# Almost Hermitian structure gives symplectic form

## Statement

Let $M$ be a differential manifold, $J$ be an almost complex structure on $M$ (viz a smooth choice of map from each tangent space to itself such that $J^2 = -I$). Let $g$ be a Riemannian metric on $M$, such that:

$g(v,Jw) + g(Jv,w) = 0$

The tripe $(M,g,J)$ is an almost Hermitian structure on $M$. This gives rise to the following almost symplectic structure: the bilinear form on a tangent space is defined as:

$b(v,w) = g(v,Jw)$

## Interpretation in terms of structure groups

An almost Hermitian structure is a reduction ofthe structure group of the $2n$-dimensional manifold to $U(n,\mathbb{C})$, while an almost symplectic structure is a reduction of the stucture group to $Sp(2n,\mathbb{R})$. Since $U(n,\mathbb{C})$ is a subgroup of $Sp(2n,\R)$ an almost Hermitian structure gives an almost symplectic structure.