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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“Thy love is such I can no way repay,
The heavens reward thee manifold I pray.
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That when we live no more, we may live ever.”
—Anne Bradstreet (c. 1612–1672)