almost Hermitian structure
An almost Hermitian structure a reduction of the structure group along the inclusion of the unitary group into the complex general linear group.
Under further embedding an almost hermitian structure on the frame bundle of a smooth manifold, hence a G-structure for , is first of all the choice of an almost complex structure and then an almost Hermitian manifold structure.
An first-order intgrable -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 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 factors through the symplectic group via the maximal compact subgroup inclusion
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