Vector-Valued Differential Forms
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.
KeywordsVector Field Vector Bundle Linear Representation Tensor Field Canonical Representation
Unable to display preview. Download preview PDF.