Hermitian connection
In mathematics, a Hermitian connection is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric
on, meaning that
for all smooth vector fields and all smooth sections of.
If is a complex manifold, and the Hermitian vector bundle on is equipped with a holomorphic structure, then there is a unique Hermitian connection whose -part coincides with the Dolbeault operator on associated to the holomorphic structure.
This is called the Chern connection on. The curvature of the Chern connection is a -form. 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.
