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Concept
Hermitian manifold
Summary
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.
A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields a unitary structure (U(n) structure) on the manifold. By dropping this condition, we get an almost Hermitian manifold.
On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it is closed (i.e., it is a symplectic form), we get an almost Kähler structure. If both the almost complex structure and the fundamental form are integrable, then we have a Kähler structure.
A Hermitian metric on a complex vector bundle E over a smooth manifold M is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be viewed as a smooth global section h of the vector bundle such that for every point p in M,
for all ζ, η in the fiber Ep and
for all nonzero ζ in Ep.
A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent bundle. Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent bundle.
On a Hermitian manifold the metric can be written in local holomorphic coordinates (zα) as
where are the components of a positive-definite Hermitian matrix.
A Hermitian metric h on an (almost) complex manifold M defines a Riemannian metric g on the underlying smooth manifold. The metric g is defined to be the real part of h:
The form g is a symmetric bilinear form on TMC, the complexified tangent bundle.
Official source
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