Hermitian connection
Encyclopedia
In mathematics, the Hermitian connection 
, also called the Chern connection, is the unique connection
on a Hermitian manifold
that satisfies the following conditions,
If the Hermitian manifold admits a symplectic structure, i.e., if it is a Kähler manifold
, then the Hermitian connection and the Levi-Civita connection
coincide.
, also called the Chern connection, is the unique connectionConnection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
on a Hermitian manifold
Hermitian manifold
In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each tangent space...
that satisfies the following conditions,
- It preserves the metricMetric (mathematics)In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
g, i.e.,
. - It preserves the complex structure J, i.e.,
. - The torsion is pure in its indices.
If the Hermitian manifold admits a symplectic structure, i.e., if it is a Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
, then the Hermitian connection and the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
coincide.