Almost Complex Structures

Almost Complex Structures on Symplectic Manifolds

Almost Complex Structure

An almost complex structure on a smooth manifold is a smooth bundle map such that . The pair is called an almost complex manifold.

Compatible Almost Complex Structures

Let be a symplectic manifold. An almost complex structure on is said to be compatible (with , or -compatible) if and for all non-zero . That is admits a Riemannian metric on .