Hermitian connection

From Wikipedia, the free encyclopedia

In mathematics, a Hermitian connection \nabla , 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

  1. Its (0, 1)-part coincides with the Cauchy-Riemann operator associated to the holomorphic structure.
  2. 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.



This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.