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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

\( {\displaystyle v\langle s,t\rangle =\langle \nabla _{v}s,t\rangle +\langle s,\nabla _{v}t\rangle } \)

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 \( {\displaystyle {\bar {\partial }}_{E}} \) 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.
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.

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