Editing Connection (vector bundle) (section)

==Affine properties of the set of connections==

Every vector bundle over a manifold admits a connection, which can be proved using [[partitions of unity]]. However, connections are not unique. If <math>\nabla_1</math> and <math>\nabla_2</math> are two connections on <math>E\to M</math> then their difference is a [[Smooth function|<math>C^{\infty}(M)</math>]]-linear operator. That is,
:<math>(\nabla_1 - \nabla_2)(fs) = f(\nabla_1s - \nabla_2s)</math>
for all smooth functions <math>f</math> on <math>M</math> and all smooth sections <math>s</math> of <math>E</math>. It follows that the difference <math>\nabla_1-\nabla_2</math> can be uniquely identified with a one-form on <math>M</math> with values in the endomorphism bundle <math>\operatorname{End}(E) = E^* \otimes E</math>:
:<math>\nabla_1 - \nabla_2 \in \Omega^1(M; \mathrm{End}\,E).</math>
Conversely, if <math>\nabla</math> is a connection on <math>E</math> and <math>A</math> is a one-form on <math>M</math> with values in <math>\operatorname{End}(E)</math>, then <math>\nabla + A</math> is a connection on <math>E</math>.

In other words, the space of connections on <math>E</math> is an [[affine space]] for <math>\Omega^1(\operatorname{End}(E))</math>. This affine space is commonly denoted <math>\mathcal{A}</math>.