Hermitian structure

on a manifold

A pair consisting of a complex structure on and a Hermitian metric in the tangent space , that is, a Riemannian metric that is invariant under :

for any vector fields and on . A Hermitian structure specifies in any tangent space the structure of a Hermitian vector space (see Hermitian metric). A manifold with a Hermitian structure is called a Hermitian manifold. A Hermitian structure defines on a differential -form , which is called the canonical -form of the Hermitian manifold. Any structure on can be completed by some Riemannian metric to a Hermitian structure : for one can take the metric , where is an arbitrary metric. The canonical Hermitian connection of a Hermitian metric can be regarded as an affine connection with torsion on relative to which the fields and are parallel. Among all affine connections satisfying these conditions it is uniquely characterized by the identity , which is valid for its torsion tensor and any vector fields and . The curvature tensor of the canonical connection satisfies the condition . A Hermitian manifold is a Kähler manifold if and only if the canonical Hermitian connection has no torsion and hence is the same as the Levi-Civita connection of .

A natural generalization of the concept of a Hermitian structure is that of an almost-Hermitian structure, which is a pair consisting of an almost-complex structure on and a Riemannian metric that is invariant under . If the fundamental -form is closed, then an almost-Hermitian structure is called almost Kählerian. The specification of an almost-Hermitian structure is equivalent to a reduction of the structure group of the tangent bundle to the group , where . Any non-degenerate differential -form on a manifold is the fundamental -form of some almost-Hermitian structure.

For references see Hermitian metric.