Hermitian connection

In mathematics, a Hermitian connection $$\nabla$$ is a connection on a Hermitian vector bundle $$E$$ over a smooth manifold $$M$$ which is compatible with the Hermitian metric $$\langle \cdot, \cdot \rangle$$ on $$E$$, meaning that
 * $$ v \langle s,t\rangle = \langle \nabla_v s, t \rangle + \langle s, \nabla_v t \rangle $$

for all smooth vector fields $$v$$ and all smooth sections $$s,t$$ of $$E$$.

If $$X$$ is a complex manifold, and the Hermitian vector bundle $$E$$ on $$X$$ is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator $$\bar{\partial}_E$$ on $$E$$ associated to the holomorphic structure. This is called the Chern connection on $$E$$. 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.