Vector-Valued Differential Forms

  • Luis A. Cordero
  • C. T. J. Dodson
  • Manuel de León
Part of the Mathematics and Its Applications book series (MAIA, volume 47)


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 MTV. 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.


Vector Field Vector Bundle Linear Representation Tensor Field Canonical Representation 
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Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Luis A. Cordero
    • 1
  • C. T. J. Dodson
    • 2
  • Manuel de León
    • 3
  1. 1.Department of Geometry and TopologyUniversity of Santiago de CompostelaSpain
  2. 2.Department of MathematicsUniversity of LancasterUK
  3. 3.C.E.C.I.M.E.-C.S.I.C.MadridSpain

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