Multiform and Extensor Calculus on Manifolds

Abstract

In this chapter, we will introduce multiform and extensor fields on an arbitrary manifold M. In order to use the full algebraic machinery presented in the previous chapters, we recall from Chapter 1 that a manifold M may in general support many different connections (and many different metrics). Then, given an open set $\mathcal{U}\subset$ M and chart $(\mathcal{U},\phi )$ and a fixed point $O \in \mathcal{U}$, we define in an appropriate way a teleparallel connection¹ on $\mathcal{U}$ $\subset$ M,.which permits us to construct a vector space $\mathbf{U}$ and its dual U, called the canonical vector space. This permits us to introduce on U, in a thoughtful way, different parallelism structures which are in a precise sense the representatives on U of the restriction to $\mathcal{U}$ of parallelism structures defined by corresponding connections defined on M. The main object in the construction of a parallelism structure on U is a connection 2-extensor field (and some other associated extensor fields)which permits the calculation of covariant derivative representatives on U of multiform and extensor fields defined on $\mathcal{U}\subset$ M. Moreover, given a metric structure for M, we introduce the concept of a metrical compatible parallelism structure (MCPS), present a particular MCPS characterized by the Christofell operator, and introduce, moreover the 2-exform torsion field and the 4-extensor curvature field associated with a general MCPS and then specialize those concepts for the case of Riemannian and Lorentzian MCPS. Next, we will introduce a crucial ingredient for our theory of the gravitational field, namely the concept of elastic and plastic deformations of a MCPS into a new one metrical compatible parallelism structure generated by a (1, 1)-extensor field h that transforms the metric extensor field of the first structure into the metric extensor field of the second structure. We will study the conditions that h must satisfy in order to generate an elastic or plastic deformation. We will prove some key theorems which relate the torsion and curvature extensor fields of two structures, in which one is the deformation of the other. Particularly important for our purposes are the gauge fields, associated with what we call a Lorentz-Cartan metric compatible structure, which permits us to interpret a Lorentz metric compatible structure as a plastic h-deformation of what we call a Minkowski-Cartan parallelism structure. All concepts have been presented with enough details in order to help the reader become conversant with the subject.