Almost Hermitian manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: unitary group

Definition

An almost Hermitian manifold is the following data:

• A differential manifold $M$
• A Riemannian metric $g$ on $M$
• A complex structure viz a map $J:TM\to TM$ such that $J^2=-I$ with the further property that $g(Jv,w)+g(v,Jw)=0$

Note that this gives a reduction of the structure group to the unitary group over complex numbers.

Relation with other structures

Stronger structures

• Hermitian manifold

Weaker structures

• Almost complex manifold
• Almost symplectic manifold
• Riemannian manifold