Difference between revisions of "Formula for curvature of dual connection"

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(Created page with '==Statement== Suppose <math>M</math> is a differential manifold, <math>E</math> is a vector bundle over <math>M</math>, and <math>\nabla</math> is a [[fact about::connec…')
 
(Applications)
 
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===Applications===
 
===Applications===
  
* [[Duall connection to flat connection is flat]]
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* [[Dual connection to flat connection is flat]]

Latest revision as of 22:13, 24 July 2009

Statement

Suppose M is a differential manifold, E is a vector bundle over M, and \nabla is a connection on E. Suppose E^* is the dual bundle and \nabla^* is the dual connection to \nabla. If R_\nabla and R_{\nabla^*} denote respectively the Riemann curvature tensors of \nabla and \nabla^*, then we have:

R_{\nabla^*}(l) = s \mapsto -l(R(X,Y)(s)).

Related facts

Applications