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In mathematics, a '''Hermitian connection <math>\nabla</math>''' is a connection on a Hermitian vector bundle <math>E</math> over a smooth manifold <math>M</math> which is compatible with the Hermitian metric
<math>\langle \cdot, \cdot \rangle</math> on <math>E</math>, meaning that
:<math>  v \langle s,t\rangle = \langle \nabla_v s, t \rangle + \langle s, \nabla_v t \rangle </math>
for all smooth vector fields <math>v</math> and all smooth sections <math>s,t</math> of <math>E</math>.

If <math>X</math> is a [[Complex manifold|complex manifold]], and the Hermitian vector bundle <math>E</math> on <math>X</math> is equipped with a [[Holomorphic vector bundle|holomorphic structure]], then there is a unique Hermitian connection whose (0, 1)-part coincides with the [[Holomorphic vector bundle#Dolbeault operators|Dolbeault operator]] <math>\bar{\partial}_E</math> on <math>E</math> associated to the holomorphic structure.
This is called the '''Chern connection''' on <math>E</math>.  The curvature of the Chern connection is a (1, 1)-form.  For details, see [[Holomorphic vector bundle#Hermitian metrics on a holomorphic vector bundle|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.

== References ==
* Shiing-Shen Chern, ''Complex Manifolds Without Potential Theory''. 
* Shoshichi Kobayashi, ''Differential geometry of complex vector bundles''. Publications of the Mathematical Society of Japan, 15. ''Princeton University Press, Princeton, NJ'', 1987. xii+305 pp. {{ISBN|0-691-08467-X}}.

[[Category:Complex manifolds]]
[[Category:Structures on manifolds]]
[[Category:Riemannian geometry]]



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