Hermitian connection

is a connection on a Hermitian vector bundle

, meaning that for all smooth vector fields

is a complex manifold, and the Hermitian vector bundle

is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator

This is called the Chern connection on

For details, see Hermitian metrics on a holomorphic vector bundle.

In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection coincides with the Levi-Civita connection of the associated Riemannian metric.

This Riemannian geometry-related article is a stub.