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

→ Main article : Newlander-Nirenberg theorem

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 . ${\displaystyle J}$ ${\displaystyle d\Omega =0}$

literature

• Klaus Fritzsche , Hans Grauert : From Holomorphic Functions to Complex Manifolds (= Graduate Texts in Mathematics . 213). Springer, New York NY et al. 2002, ISBN 0-387-95395-7 .

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 .