Dual connection to flat connection is flat

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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