A hermitian metric on a complex vector bundle E over a smooth manifold M, is a positive-definite, hermitian inner product on each fiber E_p, that varies smoothly with the point p in M.

An important special case is that of a hermitian metric on the complexified tangent bundle of a complex manifold M. This is the hermitian analogue of a Riemannian metric. Every complex manifold admits such a hermitian metric.

See also: Kähler metric