A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every fiber bundle. The basic example is the trivial bundle , where is an open set in . Then a positive definite Hermitian matrix defines a Hermitian metric by
where is the complex conjugate of . By a partition of unity, any complex vector bundle has a Hermitian metric.
In the special case of a complex manifold, the complexified tangent bundle may have a Hermitian metric, in which case its real part is a Riemannian metric and its imaginary part is a nondegenerate alternating multilinear form . When is closed, i.e., in this case a symplectic form, then is a Kähler form.
On a holomorphic vector bundle with a Hermitian metric , there is a unique connection compatible with and the complex structure. Namely, it must be , where in a trivialization.