Hermitian metric on a vector bundle

Summary

A Hermitian metric on a complex vector bundle is a choice of Hermitian inner product on each fibre that varies smoothly with the base point. It lets one measure the length of sections and define the Chern connection and its curvature. Every complex vector bundle on a paracompact manifold has one, by a partition of unity. A Hermitian metric on the holomorphic tangent bundle of a complex manifold is what makes it a Hermitian manifold.