SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Almost complex manifold</title>

</head>

Almost complex manifold

</h1>

</header>

In <a href="Mathematics" title="Mathematics">mathematics</a>, an almost complex manifold is a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> equipped with a smooth <a href="Linear_complex_structure" title="Linear complex structure">linear complex structure</a> on each <a href="Tangent_space" title="Tangent space">tangent space</a>. Every <a href="Complex_manifold" title="Complex manifold">complex manifold</a> is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in <a href="Symplectic_geometry" title="Symplectic geometry">symplectic geometry</a>.

The concept is due to <a href="Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a> and <a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a> in the 1940s.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a <a href="Linear_map" title="Linear map">linear map</a> which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> <a href="Tensor_field" title="Tensor field">tensor field</a> J of <a href="Tensor#Tensor_degree" title="Tensor">degree</a> (1, 1) such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}=-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{2}=-1}</annotation>

</semantics>

</math><img src="./82d2bb45f5120f5c885585143e126eb044268126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.649ex; height:2.843ex;" alt="{\displaystyle J^{2}=-1}" loading="lazy"> when regarded as a <a href="Vector_bundle" title="Vector bundle">vector bundle</a> <a href="Isomorphism" title="Isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\colon TM\to TM}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J\colon TM\to TM}</annotation>

</semantics>

</math><img src="./8c351c896b9473e99d2b3a9702e49f5a13880eae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.276ex; height:2.176ex;" alt="{\displaystyle J\colon TM\to TM}" loading="lazy"> on the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a>. A manifold equipped with an almost complex structure is called an almost complex manifold.

If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose M is n-dimensional, and let J : TM → TM be an almost complex structure. If J2 = −1 then (det J)2 = (−1)n. But if M is a real manifold, then det J is a real number – thus n must be even if M has an almost complex structure. One can show that it must be <a href="Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable</a> as well.

An easy exercise in <a href="Linear_algebra" title="Linear algebra">linear algebra</a> shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a (1, 1)-rank tensor pointwise (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold M is equivalent to a <a href="Reduction_of_the_structure_group" class="mw-redirect" title="Reduction of the structure group">reduction of the structure group</a> of the tangent bundle from GL(2n, R) to GL(n, C). The existence question is then a purely <a href="Algebraic_topology" title="Algebraic topology">algebraic topological</a> one and is fairly well understood.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

For every integer n, the flat space R2n admits an almost complex structure. An example for such an almost complex structure is (1 ≤ j, k ≤ 2n): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{jk}=-i\delta _{j,k-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{jk}=-i\delta _{j,k-1}}</annotation>

</semantics>

</math><img src="./b1c4f2be0bd7a4d2e2330d86e8e125ac6afd9c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.122ex; height:3.009ex;" alt="{\displaystyle J_{jk}=-i\delta _{j,k-1}}" loading="lazy"> for odd j, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{jk}=i\delta _{j,k+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{jk}=i\delta _{j,k+1}}</annotation>

</semantics>

</math><img src="./117d0680848d4f2c7ae45cda16544745dcb44260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.314ex; height:3.009ex;" alt="{\displaystyle J_{jk}=i\delta _{j,k+1}}" loading="lazy"> for even j.

The only <a href="Sphere" title="Sphere">spheres</a> which admit almost complex structures are S2 and S6 (<a href="#CITEREFBorelSerre1953">Borel & Serre (1953)</a>). In particular, S4 cannot be given an almost complex

structure (Ehresmann and Hopf). In the case of S2, the almost complex structure comes from an honest complex structure on the <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>. The 6-sphere, S6, when considered as the set of unit norm imaginary <a href="Octonion" title="Octonion">octonions</a>, inherits an almost complex structure from the octonion multiplication; the question of whether it has a <a href="#Integrable_almost_complex_structures">complex structure</a> is known as the Hopf problem, after <a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a>.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Differential_topology_of_almost_complex_manifolds">Differential topology of almost complex manifolds</h2></div>

Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V− (the <a href="Eigenspace" class="mw-redirect" title="Eigenspace">eigenspaces</a> of J corresponding to +i and −i, respectively), so an almost complex structure on M allows a decomposition of the complexified tangent bundle TMC (which is the vector bundle of complexified tangent spaces at each point) into TM+ and TM−. A section of TM+ is called a <a href="Vector_field" title="Vector field">vector field</a> of type (1, 0), while a section of TM− is a vector field of type (0, 1). Thus J corresponds to multiplication by <a href="Imaginary_unit" title="Imaginary unit">i</a> on the (1, 0)-vector fields of the complexified tangent bundle, and multiplication by −i on the (0, 1)-vector fields.

Just as we build <a href="Differential_form" title="Differential form">differential forms</a> out of <a href="Exterior_power" class="mw-redirect" title="Exterior power">exterior powers</a> of the <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>, we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle). The almost complex structure induces the decomposition of each space of r-forms

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega ^{r}(M)^{\mathbf {C} }=\bigoplus _{p+q=r}\Omega ^{(p,q)}(M).\,}</annotation>

</semantics>

</math><img src="./f96a7c7ef45f16c3416b475238f36a57f5a84bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.384ex; height:5.676ex;" alt="{\displaystyle \Omega ^{r}(M)^{\mathbf {C} }=\bigoplus _{p+q=r}\Omega ^{(p,q)}(M).\,}" loading="lazy"></dd></dl>

In other words, each Ωr(M)C admits a decomposition into a sum of Ω(p, q)(M), with r = p + q.

As with any <a href="Direct_sum_of_vector_bundles" class="mw-redirect" title="Direct sum of vector bundles">direct sum</a>, there is a canonical projection πp,q from Ωr(M)C to Ω(p,q). We also have the <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> d which maps Ωr(M)C to Ωr+1(M)C. Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial =\pi _{p+1,q}\circ d}</annotation>

</semantics>

</math><img src="./34affb72c5fa2f63792cb8e530ce09efd71f4649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.526ex; height:2.843ex;" alt="{\displaystyle \partial =\pi _{p+1,q}\circ d}" loading="lazy"></dd>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}=\pi _{p,q+1}\circ d}</annotation>

</semantics>

</math><img src="./11f0b45276a3fc7b751bab81421fb770bb688d67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.695ex; height:3.676ex;" alt="{\displaystyle {\overline {\partial }}=\pi _{p,q+1}\circ d}" loading="lazy"></dd></dl>

so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial }</annotation>

</semantics>

</math><img src="./62b4e7c1cedb9564609aefd2aa2309972f455c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }" loading="lazy"> is a map which increases the holomorphic part of the type by one (takes forms of type (p, q) to forms of type (p+1, q)), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\partial }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}}</annotation>

</semantics>

</math><img src="./6bd5dfe63f32552004b0d29d5ef7e8b046dda0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.487ex; height:3.009ex;" alt="{\displaystyle {\overline {\partial }}}" loading="lazy"> is a map which increases the antiholomorphic part of the type by one. These operators are called the <a href="Dolbeault_operator" class="mw-redirect" title="Dolbeault operator">Dolbeault operators</a>.

Since the sum of all the projections must be the <a href="Identity_function" title="Identity function">identity map</a>, we note that the exterior derivative can be written

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=\sum _{r+s=p+q+1}\pi _{r,s}\circ d=\partial +{\overline {\partial }}+\cdots .}</annotation>

</semantics>

</math><img src="./b651145a63bbd8fb08b8b2bed7137429df174a72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.013ex; height:6.009ex;" alt="{\displaystyle d=\sum _{r+s=p+q+1}\pi _{r,s}\circ d=\partial +{\overline {\partial }}+\cdots .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Integrable_almost_complex_structures">Integrable almost complex structures</h2></div>

Every <a href="Complex_manifold" title="Complex manifold">complex manifold</a> is itself an almost complex manifold. In local holomorphic coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{\mu }=x^{\mu }+iy^{\mu }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{\mu }=x^{\mu }+iy^{\mu }}</annotation>

</semantics>

</math><img src="./44c7dc31bcbd14d4ec15cd4ef50f97be00f61ae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.992ex; height:2.676ex;" alt="{\displaystyle z^{\mu }=x^{\mu }+iy^{\mu }}" loading="lazy"> one can define the maps

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J{\frac {\partial }{\partial x^{\mu }}}={\frac {\partial }{\partial y^{\mu }}}\qquad J{\frac {\partial }{\partial y^{\mu }}}=-{\frac {\partial }{\partial x^{\mu }}}}</annotation>

</semantics>

</math><img src="./d54106a8e6110899cc8ae4c9f462cd37eb9d1f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.084ex; height:6.009ex;" alt="{\displaystyle J{\frac {\partial }{\partial x^{\mu }}}={\frac {\partial }{\partial y^{\mu }}}\qquad J{\frac {\partial }{\partial y^{\mu }}}=-{\frac {\partial }{\partial x^{\mu }}}}" loading="lazy"></dd></dl>

(just like a counterclockwise rotation of π/2) or

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J{\frac {\partial }{\partial z^{\mu }}}=i{\frac {\partial }{\partial z^{\mu }}}\qquad J{\frac {\partial }{\partial {\bar {z}}^{\mu }}}=-i{\frac {\partial }{\partial {\bar {z}}^{\mu }}}.}</annotation>

</semantics>

</math><img src="./32bbbbcfa3d0d10ae15b80d2db78285ed7c66095.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.128ex; height:5.509ex;" alt="{\displaystyle J{\frac {\partial }{\partial z^{\mu }}}=i{\frac {\partial }{\partial z^{\mu }}}\qquad J{\frac {\partial }{\partial {\bar {z}}^{\mu }}}=-i{\frac {\partial }{\partial {\bar {z}}^{\mu }}}.}" loading="lazy"></dd></dl>

One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.

The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point p. In general, however, it is not possible to find coordinates so that J takes the canonical form on an entire <a href="Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> of p. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If M admits local holomorphic coordinates for J around every point then these patch together to form a <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> <a href="Atlas_(topology)" title="Atlas (topology)">atlas</a> for M giving it a complex structure, which moreover induces J. J is then said to be '<a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">integrable</a>'. If J is induced by a complex structure, then it is induced by a unique complex structure.

Given any linear map A on each tangent space of M; i.e., A is a tensor field of rank (1, 1), then the Nijenhuis tensor is a tensor field of rank (1,2) given by

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{A}(X,Y)=-A^{2}[X,Y]+A([AX,Y]+[X,AY])-[AX,AY].\,}</annotation>

</semantics>

</math><img src="./8ef85d80e47550bd281b53d4f3f076cd239317ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.035ex; height:3.176ex;" alt="{\displaystyle N_{A}(X,Y)=-A^{2}[X,Y]+A([AX,Y]+[X,AY])-[AX,AY].\,}" loading="lazy"></dd></dl>

or, for the usual case of an almost complex structure A=J such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}=-Id}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{2}=-Id}</annotation>

</semantics>

</math><img src="./726a50241d3309b088b10678de5779952b423122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.874ex; height:2.843ex;" alt="{\displaystyle J^{2}=-Id}" loading="lazy">,

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{J}(X,Y)=[X,Y]+J([JX,Y]+[X,JY])-[JX,JY].\,}</annotation>

</semantics>

</math><img src="./14341fbc9c30900b1ea4ae6c99bbdc07a505fd02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.879ex; height:2.843ex;" alt="{\displaystyle N_{J}(X,Y)=[X,Y]+J([JX,Y]+[X,JY])-[JX,JY].\,}" loading="lazy"></dd></dl>

The individual expressions on the right depend on the choice of the smooth vector fields X and Y, but the left side actually depends only on the pointwise values of X and Y, which is why NA is a tensor. This is also clear from the component formula

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -(N_{A})_{ij}^{k}=A_{i}^{m}\partial _{m}A_{j}^{k}-A_{j}^{m}\partial _{m}A_{i}^{k}-A_{m}^{k}(\partial _{i}A_{j}^{m}-\partial _{j}A_{i}^{m}).}</annotation>

</semantics>

</math><img src="./2228f4457756d152fff99aa264ccf725fd4a1eb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:55.253ex; height:3.509ex;" alt="{\displaystyle -(N_{A})_{ij}^{k}=A_{i}^{m}\partial _{m}A_{j}^{k}-A_{j}^{m}\partial _{m}A_{i}^{k}-A_{m}^{k}(\partial _{i}A_{j}^{m}-\partial _{j}A_{i}^{m}).}" loading="lazy"></dd></dl>

In terms of the <a href="Fr%C3%B6licher%E2%80%93Nijenhuis_bracket" title="Frölicher–Nijenhuis bracket">Frölicher–Nijenhuis bracket</a>, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor NA is just one-half of [A, A].

The Newlander–Nirenberg theorem states that an almost complex structure J is integrable if and only if NJ = 0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.

There are several other criteria which are equivalent to the vanishing of the Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature):

<ul><li>The Lie bracket of any two (1, 0)-vector fields is again of type (1, 0)</li>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=\partial +{\bar {\partial }}}</annotation>

</semantics>

</math><img src="./264aeafeda37713079c1295ca1f906690de91a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.961ex; height:2.843ex;" alt="{\displaystyle d=\partial +{\bar {\partial }}}" loading="lazy"></li>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\partial }}^{2}=0.}</annotation>

</semantics>

</math><img src="./fca936f401cc3c22d5a52b9ed4d818f8fcdbbfa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.45ex; height:3.009ex;" alt="{\displaystyle {\bar {\partial }}^{2}=0.}" loading="lazy"></li></ul>

Any of these conditions implies the existence of a unique compatible complex structure.

The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether S6 admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For <a href="Real-analytic" class="mw-redirect" title="Real-analytic">real-analytic</a> J, the Newlander–Nirenberg theorem follows from the <a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius theorem</a>; for C∞ (and less smooth) J, analysis is required (with more difficult techniques as the regularity hypothesis weakens).

<div class="mw-heading mw-heading2"><h2 id="Compatible_triples">Compatible triples</h2></div>

Suppose M is equipped with a <a href="Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a> ω, a <a href="Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> g, and an almost complex structure J. Since ω and g are <a href="Degenerate_form" class="mw-redirect" title="Degenerate form">nondegenerate</a>, each induces a bundle isomorphism TM → T*M, where the first map, denoted φω, is given by the <a href="Interior_product" title="Interior product">interior product</a> φω(u) = iuω = ω(u, •) and the other, denoted φg, is given by the analogous operation for g. With this understood, the three structures (g, ω, J) form a compatible triple when each structure can be specified by the two others as follows:

In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified. For example, ω and J are compatible if and only if ω(•, J•) is a Riemannian metric. The bundle on M whose sections are the almost complex structures compatible to ω has contractible fibres: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.

Using elementary properties of the symplectic form ω, one can show that a compatible almost complex structure J is an <a href="Almost_K%C3%A4hler_manifold" class="mw-redirect" title="Almost Kähler manifold">almost Kähler structure</a> for the Riemannian metric ω(u, Jv). Also, if J is integrable, then (M, ω, J) is a <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a>.

These triples are related to the <a href="Unitary_group#2-out-of-3_property" title="Unitary group">2 out of 3 property of the unitary group</a>.

<div class="mw-heading mw-heading2"><h2 id="Generalized_almost_complex_structure">Generalized almost complex structure</h2></div>

<a href="Nigel_Hitchin" title="Nigel Hitchin">Nigel Hitchin</a> introduced the notion of a <a href="Generalized_almost_complex_structure" class="mw-redirect" title="Generalized almost complex structure">generalized almost complex structure</a> on the manifold M, which was elaborated in the doctoral dissertations of his students Marco Gualtieri and Gil Cavalcanti. An ordinary almost complex structure is a choice of a half-dimensional <a href="Linear_subspace" title="Linear subspace">subspace</a> of each fiber of the complexified <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> TM. A generalized almost complex structure is a choice of a half-dimensional <a href="Isotropic_manifold" class="mw-redirect" title="Isotropic manifold">isotropic</a> subspace of each fiber of the <a href="Direct_sum_of_vector_bundles" class="mw-redirect" title="Direct sum of vector bundles">direct sum</a> of the complexified tangent and <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundles</a>. In both cases one demands that the direct sum of the <a href="Subbundle" title="Subbundle">subbundle</a> and its <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> yield the original bundle.

An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the <a href="Lie_derivative" title="Lie derivative">Lie bracket</a>. A generalized almost complex structure integrates to a <a href="Generalized_complex_structure" title="Generalized complex structure">generalized complex structure</a> if the subspace is closed under the <a href="Courant_bracket" title="Courant bracket">Courant bracket</a>. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing <a href="Pure_spinor" title="Pure spinor">pure spinor</a> then M is a <a href="Generalized_Calabi%E2%80%93Yau_manifold" class="mw-redirect" title="Generalized Calabi–Yau manifold">generalized Calabi–Yau manifold</a>.

<ul><li><a href="Almost_quaternionic_manifold" class="mw-redirect" title="Almost quaternionic manifold">Almost quaternionic manifold</a> – Concept in geometryPages displaying short descriptions of redirect targets</li>

<li><a href="Chern_class" title="Chern class">Chern class</a> – Characteristic classes of vector bundles</li>

<li><a href="Fr%C3%B6licher%E2%80%93Nijenhuis_bracket" title="Frölicher–Nijenhuis bracket">Frölicher–Nijenhuis bracket</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a> – Manifold with Riemannian, complex and symplectic structure</li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson manifold</a> – Mathematical structure in differential geometry</li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza manifold</a></li>

<li><a href="Symplectic_manifold" title="Symplectic manifold">Symplectic manifold</a> – Type of manifold in differential geometry</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFVan_de_Ven1966" class="citation journal cs1">Van de Ven, A. (June 1966). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC224368">"On the Chern numbers of certain complex and almost complex manifolds"</a>. <a href="Proceedings_of_the_National_Academy_of_Sciences" class="mw-redirect" title="Proceedings of the National Academy of Sciences">Proceedings of the National Academy of Sciences</a>. 55 (6): 1624–1627. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966PNAS...55.1624V">1966PNAS...55.1624V</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.55.6.1624">10.1073/pnas.55.6.1624</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC224368">224368</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578639">16578639</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFAgricolaBazzoniGoertschesKonstantis2018" class="citation journal cs1"><a href="Ilka_Agricola" title="Ilka Agricola">Agricola, Ilka</a>; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications. 57: 1–9. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1708.01068">1708.01068</a>. <a href="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="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>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFNewlanderNirenberg1957" class="citation journal cs1">Newlander, August; <a href="Louis_Nirenberg" title="Louis Nirenberg">Nirenberg, Louis</a> (1957). "Complex analytic coordinates in almost complex manifolds". <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. Second Series. 65 (3): 391–404. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970051">10.2307/1970051</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970051">1970051</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0088770">0088770</a>.</cite></li>

<li><cite id="CITEREFCannas_da_Silva2001" class="citation book cs1"><a href="Ana_Cannas_da_Silva" title="Ana Cannas da Silva">Cannas da Silva, Ana</a> (2001). Lectures on Symplectic Geometry. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-42195-5</bdi>.</cite> Information on compatible triples, Kähler and Hermitian manifolds, etc.</li>

<li><cite id="CITEREFWells1980" class="citation book cs1"><a href="Raymond_O._Wells_Jr." title="Raymond O. Wells Jr.">Wells, Raymond O.</a> (1980). Differential Analysis on Complex Manifolds. New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90419-0</bdi>.</cite> Short section which introduces standard basic material.</li>

<li><cite id="CITEREFRubei2014" class="citation book cs1">Rubei, Elena (2014). Algebraic Geometry, a concise dictionary. Berlin/Boston: Walter De Gruyter. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-11-031622-3</bdi>.</cite></li>

<li><cite id="CITEREFBorelSerre1953" class="citation journal cs1"><a href="Armand_Borel" title="Armand Borel">Borel, Armand</a>; <a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1953). "Groupes de Lie et puissances réduites de Steenrod". <a href="American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a>. 75 (3): 409–448. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2372495">10.2307/2372495</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2372495">2372495</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0058213">0058213</a>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary,_List,_Category)274" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Manifolds_(Glossary,_List,_Category)274" style="font-size:114%;margin:0 4em"><a href="Manifold" title="Manifold">Manifolds</a> (<a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="List_of_manifolds" title="List of manifolds">List</a>, Category)</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="Topological_manifold" title="Topological manifold">Topological manifold</a>

<ul><li><a href="Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li>

<li><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a>

<ul><li><a href="Differential_structure" title="Differential structure">Differential structure</a></li>

<li><a href="Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li>

<li><a href="Submanifold" title="Submanifold">Submanifold</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li>

<li><a href="Smoothness" title="Smoothness">Smooth map</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="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 (list)</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="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li>

<li><a href="Darboux's_theorem" title="Darboux's theorem">Darboux's</a></li>

<li><a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li>

<li><a href="Noether's_theorem" title="Noether's theorem">Noether's</a></li>

<li><a href="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="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="Differentiable_curve" title="Differentiable curve">Curve</a></li>

<li><a href="Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a>

<ul><li><a href="Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a>

<ul><li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li>

<li><a href="Foliation" title="Foliation">Foliation</a></li>

<li><a href="Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li>

<li><a href="Integral_curve" title="Integral curve">Integral curve</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Closed_manifold" title="Closed manifold">Closed</a></li>

<li><a href="Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li>

<li><a href="Complete_manifold" title="Complete manifold">Complete</a></li>

<li>() <a href="Complex_manifold" title="Complex manifold">Complex</a></li>

<li>(<a href="Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li>

<li><a href="Fibered_manifold" title="Fibered manifold">Fibered</a></li>

<li><a href="Finsler_manifold" title="Finsler manifold">Finsler</a></li>

<li>(<a href="Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="Flat_manifold" title="Flat manifold">Flat</a></li>

<li><a href="G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li>

<li><a href="Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li>

<li><a href="Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li>

<li><a href="Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li>

<li><a href="Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li>

<li><a href="Lie_group" title="Lie group">Lie group</a>

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li>

<li><a href="Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li>

<li><a href="Nilmanifold" title="Nilmanifold">Nilmanifold</a></li>

<li><a href="Orientability" title="Orientability">Oriented</a></li>

<li><a href="Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson</a></li>

<li><a href="Prime_manifold" title="Prime manifold">Prime</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li>

<li><a href="Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li>

<li>(<a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-</a>, <a href="Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-</a>) <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza</a></li>

<li><a href="Stein_manifold" title="Stein manifold">Stein</a></li>

<li>(<a href="Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li>

<li><a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a>

<ul><li><a href="Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li>

<li><a href="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="Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li>

<li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Cotangent_space" title="Cotangent space">Cotangent space</a>

<ul><li><a href="Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a>

<ul><li><a href="Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Interior_product" title="Interior product">Interior product</a></li>

<li><a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a>

<ul><li><a href="Tensor_density" title="Tensor density">density</a></li></ul></li>

<li><a href="Volume_form" title="Volume form">Volume form</a></li>

<li><a href="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="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="Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li>

<li><a href="Affine_bundle" title="Affine bundle">Affine</a></li>

<li><a href="Associated_bundle" title="Associated bundle">Associated</a></li>

<li><a href="Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber</a></li>

<li>(<a href="Cofibration" title="Cofibration">Co-</a>) <a href="Fibration" title="Fibration">Fibration</a></li>

<li><a href="Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li>

<li>(<a href="Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="Normal_bundle" title="Normal bundle">Normal</a></li>

<li><a href="Principal_bundle" title="Principal bundle">Principal</a></li>

<li><a href="Spinor_bundle" title="Spinor bundle">Spinor</a></li>

<li><a href="Subbundle" title="Subbundle">Subbundle</a></li>

<li><a href="Tangent_bundle" title="Tangent bundle">Tangent</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor</a></li>

<li><a href="Vector_bundle" title="Vector bundle">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Affine_connection" title="Affine connection">Affine</a></li>

<li><a href="Cartan_connection" title="Cartan connection">Cartan</a></li>

<li><a href="Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li>

<li><a href="Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li>

<li><a href="Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li>

<li><a href="Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li>

<li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li>

<li><a href="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="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li>

<li><a href="Morse_theory" title="Morse theory">Morse theory</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame</a></li>

<li><a href="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="Banach_manifold" title="Banach manifold">Banach manifold</a></li>

<li><a href="Diffeology" title="Diffeology">Diffeology</a></li>

<li><a href="Diffiety" title="Diffiety">Diffiety</a></li>

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="K-theory" title="K-theory">K-theory</a></li>

<li><a href="Orbifold" title="Orbifold">Orbifold</a></li>

<li><a href="Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a>

<ul><li><a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li>

<li><a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li>

<li><a href="Stratifold" title="Stratifold">Stratifold</a></li>

<li><a href="Supermanifold" title="Supermanifold">Supermanifold</a></li>

<li><a href="Stratified_space" title="Stratified space">Stratified space</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-18" href="https://en.wikipedia.org/wiki/?title=Almost_complex_manifold&oldid=1281128267">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>