MULTIVECTOR AND EXTENSOR FIELDS ON SMOOTH MANIFOLDS

Abstract

The main objective of this paper (second in a series of four) is to show how the Clifford and extensor algebras methods introduced in a previous paper of the series are indeed powerful tools for performing sophisticated calculations appearing in the study of the differential geometry of a n-dimensional manifold M of arbitrary topology, supporting a metric field g (of given signature (p,q)) and an arbitrary connection ∇. Specifically, we deal here with the theory of multivector and extensor fields on M. Our approach does not suffer the problems of earlier attempts which are restricted to vector manifolds. It is based on the existence of canonical algebraic structures over the canonical (vector) space associated to a local chart (Uo, ϕo) of a given atlas of M. The key concepts of a-directional ordinary derivatives of multivector and extensor fields are defined and their properties studied. Also, we recall the Lie algebra of smooth vector fields in our formalism, the concept of Hestenes derivatives and present some illustrative applications.