Editing Connection (vector bundle) (section)

==Exterior covariant derivative and vector-valued forms==

{{See also|Exterior covariant derivative}}

Let <math>E\to M</math> be a vector bundle. An [[vector-valued form|<math>E</math>-valued differential form]] of degree <math>r</math> is a section of the [[tensor product]] bundle:

:<math>\bigwedge^rT^*M \otimes E.</math>

The space of such forms is denoted by

:<math>\Omega^r(E) = \Omega^r(M;E) = \Gamma \left (\bigwedge^rT^*M  \otimes E \right ) = \Omega^r(M) \otimes_{C^{\infty}(M)} \Gamma(E),</math>
where the last tensor product denotes the tensor product of [[module (mathematics)|modules]] over the [[ring (mathematics)|ring]] of smooth functions on <math>M</math>.

An <math>E</math>-valued 0-form is just a section of the bundle <math>E</math>. That is,

:<math>\Omega^0(E) = \Gamma(E).</math>

In this notation a connection on  <math>E\to M</math> is a linear map

:<math>\nabla:\Omega^0(E) \to \Omega^1(E).</math>

A connection may then be viewed as a generalization of the [[exterior derivative]] to vector bundle valued forms. In fact, given a connection <math>\nabla</math> on <math>E</math> there is a unique way to extend <math>\nabla</math> to an '''[[exterior covariant derivative]]'''

:<math>d_{\nabla}: \Omega^r(E) \to \Omega^{r+1}(E).</math>

This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the form <math>\omega\otimes s</math> and extended linearly:

:<math>d_\nabla (\omega \otimes s) = d\omega \otimes s + (-1)^{\deg \omega} \omega \wedge \nabla s</math>

where <math>\omega \in \Omega^r(M)</math> so that <math>\deg \omega = r</math>, <math>s\in \Gamma(E)</math> is a section, and <math>\omega \wedge \nabla s</math> denotes the <math>(r+1)</math>-form with values in <math>E</math> defined by wedging <math>\omega</math> with the one-form part of <math>\nabla s</math>. Notice that for <math>E</math>-valued 0-forms, this recovers the normal Leibniz rule for the connection <math>\nabla</math>.

Unlike the ordinary exterior derivative, one generally has <math>d_{\nabla}^2 \ne 0</math>. In fact, <math>d_{\nabla}^2</math> is directly related to the curvature of the connection <math>\nabla</math> (see [[#Curvature|below]]).