A Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each holomorphic tangent space. Hermitian manifolds were introduced by Cartan in 1922.
A Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each holomorphic tangent space. Hermitian manifolds were introduced by Cartan in 1922.
On a Hermitian manifold the metric can be written in local holomorphic coordinates ($z^\alpha$) as
$$h = h_{\alpha\bar\beta}dz^{\alpha}\otimes d\bar z^\beta$$
where $h_{\alpha\bar\beta}$ are the components of a positive-definite Hermitian matrix.
