Differential Geometry of Frame Bundles pp 39-55 | Cite as

# Vector-Valued Differential Forms

## Abstract

We propose to build on the development in Sections 1.3 and 1.4 concerning the object *J* _{ p } ^{1} *V* for a vector space *V*. As a vector space, it is isomorphic to (*p* + 1) copies of *V* and it induces a canonical embedding of Lie groups (cf. 1.4): *j*_{ p }: *J* _{ p } ^{1} *Gl*(*V*)→ *Gl*(*J* _{ p } ^{1} *V*). This embedding allows a lifting of linear representation of a Lie group *G* in *V* by the functor *J* _{ p } ^{1} . As candidates for *V*, we are particularly interested in tensor products of **R**^{ n } and its dual; these being the fibre types of the geometrically interesting tangent tensor bundles on an *n*-dimensional manifold. A *V*-valued function on a manifold *M* is a section of *M × V;* a *V*-valued *r*-form on *M* is a section of ⋀^{ r }*M* ⊗ *TV*. We see how these are lifted by *J* _{ p } ^{1} . Similarly, we consider the lifting of *V*-valued functions on *FM* and their associated *G-*structures.

## Keywords

Vector Field Vector Bundle Linear Representation Tensor Field Canonical Representation## Preview

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