almost Hermitian structure


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Complex geometry



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.


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 (