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.

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.