# nLab almost Hermitian structure

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

complex geometry

# Contents

## 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.

Last revised on January 22, 2015 at 01:06:37. See the history of this page for a list of all contributions to it.