Almost complex structure

In differential Geometry, a almost complex structure on a differential Variété real is the data of a complex structure on each tangent space. Its importance is of size.

Formal definition

An almost complex structure J on a differential variety M is a field of endomorphisms J , id is a total section of the vectorial Fibré $End (TM)$ , checking:

\ forall X \ in M, J_x^2=-Id

A differential variety provided with an almost complex structure is called an almost complex variety.

Theorem : The existence of an almost complex structure J on a differential variety M implies that M is of even size, say 2n . Moreover, there exists a single orientation on M such as…

Therefore, so that there exists an almost complex structure, it is necessary that the variety is of even and directed size. But this condition with it-only is not enough:

Theorem : The existence of an almost complex structure on a differential variety of directional even size is equivalent to the Réduction of the structural group of fiber tangent of $GL (2n, R)$ with $GL (N, C)$ .

Examples

The Sphere S to only admit an almost complex structure is:

the sphere $S^2$ , seen like compactifié of ℂ.
the sphere $S^6$ , seen like the sphere unit of the imaginary Octonion S.

Differential forms

Linear algebra: a linear operator $A \ in GL (N, R)$ checking the identity $A^2=-Id$ is reduced on $C^n=R^n \ otimes C$ . He admits two clean spaces, $E^+$ and $E^-$ , of respective eigenvalues $i$ and $-i$ .

Almost complex structures: The differential forms are the sections of the produced external of the Fibré cotangent.

See too

Random links: