Almost Hermitian structure gives symplectic form

From Diffgeom
Jump to: navigation, search

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.