# Hermitian connection

In mathematics, a **Hermitian connection **, is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric. If the base manifold is a complex manifold, and the Hermitian vector bundle admits a holomorphic structure, then there is a canonical Hermitian connection, which is called the **Chern connection** which satisfies the following conditions

- Its (0, 1)-part coincides with the Cauchy-Riemann operator associated to the holomorphic structure.
- Its curvature form is a (1, 1)-form.

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*.

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