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 ${\displaystyle E}$ over a smooth manifold ${\displaystyle M}$ is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be viewed as a smooth global section ${\displaystyle h}$ of the vector bundle ${\displaystyle (E\otimes {\overline {E}})^{*}}$ such that for every point ${\displaystyle p}$ in ${\displaystyle M}$, $${\displaystyle h_{p}{\mathord {\left(\eta ,{\bar {\zeta }}\right)}}={\overline {h_{p}{\mathord {\left(\zeta ,{\bar {\eta }}\right)}}}}}$$ for all ${\displaystyle \zeta }$, ${\displaystyle \eta }$ in the fiber ${\displaystyle E_{p}}$ and $${\displaystyle h_{p}{\mathord {\left(\zeta ,{\bar {\zeta }}\right)}}>0}$$ for all nonzero ${\displaystyle \zeta }$ in ${\displaystyle E_{p}}$.

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 ${\displaystyle (z^{\alpha })}$ as $${\displaystyle h=h_{\alpha {\bar {\beta }}}\,dz^{\alpha }\otimes d{\bar {z}}^{\beta }}$$ where ${\displaystyle h_{\alpha {\bar {\beta }}}}$ 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: $${\displaystyle g={1 \over 2}\left(h+{\bar {h}}\right).}$$

The form g is a symmetric bilinear form on TM^C, the complexified tangent bundle. Since g is equal to its conjugate it is the complexification of a real form on TM. The symmetry and positive-definiteness of g on TM follow from the corresponding properties of h. In local holomorphic coordinates the metric g can be written $${\displaystyle g={1 \over 2}h_{\alpha {\bar {\beta }}}\,\left(dz^{\alpha }\otimes d{\bar {z}}^{\beta }+d{\bar {z}}^{\beta }\otimes dz^{\alpha }\right).}$$