Connection (vector Bundle) - Vector-valued Forms

Vector-valued Forms

Let E → M be a vector bundle. An E-valued differential form of degree r is a section of the tensor product bundle E ⊗ ΛrT*M. The space of such forms is denoted by

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

In this notation a connection on E → M is a linear map

A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms. In fact, given a connection ∇ on E there is a unique way to extend ∇ to a covariant exterior derivative or exterior covariant derivative

Unlike the ordinary exterior derivative one need not have (d∇)2 = 0. In fact, (d∇)2 is directly related to the curvature of the connection ∇ (see below).