Properties
Every (almost) complex manifold admits a Hermitian metric. This follows directly from the analogous statement for Riemannian metric. Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g′ compatible with the almost complex structure J in an obvious manner:
Choosing a Hermitian metric on an almost complex manifold M is equivalent to a choice of U(n)-structure on M; that is, a reduction of the structure group of the frame bundle of M from GL(n,C) to the unitary group U(n). A unitary frame on an almost Hermitian manifold is complex linear frame which is orthonormal with respect to the Hermitian metric. The unitary frame bundle of M is the principal U(n)-bundle of all unitary frames.
Every almost Hermitian manifold M has a canonical volume form which is just the Riemannian volume form determined by g. This form is given in terms of the associated (1,1)-form ω by
where ωn is the wedge product of ω with itself n times. The volume form is therefore a real (n,n)-form on M. In local holomorphic coordinates the volume form is given by
Read more about this topic: Hermitian Manifold
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