Vipul: Created page with '==Statement== ===Statement in the language of connections=== Suppose $M$ is a differential manifold, $E$ is a vector bundle over $M$ , a…'

2009-07-24T22:15:50Z

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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==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>, and <math>\nabla</math> is a [[connection on a vector bundle|connection]] on <math>M</math>. Suppose <math>E^*</math> is the [[dual bundle]] to <math>E</math> and <math>\nabla^*</math> is the [[fact about::dual connection]] to <math>\nabla</math>.

Then, if <math>\nabla</math> is a [[fact about::flat connection]] (i.e., the [[Riemann curvature tensor]] <math>R_\nabla</math> is identically zero), <math>\nabla^*</math> is also a flat connection.

===Statement in the language of modules over the connection algebra===

{{fillin}}

==Facts used==

# [[uses::Formula for curvature of dual connection]]

Dual connection to flat connection is flat - Revision history

Vipul: Created page with '==Statement== ===Statement in the language of connections=== Suppose M is a differential manifold, E is a vector bundle over M, a…'

Vipul: Created page with '==Statement== ===Statement in the language of connections=== Suppose $M$ is a differential manifold, $E$ is a vector bundle over $M$ , a…'