Hermitian connection
In mathematics, a Hermitian connection [math]\displaystyle{ \nabla }[/math] is a connection on a Hermitian vector bundle [math]\displaystyle{ E }[/math] over a smooth manifold [math]\displaystyle{ M }[/math] which is compatible with the Hermitian metric [math]\displaystyle{ \langle \cdot, \cdot \rangle }[/math] on [math]\displaystyle{ E }[/math], meaning that
- [math]\displaystyle{ v \langle s,t\rangle = \langle \nabla_v s, t \rangle + \langle s, \nabla_v t \rangle }[/math]
for all smooth vector fields [math]\displaystyle{ v }[/math] and all smooth sections [math]\displaystyle{ s,t }[/math] of [math]\displaystyle{ E }[/math].
If [math]\displaystyle{ X }[/math] is a complex manifold, and the Hermitian vector bundle [math]\displaystyle{ E }[/math] on [math]\displaystyle{ X }[/math] is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator [math]\displaystyle{ \bar{\partial}_E }[/math] on [math]\displaystyle{ E }[/math] associated to the holomorphic structure. This is called the Chern connection on [math]\displaystyle{ E }[/math]. 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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