Hermitian Manifold

A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent space. Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent space.

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.

Read more about Hermitian Manifold:  Riemannian Metric and Associated Form, Properties, Kähler Manifolds

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