Hermitian connection

In mathematics, a Hermitian connection formula_1 is a connection on a Hermitian vector bundle formula_2 over a smooth manifold formula_3 which is compatible with the Hermitian metric
formula_4 on formula_2, meaning that
for all smooth vector fields formula_7 and all smooth sections formula_8 of formula_2.
If formula_10 is a complex manifold, and the Hermitian vector bundle formula_2 on formula_10 is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator formula_13 on formula_2 associated to the holomorphic structure.
This is called the Chern connection on formula_2. The curvature of the Chern connection is a (1, 1)-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.