nLab almost Hermitian structure

Definition

An almost Hermitian structure a reduction of the structure group along the inclusion $U(n) \hookrightarrow GL(n,\mathbb{C})$ of the unitary group into the complex general linear group.

Under further embedding $U(n) \hookrightarrow GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$ an almost hermitian structure on the frame bundle of a smooth manifold, hence a G-structure for $G = U(n)$, is first of all the choice of an almost complex structure and then an almost Hermitian manifold structure.

An first-order intgrable $U(n)$-structure (almost Hermitian manifold) structure is Kähler manifold structure.

By the fact (see at unitary group – relation to orthogonal, symplectic and general linear group) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that an almost Hermitian structure is precisely a joint orthogonal structure, almost symplectic structure and almost complex manifold.

Properties

Relation to almost complex structure

Since the inclusion $U(n) \hookrightarrow GL(2n,\mathbb{R})$ factors through the symplectic group via the maximal compact subgroup inclusion

$$U(n) \hookrightarrow Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$$

an almost Hermitian manifold structure is in particular an almost complex structure. Conversely, since the maximal compact subgroup inclusion is a homotopy equivalence, there is no obstruction to lifting an almost complex structure to an almost Hermitian structure.

Relation to Kähler manifolds

An first-order integrable almost Hermitian structure is a Kähler manifold structure.

Related concepts

• Kähler manifold

• Hermitean Yang-Mills connection