Connection is module structure over connection algebra

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Revision as of 21:38, 5 April 2008 by Vipul (talk | contribs) (New page: ==Statement== Let <math>E</math> be a vector bundle over a differential manifold <math>M</math>. Then, a connection on <math>E</math> is equivalent to giving <math>E</math> th...)
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Statement

Let E be a vector bundle over a differential manifold M. Then, a connection on E is equivalent to giving E the structure of a module over the connection algebra over M.

Definitions used

Connection

Further information: Connection

Connection algebra

Further information: Connection algebra

Proof

We start with a connection \nabla on E and show how \nabla naturally equips E with the structure of a module over \mathcal{C}(M).

First, observe that a connection gives a rule for the Lie algebra of first-order differential operators to act on E, hence the tensor algebra generated by it as a vector space, acts on E. We need to check that under this action m(1) - 1 acts trivially on E.