nLab
almost Hermitian structure

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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)GL(n,)U(n) \hookrightarrow GL(n,\mathbb{C}) of the unitary group into the complex general linear group.

Under further embedding U(n)GL(n,)GL(2n,)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)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)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)O(2n)×GL(2n,)Sp(2n,)×GL(2n,)GL(n,)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)GL(2n,)U(n) \hookrightarrow GL(2n,\mathbb{R}) factors through the symplectic group via the maximal compact subgroup inclusion

U(n)Sp(2n,)GL(2n,) 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.

Revised on January 22, 2015 01:06:37 by Urs Schreiber (88.100.66.95)