Almost complex manifold

In mathematics, the notion of almost complex manifold is a weakening of the concept of complex manifold. While complex manifolds look like the complex space locally, do so almost complex only " infinitesimal ", ie the tangent spaces are complex (on each other acceptable way ) vector spaces. To make a real vector space to a complex, you must define what should be the product of a vector with the imaginary unit. This is the task of mapping in the case of the tangent space.

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 at each point is a bijective linear map, the

Met. ( This corresponds to the equality. )

Almost complex manifold

An almost complex manifold is a smooth manifold on together with an almost complex structure.

Properties

  • Let and be two almost complex manifolds with almost complex structures and the respective. A continuously differentiable mapping is called holomorphic (or pseudo- holomorphic ) if the push forward of the almost complex structures and is compatible, ie, it must
  • A complex manifold is automatically also an almost complex. Due to the complex structure of the tangent spaces are complex vector spaces and is defined for an almost complex structure. Conversely needs an almost complex manifold in general do not have complex structure. But if it is an atlas map with which the target area is a complex vector space and which are in accordance with the almost holomorphic complex structure, then this is a complex Atlas Atlas induces nearly complex structure. It can therefore be complex manifolds also define as almost complex manifolds which possess a holomorphic atlas.

Integrability

Almost a complex structure is integrated, if it has a holomorphic Atlas which is a complex structure. The set of Newlander - Nirenberg says that an almost complex structure is integrable if the Nijenhuis tensor vanishes.

Examples

  • For every natural number n there are complex structures on R2n, for example, ( 1 ≤ i, j ≤ 2n ) for odd i, for i just
  • Almost complex structures, there is only on manifolds of even dimension. ( Otherwise, if at least one real eigenvalue contradict. )
  • In the real two-dimensional ( ie, in the complex - dimensional ) any almost complex manifold is a complex manifold, ie a Riemann surface. This can be demonstrated by the release of the Beltrami equation.
  • The only spheres with almost complex structures are and. The well-known almost complex structure is not integrable. It is not known whether there on the complex structure.
  • Every symplectic manifold is almost complex.

Hermitian metric

A Hermitian metric on an almost complex manifold is an invariant Riemannian metric, ie

The 2- form

Is called the fundamental 2- form of the almost - Hermitian manifold. is called almost - Kählersch when.

Is called Hermitian manifold if it is integrable. A Hermitian manifold with a Kählermannigfaltigkeit.

Pushforward (differential) Quasiconformal mapping Hans Grauert
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