# Almost Hermitian structure gives symplectic form

From Diffgeom

## Statement

Let be a differential manifold, be an almost complex structure on (viz a smooth choice of map from each tangent space to itself such that ). Let be a Riemannian metric on , such that:

The tripe is an almost Hermitian structure on . This gives rise to the following almost symplectic structure: the bilinear form on a tangent space is defined as:

## Interpretation in terms of structure groups

An almost Hermitian structure is a reduction ofthe structure group of the -dimensional manifold to , while an almost symplectic structure is a reduction of the stucture group to . Since is a subgroup of an almost Hermitian structure gives an almost symplectic structure.