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).
Read more about this topic: Connection (vector Bundle)
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