Almost complex manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: general linear group over complex numbers

Definition

An almost complex manifold is the following data:

• A differential manifold $M$
• At each point, a complex structure to the vector space at that point. In other words, for every point, we give a linear map $J$ such that $J^2=-I$, with $J$ varying smoothly with the point.

Relation with other structures

Stronger structures

• Complex manifold
• Kahler manifold
• Hermitian manifold

Weaker structures

• Almost symplectic manifold
• Oriented manifold