Hermitian connection

In mathematics, a Hermitian connection $\mathrm{\nabla }$ is a connection on a Hermitian vector bundle $E$ over a smooth manifold $M$ which is compatible with the Hermitian metric $⟨\cdot ,\cdot ⟩$ on $E$, meaning that

$v⟨s,t⟩=⟨{\mathrm{\nabla }}_{v}s,t⟩+⟨s,{\mathrm{\nabla }}_{v}t⟩$

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 ${\overline{\partial }}_E$ 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.