Difference between revisions of "Dual connection to flat connection is flat"

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(Created page with '==Statement== ===Statement in the language of connections=== Suppose <math>M</math> is a differential manifold, <math>E</math> is a vector bundle over <math>M</math>, a…')
 
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Latest revision as of 22:15, 24 July 2009

Statement

Statement in the language of connections

Suppose M is a differential manifold, E is a vector bundle over M, and \nabla is a connection on M. Suppose E^* is the dual bundle to E and \nabla^* is the dual connection to \nabla.

Then, if \nabla is a flat connection (i.e., the Riemann curvature tensor R_\nabla is identically zero), \nabla^* is also a flat connection.

Statement in the language of modules over the connection algebra

Fill this in later

Facts used

  1. Formula for curvature of dual connection