# Connection is module structure over connection algebra

From Diffgeom

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...)

## Statement

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

## Definitions used

### Connection

`Further information: Connection`

### Connection algebra

`Further information: Connection algebra`

## Proof

We start with a connection on and show how naturally equips with the structure of a module over .

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