notice 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 x_{i}+\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}+\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4706d353746e9f2384fbd9129d7f3402351932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.457ex; height:2.343ex;" alt="{\displaystyle x_{i}+\alpha }" /></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7988141e89a37e7f4deb883dbd74d9bbd6d11317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {R} }" /></span> is not an invariant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /></span>-form. This with injectivity implies 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 [dx_{i}]\in H_{dR}^{1}(T^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [dx_{i}]\in H_{dR}^{1}(T^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d4beb796ad364f1bd1f3a4b76a5a22e12e0399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.498ex; height:3.176ex;" alt="{\displaystyle [dx_{i}]\in H_{dR}^{1}(T^{n})}" /></span></dd></dl> <p>Since the cohomology ring of a torus is generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246d198ccb2f5e5488a7afd13093aab7b005139b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.158ex; height:2.676ex;" alt="{\displaystyle H^{1}}" /></span>, taking the exterior products of these forms gives all of the explicit <a href="/wiki/Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">representatives</a> for the de Rham cohomology of a torus. </p> <div class="mw-heading mw-heading3"><h3 id="Punctured_Euclidean_space">Punctured Euclidean space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=5" title="Edit section: Punctured Euclidean space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Punctured Euclidean space is simply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" /></span> with the origin removed. </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 H_{\text{dR}}^{k}(\mathbb {R} ^{n}\setminus \{0\})\cong {\begin{cases}\mathbb {R} ^{2}&n=1,k=0\\\mathbb {R} &n>1,k=0,n-1\\0&{\text{otherwise}}\end{cases}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>dR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> <mtd> <mi>n</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{dR}}^{k}(\mathbb {R} ^{n}\setminus \{0\})\cong {\begin{cases}\mathbb {R} ^{2}&n=1,k=0\\\mathbb {R} &n>1,k=0,n-1\\0&{\text{otherwise}}\end{cases}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a507bc1635d6d66f77cd317883f39036c447d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:44.886ex; height:8.509ex;" alt="{\displaystyle H_{\text{dR}}^{k}(\mathbb {R} ^{n}\setminus \{0\})\cong {\begin{cases}\mathbb {R} ^{2}&n=1,k=0\\\mathbb {R} &n>1,k=0,n-1\\0&{\text{otherwise}}\end{cases}}.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_Möbius_strip"><span id="The_M.C3.B6bius_strip"></span>The Möbius strip</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=6" title="Edit section: The Möbius strip"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We may deduce from the fact that the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a>, <span class="texhtml mvar" style="font-style:italic;">M</span>, can be <a href="/wiki/Deformation_retract" class="mw-redirect" title="Deformation retract">deformation retracted</a> to the <span class="texhtml">1</span>-sphere (i.e. the real unit circle), 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 H_{\mathrm {dR} }^{k}(M)\simeq H_{\mathrm {dR} }^{k}(S^{1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{k}(M)\simeq H_{\mathrm {dR} }^{k}(S^{1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0b496a4028fe7564743c7377791f41cd80dd34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.956ex; height:3.343ex;" alt="{\displaystyle H_{\mathrm {dR} }^{k}(M)\simeq H_{\mathrm {dR} }^{k}(S^{1}).}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="De_Rham_theorem">De Rham theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=7" title="Edit section: De Rham theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/De_Rham_theorem" title="De Rham theorem">de Rham theorem</a></div> <p><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Stokes' theorem</a> is an expression of <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">duality</a> between de Rham cohomology and the <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> of <a href="/wiki/Chain_(algebraic_topology)" title="Chain (algebraic topology)">chains</a>. It says that the pairing of differential forms and chains, via integration, gives a <a href="/wiki/Group_homomorphism" title="Group homomorphism">homomorphism</a> from de Rham cohomology <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathrm {dR} }^{k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{k}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb5996549a160b0bed09ef61e14b9e6da2c830c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.539ex; height:3.176ex;" alt="{\displaystyle H_{\mathrm {dR} }^{k}(M)}" /></span> to <a href="/wiki/Singular_cohomology" class="mw-redirect" title="Singular cohomology">singular cohomology groups</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 H^{k}(M;\mathbb {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{k}(M;\mathbb {R} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd13249e868a14ffea27a599b0d79e3ef1c58327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.803ex; height:3.176ex;" alt="{\displaystyle H^{k}(M;\mathbb {R} ).}" /></span> de Rham's theorem, proved by <a href="/wiki/Georges_de_Rham" title="Georges de Rham">Georges de Rham</a> in 1931, states that for a smooth manifold <span class="texhtml mvar" style="font-style:italic;">M</span>, this map is in fact an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. </p><p>More precisely, consider the map </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 I:H_{\mathrm {dR} }^{p}(M)\to H^{p}(M;\mathbb {R} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I:H_{\mathrm {dR} }^{p}(M)\to H^{p}(M;\mathbb {R} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e497ac7b10d68134e141b2040fe2cbd2403980a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.034ex; height:3.176ex;" alt="{\displaystyle I:H_{\mathrm {dR} }^{p}(M)\to H^{p}(M;\mathbb {R} ),}" /></span></dd></dl> <p>defined as follows: for any <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 [\omega ]\in H_{\mathrm {dR} }^{p}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>ω</mi> <mo stretchy="false">]</mo> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\omega ]\in H_{\mathrm {dR} }^{p}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f95d7e031a14109b2ba0137becb192dafde7eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.119ex; height:3.176ex;" alt="{\displaystyle [\omega ]\in H_{\mathrm {dR} }^{p}(M)}" /></span>, let <span class="texhtml"><i>I</i>(<i>ω</i>)</span> be the element 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{Hom}}(H_{p}(M),\mathbb {R} )\simeq H^{p}(M;\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hom</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>≃</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</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 {\text{Hom}}(H_{p}(M),\mathbb {R} )\simeq H^{p}(M;\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff356b786082ba4efc4e3e5d975e8c7ac4accb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.829ex; height:3.009ex;" alt="{\displaystyle {\text{Hom}}(H_{p}(M),\mathbb {R} )\simeq H^{p}(M;\mathbb {R} )}" /></span> that acts as follows: </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 H_{p}(M)\ni [c]\longmapsto \int _{c}\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>∋</mo> <mo stretchy="false">[</mo> <mi>c</mi> <mo stretchy="false">]</mo> <mo stretchy="false">⟼</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{p}(M)\ni [c]\longmapsto \int _{c}\omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6726930758d2e613d3f1396c418058de95a983af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.195ex; height:5.676ex;" alt="{\displaystyle H_{p}(M)\ni [c]\longmapsto \int _{c}\omega .}" /></span></dd></dl> <p>The theorem of de Rham asserts that this is an isomorphism between de Rham cohomology and singular cohomology. </p><p>The <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a> endows the <a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">direct sum</a> of these groups with a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> structure. A further result of the theorem is that the two <a href="/wiki/Cohomology_ring" title="Cohomology ring">cohomology rings</a> are isomorphic (as <a href="/wiki/Graded_ring" title="Graded ring">graded rings</a>), where the analogous product on singular cohomology is the <a href="/wiki/Cup_product" title="Cup product">cup product</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Sheaf-theoretic_de_Rham_isomorphism">Sheaf-theoretic de Rham isomorphism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=8" title="Edit section: Sheaf-theoretic de Rham isomorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any smooth manifold <i>M</i>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\underline {\mathbb {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\underline {\mathbb {R} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc23919c72ccf94a21a6e0c3c6d340f4ecc49659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-bottom: -0.825ex; width:1.68ex; height:3.176ex;" alt="{\textstyle {\underline {\mathbb {R} }}}" /></span> be the <a href="/wiki/Constant_sheaf" title="Constant sheaf">constant sheaf</a> on <i>M</i> associated to the abelian group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2031b33f0faf73d2d73a1da12544f00a37474f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\textstyle \mathbb {R} }" /></span>; in other words, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\underline {\mathbb {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\underline {\mathbb {R} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc23919c72ccf94a21a6e0c3c6d340f4ecc49659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-bottom: -0.825ex; width:1.68ex; height:3.176ex;" alt="{\textstyle {\underline {\mathbb {R} }}}" /></span> is the sheaf of locally constant real-valued functions on <i>M.</i> Then we have a <a href="/wiki/Natural_isomorphism" class="mw-redirect" title="Natural isomorphism">natural isomorphism</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 H_{\mathrm {dR} }^{*}(M)\cong H^{*}(M,{\underline {\mathbb {R} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{*}(M)\cong H^{*}(M,{\underline {\mathbb {R} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa08686f69c116f05503c2adcffdf93d6138bd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.799ex; margin-bottom: -0.539ex; width:21.76ex; height:3.343ex;" alt="{\displaystyle H_{\mathrm {dR} }^{*}(M)\cong H^{*}(M,{\underline {\mathbb {R} }})}" /></span></dd></dl> <p>between the de Rham cohomology and the <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</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="{\textstyle {\underline {\mathbb {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\underline {\mathbb {R} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc23919c72ccf94a21a6e0c3c6d340f4ecc49659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-bottom: -0.825ex; width:1.68ex; height:3.176ex;" alt="{\textstyle {\underline {\mathbb {R} }}}" /></span>. (Note that this shows that de Rham cohomology may also be computed in terms of <a href="/wiki/%C4%8Cech_cohomology" title="Čech cohomology">Čech cohomology</a>; indeed, since every smooth manifold is paracompact Hausdorff we have that sheaf cohomology is isomorphic to the Čech cohomology <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\check {H}}^{*}({\mathcal {U}},{\underline {\mathbb {R} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\check {H}}^{*}({\mathcal {U}},{\underline {\mathbb {R} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc2a30db3dfc7dc62f86fd19833a9f008471374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:9.238ex; height:3.843ex;" alt="{\textstyle {\check {H}}^{*}({\mathcal {U}},{\underline {\mathbb {R} }})}" /></span> for any <a href="/wiki/Good_cover_(algebraic_topology)" title="Good cover (algebraic topology)">good cover</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="{\textstyle {\mathcal {U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049c010c227beca6ffa646109b5b80386644ed9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.038ex; width:1.635ex; height:2.176ex;" alt="{\textstyle {\mathcal {U}}}" /></span> of <i>M</i>.) </p> <div class="mw-heading mw-heading3"><h3 id="Proof">Proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=9" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard proof proceeds by showing that the de Rham complex, when viewed as a complex of sheaves, is an <a href="/wiki/Acyclic_resolution" class="mw-redirect" title="Acyclic resolution">acyclic resolution</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="{\textstyle {\underline {\mathbb {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\underline {\mathbb {R} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc23919c72ccf94a21a6e0c3c6d340f4ecc49659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-bottom: -0.825ex; width:1.68ex; height:3.176ex;" alt="{\textstyle {\underline {\mathbb {R} }}}" /></span>. In more detail, let <i>m</i> be the dimension of <i>M</i> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Omega ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Omega ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5711c4dec953c9caa87a39691df117b9976db2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.509ex;" alt="{\textstyle \Omega ^{k}}" /></span> denote the <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf of germs</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 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>-forms on <i>M</i> (with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Omega ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Omega ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa4e1f317fa5767f3bd182b86c2afaa1c54b09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.509ex;" alt="{\textstyle \Omega ^{0}}" /></span> the sheaf 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="{\textstyle C^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle C^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20cef794141832e12bf48f02e2ef29fbc618d7c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.176ex;" alt="{\textstyle C^{\infty }}" /></span> functions on <i>M</i>). By the <a href="/wiki/Poincar%C3%A9_lemma" title="Poincaré lemma">Poincaré lemma</a>, the following sequence of sheaves is exact (in the <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a> of sheaves): </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 0\to {\underline {\mathbb {R} }}\to \Omega ^{0}\,\xrightarrow {d_{0}} \,\Omega ^{1}\,\xrightarrow {d_{1}} \,\Omega ^{2}\,\xrightarrow {d_{2}} \dots \xrightarrow {d_{m-1}} \,\Omega ^{m}\to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mpadded> </mover> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mpadded> </mover> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mpadded> </mover> <mo>⋯</mo> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mpadded> </mover> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to {\underline {\mathbb {R} }}\to \Omega ^{0}\,\xrightarrow {d_{0}} \,\Omega ^{1}\,\xrightarrow {d_{1}} \,\Omega ^{2}\,\xrightarrow {d_{2}} \dots \xrightarrow {d_{m-1}} \,\Omega ^{m}\to 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb12fcfb29d29455bfef754af52b07c0163e8ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-top: -0.416ex; margin-bottom: -0.825ex; width:51.501ex; height:5.009ex;" alt="{\displaystyle 0\to {\underline {\mathbb {R} }}\to \Omega ^{0}\,\xrightarrow {d_{0}} \,\Omega ^{1}\,\xrightarrow {d_{1}} \,\Omega ^{2}\,\xrightarrow {d_{2}} \dots \xrightarrow {d_{m-1}} \,\Omega ^{m}\to 0.}" /></span></dd></dl> <p>This <a href="/wiki/Long_exact_sequence" class="mw-redirect" title="Long exact sequence">long exact sequence</a> now breaks up into <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequences</a> of sheaves </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 0\to \mathrm {im} \,d_{k-1}\,\xrightarrow {\subset } \,\Omega ^{k}\,\xrightarrow {d_{k}} \,\mathrm {im} \,d_{k}\to 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mo>⊂</mo> </mpadded> </mover> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mpadded> </mover> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to \mathrm {im} \,d_{k-1}\,\xrightarrow {\subset } \,\Omega ^{k}\,\xrightarrow {d_{k}} \,\mathrm {im} \,d_{k}\to 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc690bdb62f808e1c760b4ce9c85a2b0935f448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.307ex; width:35.371ex; height:4.176ex;" alt="{\displaystyle 0\to \mathrm {im} \,d_{k-1}\,\xrightarrow {\subset } \,\Omega ^{k}\,\xrightarrow {d_{k}} \,\mathrm {im} \,d_{k}\to 0,}" /></span></dd></dl> <p>where by exactness we have isomorphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {im} \,d_{k-1}\cong \mathrm {ker} \,d_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathrm {im} \,d_{k-1}\cong \mathrm {ker} \,d_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc7425b13ff4f81515855bc67864c350383b7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.323ex; height:2.509ex;" alt="{\textstyle \mathrm {im} \,d_{k-1}\cong \mathrm {ker} \,d_{k}}" /></span> for all <i>k</i>. Each of these induces a long exact sequence in cohomology. Since the sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Omega ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Omega ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa4e1f317fa5767f3bd182b86c2afaa1c54b09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.509ex;" alt="{\textstyle \Omega ^{0}}" /></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle C^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20cef794141832e12bf48f02e2ef29fbc618d7c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.176ex;" alt="{\textstyle C^{\infty }}" /></span> functions on <i>M</i> admits <a href="/wiki/Partition_of_unity" title="Partition of unity">partitions of unity</a>, any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Omega ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Omega ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa4e1f317fa5767f3bd182b86c2afaa1c54b09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.509ex;" alt="{\textstyle \Omega ^{0}}" /></span>-module is a <a href="/wiki/Fine_sheaf" class="mw-redirect" title="Fine sheaf">fine sheaf</a>; in particular, the sheaves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Omega ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Omega ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5711c4dec953c9caa87a39691df117b9976db2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.509ex;" alt="{\textstyle \Omega ^{k}}" /></span> are all fine. Therefore, the sheaf cohomology groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H^{i}(M,\Omega ^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H^{i}(M,\Omega ^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e2d270252df82fe504ff7d29afff355148e8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.955ex; height:3.009ex;" alt="{\textstyle H^{i}(M,\Omega ^{k})}" /></span> vanish for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle i>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>i</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle i>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a1d84fb7d5164c08cfe1d3b493654de3318302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\textstyle i>0}" /></span> since all fine sheaves on paracompact spaces are acyclic. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the sheaf cohomology 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="{\textstyle {\underline {\mathbb {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\underline {\mathbb {R} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc23919c72ccf94a21a6e0c3c6d340f4ecc49659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-bottom: -0.825ex; width:1.68ex; height:3.176ex;" alt="{\textstyle {\underline {\mathbb {R} }}}" /></span> and at the other lies the de Rham cohomology. </p> <div class="mw-heading mw-heading2"><h2 id="Related_ideas">Related ideas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=10" title="Edit section: Related ideas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The de Rham cohomology has inspired many mathematical ideas, including <a href="/wiki/Dolbeault_cohomology" title="Dolbeault cohomology">Dolbeault cohomology</a>, <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a>, and the <a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a> proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem. For further details see <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_forms">Harmonic forms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=11" title="Edit section: Harmonic forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Harmonic_differential" title="Harmonic differential">Harmonic differential</a></div> <p>If <span class="texhtml mvar" style="font-style:italic;">M</span> is a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, then each equivalence class 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 H_{\mathrm {dR} }^{k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {dR} }^{k}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb5996549a160b0bed09ef61e14b9e6da2c830c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.539ex; height:3.176ex;" alt="{\displaystyle H_{\mathrm {dR} }^{k}(M)}" /></span> contains exactly one <a href="/wiki/Harmonic_form" class="mw-redirect" title="Harmonic form">harmonic form</a>. That is, every member <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /></span> of a given equivalence class of closed forms can be written 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 \omega =\alpha +\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\alpha +\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b702801cc3978f81737c118769b4f2a465b95a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.135ex; height:2.509ex;" alt="{\displaystyle \omega =\alpha +\gamma }" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> is exact 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" /></span> is harmonic: <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 \Delta \gamma =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \gamma =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/278fe2da6d3e04198d94559c0436be1d93525ac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.459ex; height:2.676ex;" alt="{\displaystyle \Delta \gamma =0}" /></span>. </p><p>Any <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic function</a> on a compact connected Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold. For example, on a <span class="texhtml">2</span>-<a href="/wiki/Torus" title="Torus">torus</a>, one may envision a constant <span class="texhtml">1</span>-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1st <a href="/wiki/Betti_number" title="Betti number">Betti number</a> of a <span class="texhtml">2</span>-torus is two. More generally, on an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>-dimensional torus <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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a5182df16f38b52918786915cbaa047bb46a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.938ex; height:2.343ex;" alt="{\displaystyle T^{n}}" /></span>, one can consider the various combings 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 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>-forms on the torus. There are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> choose <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> such combings that can be used to form the basis vectors for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{dR}}^{k}(T^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>dR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{dR}}^{k}(T^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c19755ecb3e4ed38e2724c30afc3317865f611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.035ex; height:3.176ex;" alt="{\displaystyle H_{\text{dR}}^{k}(T^{n})}" /></span>; 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 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>-th Betti number for the de Rham cohomology group for 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>-torus is thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> choose <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>. </p><p>More precisely, for a <a href="/wiki/Differential_manifold" class="mw-redirect" title="Differential manifold">differential manifold</a> <span class="texhtml mvar" style="font-style:italic;">M</span>, one may equip it with some auxiliary <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a>. Then the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /></span> is defined by </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 \Delta =d\delta +\delta d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mi>d</mi> <mi>δ</mi> <mo>+</mo> <mi>δ</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =d\delta +\delta d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92a393f3c3b5c6102a33f0f0b924476ba09e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.404ex; height:2.509ex;" alt="{\displaystyle \Delta =d\delta +\delta d}" /></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> 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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" /></span> the <a href="/wiki/Codifferential" class="mw-redirect" title="Codifferential">codifferential</a>. The Laplacian is a homogeneous (in <a href="/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">grading</a>) <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> acting upon the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> of <a href="/wiki/Differential_form" title="Differential form">differential forms</a>: we can look at its action on each component of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> separately. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> is <a href="/wiki/Compact_space" title="Compact space">compact</a> and <a href="/wiki/Oriented" class="mw-redirect" title="Oriented">oriented</a>, the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of the Laplacian acting upon the space of <a href="/wiki/Differential_form" title="Differential form"><span class="texhtml mvar" style="font-style:italic;">k</span>-forms</a> is then equal (by <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a>) to that of the de Rham cohomology group in degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>: the Laplacian picks out a unique <b>harmonic</b> <b>form</b> in each cohomology class of <a href="/wiki/Closed_form_(calculus)" class="mw-redirect" title="Closed form (calculus)">closed forms</a>. In particular, the space of all harmonic <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>-forms on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> is isomorphic 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 H^{k}(M;\mathbb {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{k}(M;\mathbb {R} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd13249e868a14ffea27a599b0d79e3ef1c58327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.803ex; height:3.176ex;" alt="{\displaystyle H^{k}(M;\mathbb {R} ).}" /></span> The dimension of each such space is finite, and is given by 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 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>-th <a href="/wiki/Betti_number" title="Betti number">Betti number</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Hodge_decomposition">Hodge decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=12" title="Edit section: Hodge decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> be a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Oriented_manifold" class="mw-redirect" title="Oriented manifold">oriented</a> <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. The <i>Hodge decomposition</i> states that any <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>-form on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> uniquely splits into the sum of three <span class="texhtml"><a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>2</sup></a></span> components: </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 \omega =\alpha +\beta +\gamma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\alpha +\beta +\gamma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f65fd0a64ef3b817fdfcce73f6018060a681104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.954ex; height:2.676ex;" alt="{\displaystyle \omega =\alpha +\beta +\gamma ,}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> is exact, <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> is co-exact, 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" /></span> is harmonic. </p><p>One says that a form <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> is co-closed 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 \delta \beta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>β</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \beta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452b1988f0cc655497c48b4303c81823fe7e54e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.642ex; height:2.676ex;" alt="{\displaystyle \delta \beta =0}" /></span> and co-exact 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 \beta =\delta \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>δ</mi> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\delta \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010dba814da8786d80d3e5934a23370a3d297efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.649ex; height:2.843ex;" alt="{\displaystyle \beta =\delta \eta }" /></span> for some form <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" /></span>, and 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" /></span> is harmonic if the Laplacian is zero, <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 \Delta \gamma =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \gamma =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/278fe2da6d3e04198d94559c0436be1d93525ac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.459ex; height:2.676ex;" alt="{\displaystyle \Delta \gamma =0}" /></span>. This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to the <span class="texhtml"><a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>2</sup></a></span> inner product 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 \Omega ^{k}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{k}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e527515b7c9de608cccd789454806924363647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.018ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k}(M)}" /></span>: </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 (\alpha ,\beta )=\int _{M}\alpha \wedge {\star \beta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>α</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> <mi>β</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge {\star \beta }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a0cde5699f657827d6a137a282fb5aef6791a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.612ex; height:5.676ex;" alt="{\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge {\star \beta }.}" /></span></dd></dl> <p>By use of <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a> or <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </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=De_Rham_cohomology&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a></li> <li><a href="/wiki/Integration_along_fibers" title="Integration along fibers">Integration along fibers</a> (for de Rham cohomology, the <a href="/w/index.php?title=Pushforward_(cohomology)&action=edit&redlink=1" class="new" title="Pushforward (cohomology) (page does not exist)">pushforward</a> is given by <a href="/wiki/Integration_(mathematics)" class="mw-redirect" title="Integration (mathematics)">integration</a>)</li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf theory</a></li> <li><a href="/wiki/Ddbar_lemma" title="Ddbar lemma"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial {\bar {\partial }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial {\bar {\partial }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61471a2c013227ee2956b4a17932feebcdd39dca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.806ex; height:2.676ex;" alt="{\displaystyle \partial {\bar {\partial }}}" /></span>-lemma</a> for a refinement of exact differential forms in the case of compact <a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifolds</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=14" title="Edit section: Citations"><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-FOOTNOTELee2013440-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELee2013440_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLee2013">Lee 2013</a>, p. 440.</span> </li> <li id="cite_note-PCM-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-PCM_2-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Terence_Tao" title="Terence Tao">Tao, Terence</a> <a rel="nofollow" class="external text" href="https://www.math.ucla.edu/~tao/preprints/forms.pdf">(2007) "Differential Forms and Integration"</a> <i>Princeton Companion to Mathematics</i> 2008. Timothy Gowers, ed.</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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="CITEREFEdelen2011" class="citation book cs1">Edelen, Dominic G. B. (2011). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/56347718"><i>Applied exterior calculus</i></a> (Revised ed.). Mineola, N.Y.: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43871-9" title="Special:BookSources/978-0-486-43871-9"><bdi>978-0-486-43871-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/56347718">56347718</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+exterior+calculus&rft.place=Mineola%2C+N.Y.&rft.edition=Revised&rft.pub=Dover+Publications&rft.date=2011&rft_id=info%3Aoclcnum%2F56347718&rft.isbn=978-0-486-43871-9&rft.aulast=Edelen&rft.aufirst=Dominic+G.+B.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F56347718&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarner1983" class="citation book cs1">Warner, Frank W. (1983). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/9683855"><i>Foundations of differentiable manifolds and Lie groups</i></a>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90894-3" title="Special:BookSources/0-387-90894-3"><bdi>0-387-90894-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/9683855">9683855</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+differentiable+manifolds+and+Lie+groups&rft.place=New+York&rft.pub=Springer&rft.date=1983&rft_id=info%3Aoclcnum%2F9683855&rft.isbn=0-387-90894-3&rft.aulast=Warner&rft.aufirst=Frank+W.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F9683855&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKycia2020" class="citation journal cs1">Kycia, Radosław Antoni (2020). <a rel="nofollow" class="external text" href="https://link.springer.com/10.1007/s00025-020-01247-8">"The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator"</a>. <i>Results in Mathematics</i>. <b>75</b> (3): 122. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.02349">1908.02349</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00025-020-01247-8">10.1007/s00025-020-01247-8</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1422-6383">1422-6383</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:199472766">199472766</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Results+in+Mathematics&rft.atitle=The+Poincare+Lemma%2C+Antiexact+Forms%2C+and+Fermionic+Quantum+Harmonic+Oscillator&rft.volume=75&rft.issue=3&rft.pages=122&rft.date=2020&rft_id=info%3Aarxiv%2F1908.02349&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199472766%23id-name%3DS2CID&rft.issn=1422-6383&rft_id=info%3Adoi%2F10.1007%2Fs00025-020-01247-8&rft.aulast=Kycia&rft.aufirst=Rados%C5%82aw+Antoni&rft_id=https%3A%2F%2Flink.springer.com%2F10.1007%2Fs00025-020-01247-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Jean-Pierre Demailly, <a rel="nofollow" class="external text" href="https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf">Complex Analytic and Differential Geometry</a> Ch VIII, § 3.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Rham_cohomology&action=edit&section=15" title="Edit section: References"><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" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLee2013" class="citation book cs1">Lee, John M. (2013). <i>Introduction to Smooth Manifolds</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-9981-8" title="Special:BookSources/978-1-4419-9981-8"><bdi>978-1-4419-9981-8</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.pub=Springer-Verlag&rft.date=2013&rft.isbn=978-1-4419-9981-8&rft.aulast=Lee&rft.aufirst=John+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBottTu1982" class="citation cs2"><a href="/wiki/Raoul_Bott" title="Raoul Bott">Bott, Raoul</a>; Tu, Loring W. (1982), <i>Differential Forms in Algebraic Topology</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/978-0-387-90613-3" title="Special:BookSources/978-0-387-90613-3"><bdi>978-0-387-90613-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Forms+in+Algebraic+Topology&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-90613-3&rft.aulast=Bott&rft.aufirst=Raoul&rft.au=Tu%2C+Loring+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGriffithsHarris1994" class="citation cs2"><a href="/wiki/Phillip_Griffiths" title="Phillip Griffiths">Griffiths, Phillip</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joseph</a> (1994), <i>Principles of algebraic geometry</i>, Wiley Classics Library, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-05059-9" title="Special:BookSources/978-0-471-05059-9"><bdi>978-0-471-05059-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1288523">1288523</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+algebraic+geometry&rft.place=New+York&rft.series=Wiley+Classics+Library&rft.pub=John+Wiley+%26+Sons&rft.date=1994&rft.isbn=978-0-471-05059-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1288523%23id-name%3DMR&rft.aulast=Griffiths&rft.aufirst=Phillip&rft.au=Harris%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarner1983" class="citation cs2">Warner, Frank (1983), <i>Foundations of Differentiable Manifolds and Lie Groups</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/978-0-387-90894-6" title="Special:BookSources/978-0-387-90894-6"><bdi>978-0-387-90894-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Differentiable+Manifolds+and+Lie+Groups&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1983&rft.isbn=978-0-387-90894-6&rft.aulast=Warner&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" 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=De_Rham_cohomology&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160219043413/http://mathifold.org/en/de-rham-cohomology/cohomologia_rham">Idea of the De Rham Cohomology</a></i> in <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160204170745/http://mathifold.org/">Mathifold Project</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=De_Rham_cohomology">"De Rham cohomology"</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=De+Rham+cohomology&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDe_Rham_cohomology&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Rham+cohomology" class="Z3988"></span></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 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.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,_List,_Category)273" 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>, <a href="/wiki/List_of_manifolds" title="List of manifolds">List</a>, <a href="/wiki/Category:Manifolds" title="Category:Manifolds">Category</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 class="mw-selflink-fragment" href="#De_Rham's_theorem">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/Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li> <li><a href="/wiki/Complete_manifold" title="Complete manifold">Complete</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/Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <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/Nilmanifold" title="Nilmanifold">Nilmanifold</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 class="mw-selflink selflink">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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