Hermitian connection explained

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 ⋅ , ⋅ \rangle on E , meaning thatv \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. .