# Almost complex manifold

In mathematics , the concept of almost complex manifold is a weakening of the concept of complex manifold . While complex manifolds look locally like complex space, almost complex manifolds only do this "infinitesimally", that is, the tangent spaces are (in a mutually compatible way) complex vector spaces. In order to make a real vector space complex, one has to determine what the product of a vector with the imaginary unit should be. In the case of the tangent space, this is the task of the mapping . ${\ displaystyle i}$${\ displaystyle T_ {p} M}$${\ displaystyle J_ {p}}$

## definition

### Almost complex structure

An almost complex structure on a smooth manifold is a smooth map with the property that the restriction to the tangent space to each point is a bijective linear map, which ${\ displaystyle M}$${\ displaystyle J \ colon TM \ to TM}$${\ displaystyle J_ {p}: = J | _ {T_ {p} M}}$${\ displaystyle p \ in M}$

${\ displaystyle J_ {p} \ circ J_ {p} = - \ mathrm {id}}$

Fulfills. (This corresponds to equality .) ${\ displaystyle i ^ {2} = - 1}$

### Almost complex manifold

An almost complex manifold is a smooth manifold together with an almost complex structure . ${\ displaystyle M}$${\ displaystyle M}$

## properties

• Let and be two almost complex manifolds with the respective almost complex structures and . A continuously differentiable mapping is called holomorphic (or pseudo-holomorphic) if the pushforward of is compatible with the almost complex structures of and , that is, it must${\ displaystyle M}$${\ displaystyle N}$${\ displaystyle J_ {M}}$${\ displaystyle J_ {N}}$${\ displaystyle f \ colon M \ to N}$ ${\ displaystyle df \ colon TM \ to TN}$${\ displaystyle f}$${\ displaystyle M}$${\ displaystyle N}$
${\ displaystyle df \ circ J_ {M} = J_ {N} \ circ df}$
be valid.
• A complex manifold is automatically an almost complex one. Due to the complex structure, the tangential spaces become complex vector spaces and an almost complex structure is defined by for . Conversely, an almost complex manifold does not in general need to have a complex structure. If, however, there is an atlas with maps whose target area is a complex vector space and which are holomorphic in the sense of the almost complex structure, then this atlas is a complex atlas that induces the almost complex structure. Complex manifolds can therefore also be defined as almost complex manifolds that have a holomorphic atlas.${\ displaystyle Jv: = iv}$${\ displaystyle v \ in TM}$

## Integrability

An almost complex structure is said to be integrable if it has a holomorphic atlas, i.e. if it is a complex structure . Newlander-Nirenberg's theorem states that an almost complex structure can be integrated if and only if the Nijenhuis tensor vanishes.

## Examples

• For every natural number there are complex structures on the , for example ( ): for odd and for even .${\ displaystyle n}$${\ displaystyle \ mathbb {R} ^ {2n}}$${\ displaystyle 1 \ leq i, j \ leq 2n}$${\ displaystyle J_ {ij} = - \ delta _ {i, j-1}}$${\ displaystyle i}$${\ displaystyle J_ {ij} = \ delta _ {i, j + 1}}$${\ displaystyle i}$
• Almost complex structures only exist on manifolds of even dimension. (Otherwise at least one real eigenvalue would have contradicted .)${\ displaystyle J: TM \ rightarrow TM}$${\ displaystyle J ^ {2} = - 1}$
• In the real two-dimensional (that is, in the complex-one-dimensional) every almost complex manifold is a complex manifold, i.e. a Riemann surface . This can be shown by solving the Beltrami equation .
• The only spheres with almost complex structures are and ( Armand Borel , Jean-Pierre Serre 1953). The well-known, almost complex structure - derived from the geometry of the octonion ions - cannot be integrated into the. It is not known whether there is a complex structure on it. In general, however, it is assumed that this is not the case, although attempts have been made to construct one. Attempts to prove non-existence were made by CC Hsiung (1986) and SS Chern (2003) and in 2016 by Michael Atiyah .${\ displaystyle S ^ {2}}$${\ displaystyle S ^ {6}}$${\ displaystyle S ^ {6}}$${\ displaystyle S ^ {6}}$
• Every symplectic manifold is almost complex.

## Hermitian metric

A Hermitian metric on an almost complex manifold is an -invariant Riemannian metric , i. H. a Riemannian metric that ${\ displaystyle g}$${\ displaystyle J}$

${\ displaystyle g (JX, JY) = g (X, Y)}$

fulfilled for all . ${\ displaystyle X, Y \ in TM}$

The 2 form

${\ displaystyle \ Omega (X, Y): = g (X, JY)}$

is called the fundamental 2-form of the almost Hermitian manifold. is called almost-kählersch if . ${\ displaystyle (M, J, g)}$${\ displaystyle d \ Omega = 0}$

${\ displaystyle (M, J, g)}$is called Hermitian manifold if is integrable. A Hermitian manifold with is a Kahler manifold . ${\ displaystyle J}$${\ displaystyle d \ Omega = 0}$

## Individual evidence

1. ^ Armand Borel , Jean-Pierre Serre : Groupes de Lie et et puissances réduites de Steenrod. In: American Journal of Mathematics . Volume 75, number 3, 1953, pp. 409-448, doi : 10.2307 / 2372495 .
2. Robert Bryant : S.-S. Chern's study of almost-complex structures on the six-sphere. Arxiv 2014 .
3. Michael Atiyah : The Non-Existent Complex 6-Sphere. Arxiv 2016 .