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In mathematics, a '''Hermitian connection <math>\nabla</math>''', 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
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# Its (0, 1)-part coincides with the Cauchy-Riemann operator associated to the holomorphic structure.
# 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''.
 
 
 
[[Category:Complex manifolds]]
[[Category:Structures on manifolds]]
[[Category:Riemannian geometry]]
 
 
{{differential-geometry-stub}}

Latest revision as of 02:49, 14 September 2014

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