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vector-toc-level-2"> <a class="vector-toc-link" href="#Simply_connected"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Simply connected</span> </div> </a> <ul id="toc-Simply_connected-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Disc_vs._space_vs._polydisc" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Disc_vs._space_vs._polydisc"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Disc vs. space vs. polydisc</span> </div> </a> <ul id="toc-Disc_vs._space_vs._polydisc-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Almost_complex_structures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Almost_complex_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Almost complex structures</span> </div> </a> <ul id="toc-Almost_complex_structures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kähler_and_Calabi–Yau_manifolds" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kähler_and_Calabi–Yau_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Kähler and Calabi–Yau manifolds</span> </div> </a> <ul id="toc-Kähler_and_Calabi–Yau_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-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> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the 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href="https://cs.wikipedia.org/wiki/Komplexn%C3%AD_varieta" title="Komplexní varieta – Czech" lang="cs" hreflang="cs" data-title="Komplexní varieta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit – German" lang="de" hreflang="de" data-title="Komplexe Mannigfaltigkeit" 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/Variedad_compleja" title="Variedad compleja – Spanish" lang="es" hreflang="es" data-title="Variedad compleja" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleksa_sterna%C4%B5o" title="Kompleksa sternaĵo – Esperanto" lang="eo" hreflang="eo" data-title="Kompleksa sternaĵo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_complexe" title="Variété complexe – French" lang="fr" hreflang="fr" data-title="Variété complexe" 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%B3%B5%EC%86%8C%EB%8B%A4%EC%96%91%EC%B2%B4" title="복소다양체 – Korean" lang="ko" hreflang="ko" data-title="복소다양체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD_%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%BF%D5%AB%D5%AF_%D5%A2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B1%D6%87%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Կոմպլեքս անալիտիկ բազմաձևություն – Armenian" lang="hy" hreflang="hy" data-title="Կոմպլեքս անալիտիկ բազմաձևություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Complexe_vari%C3%ABteit" title="Complexe variëteit – Dutch" lang="nl" hreflang="nl" data-title="Complexe variëteit" 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/%E8%A4%87%E7%B4%A0%E5%A4%9A%E6%A7%98%E4%BD%93" title="複素多様体 – Japanese" lang="ja" hreflang="ja" data-title="複素多様体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Variedade_complexa" title="Variedade complexa – Portuguese" lang="pt" hreflang="pt" data-title="Variedade complexa" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Комплексное многообразие – Russian" lang="ru" hreflang="ru" data-title="Комплексное многообразие" data-language-autonym="Русский" 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.mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" 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href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">October 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Holomorphic_Maps.webm/220px--Holomorphic_Maps.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="220" data-durationhint="13" data-mwtitle="Holomorphic_Maps.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Holomorphic_Maps.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Holomorphic_Maps.webm/Holomorphic_Maps.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="480" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/6/6d/Holomorphic_Maps.webm" type="video/webm; codecs="vp8"" data-width="720" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Holomorphic_Maps.webm/Holomorphic_Maps.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="720" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Holomorphic_Maps.webm/Holomorphic_Maps.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="144" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Holomorphic_Maps.webm/Holomorphic_Maps.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="240" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Holomorphic_Maps.webm/Holomorphic_Maps.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="360" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Holomorphic_Maps.webm/Holomorphic_Maps.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="360" data-height="360" /></video></span><figcaption>Holomorphic Maps</figcaption></figure> <p>In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="/wiki/Complex_geometry" title="Complex geometry">complex geometry</a>, a <b>complex manifold</b> is a <a href="/wiki/Manifold" title="Manifold">manifold</a> with a <i>complex structure</i>, that is an <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a> of <a href="/wiki/Chart_(topology)" class="mw-redirect" title="Chart (topology)">charts</a> to the <a href="/wiki/Open_unit_disc" class="mw-redirect" title="Open unit disc">open unit disc</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in the <a href="/wiki/Complex_coordinate_space" title="Complex coordinate space">complex coordinate space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>, such that the <a href="/wiki/Transition_map" class="mw-redirect" title="Transition map">transition maps</a> are <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a>. </p><p>The term "complex manifold" is variously used to mean a complex manifold in the sense above (which can be specified as an <i>integrable</i> complex manifold) or an <a href="/wiki/Almost_complex_manifold" title="Almost complex manifold"><i>almost</i> complex manifold</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Implications_of_complex_structure">Implications of complex structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=1" title="Edit section: Implications of complex structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> are much more rigid than <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a>, the theories of <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth</a> and complex manifolds have very different flavors: <a href="/wiki/Compact_space" title="Compact space">compact</a> complex manifolds are much closer to <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> than to differentiable manifolds. </p><p>For example, the <a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding theorem</a> tells us that every smooth <i>n</i>-dimensional manifold can be <a href="/wiki/Embedding" title="Embedding">embedded</a> as a smooth submanifold of <b>R</b><sup>2<i>n</i></sup>, whereas it is "rare" for a complex manifold to have a holomorphic embedding into <b>C</b><sup><i>n</i></sup>. Consider for example any <a href="/wiki/Compact_space" title="Compact space">compact</a> connected complex manifold <i>M</i>: any holomorphic function on it is constant by <a href="/wiki/Maximum_modulus_principle" title="Maximum modulus principle">the maximum modulus principle</a>. Now if we had a holomorphic embedding of <i>M</i> into <b>C</b><sup><i>n</i></sup>, then the coordinate functions of <b>C</b><sup><i>n</i></sup> would restrict to nonconstant holomorphic functions on <i>M</i>, contradicting compactness, except in the case that <i>M</i> is just a point. Complex manifolds that can be embedded in <b>C</b><sup><i>n</i></sup> are called <a href="/wiki/Stein_manifold" title="Stein manifold">Stein manifolds</a> and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties. </p><p>The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many <a href="/wiki/Smooth_structure" title="Smooth structure">smooth structures</a>, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, two dimensional manifolds equipped with a complex structure, which are topologically classified by the <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a>, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a <a href="/wiki/Moduli_space" title="Moduli space">moduli space</a>, the structure of which remains an area of active research. </p><p>Since the transition maps between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not just <a href="/wiki/Orientable" class="mw-redirect" title="Orientable">orientable</a>: a biholomorphic map to (a subset of) <b>C</b><sup><i>n</i></sup> gives an orientation, as biholomorphic maps are orientation-preserving). </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_complex_manifolds">Examples of complex manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=2" title="Edit section: Examples of complex manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>.</li> <li><a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifolds</a>.</li> <li>The Cartesian product of two complex manifolds.</li> <li>The inverse image of any noncritical value of a holomorphic map.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Smooth_complex_algebraic_varieties">Smooth complex algebraic varieties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=3" title="Edit section: Smooth complex algebraic varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Smooth complex <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a> are complex manifolds, including: </p> <ul><li>Complex vector spaces.</li> <li><a href="/wiki/Complex_projective_space" title="Complex projective space">Complex projective spaces</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <b>P</b><sup><i>n</i></sup>(<b>C</b>).</li> <li>Complex <a href="/wiki/Grassmannian" title="Grassmannian">Grassmannians</a>.</li> <li>Complex <a href="/wiki/Lie_groups" class="mw-redirect" title="Lie groups">Lie groups</a> such as GL(<i>n</i>, <b>C</b>) or Sp(<i>n</i>, <b>C</b>).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Simply_connected">Simply connected</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=4" title="Edit section: Simply connected"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> 1-dimensional complex manifolds are isomorphic to either: </p> <ul><li>Δ, the unit <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a> in <b>C</b></li> <li><b>C</b>, the complex plane</li> <li><b>Ĉ</b>, the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li></ul> <p>Note that there are inclusions between these as Δ ⊆ <b>C</b> ⊆ <b>Ĉ</b>, but that there are no non-constant holomorphic maps in the other direction, by <a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Disc_vs._space_vs._polydisc">Disc vs. space vs. polydisc</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=5" title="Edit section: Disc vs. space vs. polydisc"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following spaces are different as complex manifolds, demonstrating the more rigid geometric character of complex manifolds (compared to smooth manifolds): </p> <ul><li>complex space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>.</li> <li>the unit disc or <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open ball</a></li></ul> <dl><dd><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 \left\{z\in \mathbb {C} ^{n}:\|z\|<1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>z</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>z</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{z\in \mathbb {C} ^{n}:\|z\|<1\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20be4e4dee08d93babec035038040bdd9a7cf0e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.795ex; height:2.843ex;" alt="{\displaystyle \left\{z\in \mathbb {C} ^{n}:\|z\|<1\right\}.}"></span></dd></dl></dd></dl> <ul><li>the <a href="/wiki/Polydisc" title="Polydisc">polydisc</a></li></ul> <dl><dd><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 \left\{z=(z_{1},\dots ,z_{n})\in \mathbb {C} ^{n}:\vert z_{j}\vert <1\ \forall j=1,\dots ,n\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{z=(z_{1},\dots ,z_{n})\in \mathbb {C} ^{n}:\vert z_{j}\vert <1\ \forall j=1,\dots ,n\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06432dea992b70b61a627e4c492ab7e037457d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.853ex; height:3.009ex;" alt="{\displaystyle \left\{z=(z_{1},\dots ,z_{n})\in \mathbb {C} ^{n}:\vert z_{j}\vert <1\ \forall j=1,\dots ,n\right\}.}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Almost_complex_structures">Almost complex structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=6" title="Edit section: Almost complex structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost complex manifold</a></div> <p>An <a href="/wiki/Almost_complex_structure" class="mw-redirect" title="Almost complex structure">almost complex structure</a> on a real 2n-manifold is a GL(<i>n</i>, <b>C</b>)-structure (in the sense of <a href="/wiki/G-structure" class="mw-redirect" title="G-structure">G-structures</a>) – that is, the tangent bundle is equipped with a <a href="/wiki/Linear_complex_structure" title="Linear complex structure">linear complex structure</a>. </p><p>Concretely, this is an <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> of the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> whose square is −<i>I</i>; this endomorphism is analogous to multiplication by the imaginary number <i>i</i>, and is denoted <i>J</i> (to avoid confusion with the identity matrix <i>I</i>). An almost complex manifold is necessarily even-dimensional. </p><p>An almost complex structure is <i>weaker</i> than a complex structure: any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. Note that every even-dimensional real manifold has an almost complex structure defined locally from the local coordinate chart. The question is whether this almost complex structure can be defined globally. An almost complex structure that comes from a complex structure is called <a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">integrable</a>, and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an <i>integrable</i> complex structure. For integrable complex structures the so-called <a href="/wiki/Nijenhuis_tensor" class="mw-redirect" title="Nijenhuis tensor">Nijenhuis tensor</a> vanishes. This tensor is defined on pairs of vector fields, <i>X</i>, <i>Y</i> 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 N_{J}(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>J</mi> <mo stretchy="false">[</mo> <mi>J</mi> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>J</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>J</mi> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>J</mi> <mi>X</mi> <mo>,</mo> <mi>J</mi> <mi>Y</mi> <mo stretchy="false">]</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{J}(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0657943fc0c667681387a682be413da0030ec1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.734ex; height:2.843ex;" alt="{\displaystyle N_{J}(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]\ .}"></span></dd></dl> <p>For example, the 6-dimensional <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">sphere</a> <b>S</b><sup>6</sup> has a natural almost complex structure arising from the fact that it is the <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">orthogonal complement</a> of <i>i</i> in the unit sphere of the <a href="/wiki/Octonion" title="Octonion">octonions</a>, but this is not a complex structure. (The question of whether it has a complex structure is known as the <i>Hopf problem,</i> after <a href="/wiki/Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>) Using an almost complex structure we can make sense of holomorphic maps and ask about the existence of holomorphic coordinates on the manifold. The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says). </p><p>Tensoring the tangent bundle with the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> we get the <i>complexified</i> tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold). The eigenvalues of an almost complex structure are ±<i>i</i> and the eigenspaces form sub-bundles denoted by <i>T</i><sup>0,1</sup><i>M</i> and <i>T</i><sup>1,0</sup><i>M</i>. The <a href="/wiki/Newlander%E2%80%93Nirenberg_theorem" class="mw-redirect" title="Newlander–Nirenberg theorem">Newlander–Nirenberg theorem</a> shows that an almost complex structure is actually a complex structure precisely when these subbundles are <i>involutive</i>, i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is called <a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">integrable</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Kähler_and_Calabi–Yau_manifolds"><span id="K.C3.A4hler_and_Calabi.E2.80.93Yau_manifolds"></span>Kähler and Calabi–Yau manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=7" title="Edit section: Kähler and Calabi–Yau manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can define an analogue of a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> for complex manifolds, called a <a href="/wiki/Hermitian_metric" class="mw-redirect" title="Hermitian metric">Hermitian metric</a>. Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point. As in the Riemannian case, such metrics always exist in abundance on any complex manifold. If the skew symmetric part of such a metric is <a href="/wiki/Symplectic_geometry" title="Symplectic geometry">symplectic</a>, i.e. closed and nondegenerate, then the metric is called <a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a>. Kähler structures are much more difficult to come by and are much more rigid. </p><p>Examples of <a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifolds</a> include smooth <a href="/wiki/Projective_varieties" class="mw-redirect" title="Projective varieties">projective varieties</a> and more generally any complex submanifold of a Kähler manifold. The <a href="/wiki/Hopf_manifold" title="Hopf manifold">Hopf manifolds</a> are examples of complex manifolds that are not Kähler. To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(<i>n</i>). The quotient is a complex manifold whose first <a href="/wiki/Betti_number" title="Betti number">Betti number</a> is one, so by the <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a>, it cannot be Kähler. </p><p>A <a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a> can be defined as a compact <a href="/wiki/Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat</a> Kähler manifold or equivalently one whose first <a href="/wiki/Chern_class" title="Chern class">Chern class</a> vanishes. </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=Complex_manifold&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Complex_dimension" title="Complex dimension">Complex dimension</a></li> <li><a href="/wiki/Complex_analytic_variety" title="Complex analytic variety">Complex analytic variety</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic manifold</a></li> <li><a href="/wiki/Real-complex_manifold" class="mw-redirect" title="Real-complex manifold">Real-complex manifold</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=9" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">One must use the open unit disc in 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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> as the model space instead 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 {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> because these are not isomorphic, unlike for real manifolds.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">This means that all complex projective spaces are <i>orientable</i>, in contrast to the real case</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><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="CITEREFAgricolaBazzoniGoertschesKonstantis2018" class="citation journal cs1"><a href="/wiki/Ilka_Agricola" title="Ilka Agricola">Agricola, Ilka</a>; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". <i><a href="/w/index.php?title=Differential_Geometry_and_Its_Applications&action=edit&redlink=1" class="new" title="Differential Geometry and Its Applications (page does not exist)">Differential Geometry and Its Applications</a></i>. <b>57</b>: 1–9. <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/1708.01068">1708.01068</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/10.1016%2Fj.difgeo.2017.10.014">10.1016/j.difgeo.2017.10.014</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:119297359">119297359</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Differential+Geometry+and+Its+Applications&rft.atitle=On+the+history+of+the+Hopf+problem&rft.volume=57&rft.pages=1-9&rft.date=2018&rft_id=info%3Aarxiv%2F1708.01068&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119297359%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.difgeo.2017.10.014&rft.aulast=Agricola&rft.aufirst=Ilka&rft.au=Bazzoni%2C+Giovanni&rft.au=Goertsches%2C+Oliver&rft.au=Konstantis%2C+Panagiotis&rft.au=Rollenske%2C+S%C3%B6nke&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+manifold" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_manifold&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKodaira2004" class="citation book cs1"><a href="/wiki/Kunihiko_Kodaira" title="Kunihiko Kodaira">Kodaira, Kunihiko</a> (17 November 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nvLCrfQFuh4C&q=%22complex+manifold%22"><i>Complex Manifolds and Deformation of Complex Structures</i></a>. Classics in Mathematics. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-22614-1" title="Special:BookSources/3-540-22614-1"><bdi>3-540-22614-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Manifolds+and+Deformation+of+Complex+Structures&rft.series=Classics+in+Mathematics&rft.pub=Springer&rft.date=2004-11-17&rft.isbn=3-540-22614-1&rft.aulast=Kodaira&rft.aufirst=Kunihiko&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnvLCrfQFuh4C%26q%3D%2522complex%2Bmanifold%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+manifold" 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 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href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a class="mw-selflink selflink">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style 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