A. M. Moya

Euclidean Clifford algebra

LetV be ann-dimensional real vector space. In this paper we introduce the concept ofeuclidean Clifford algebraCℓ (V, GE) for a given euclidean structure onV , i.e., a pair (V, GE) where GE is an euclidean metric forV (also called an euclidean scalar product). Our construction ofCℓ(V, GE) has been designed to produce a powerful computational tool. We start introducing the concept ofmultivectors overV. These objects are elements of a linear space over the real field, denoted by ΛV. We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of twocontraction operators on ΛV; and the concept of euclideaninterior algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develope a skill on the subject, preparing himself for the reading of the following papers in this series.

Metric Clifford algebra

In this paper we introduce the concept of metric Clifford algebraCℓ(V; g) for ann-dimensional real vector spaceV endowed with a metric extensor g whose signature is (p; q), withp+q=n. The metric Clifford product onCℓ (V; g) appears as a well-defined deformation (induced by g) of an euclidean Clifford product onCℓ (V). Associated with the metric extensorg; there is a gauge metric extensorh which codifies all the geometric information just contained ing: The precise form of suchh is here determined. Moreover, we present and give a proof of the so-calledgolden formula, which is important in many applications that naturally appear in our studies of multivector functions, and differential geometry and theoretical physics.

Multivector functions of a real variable

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the derivative rules involving exterior and Clifford products, and also to the rule for derivation of a composition of an ordinary scalar function with a multivector function of a real variable.

Multivector functions of a multivector variable

In this paper we develop with considerable details a theory of multivector functions of ap-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these multivector functions are introduced, as e.g., theA-directional derivative (whereA is ap-vector) and the generalized concepts of curl, divergence and gradient. The derivation rules for different types of products of multivector functions and for compositon of multivector functions are proved.

Covariant Derivatives on Minkowski Manifolds

We present a general theory of covariant derivative operators (linear connections) on a Minkowski manifold (represented as an affine space (M, M*) using the powerful multiform calculus. When a gauge metric extensor G (generated by a gauge distortion extensor h) is introduced in the Minkowski manifold, we get a theory that permits the introduction of general Riemann-Cartan-Weyl geometries. The concept of gauge covariant derivatives is introduced as the key notion necessary to generate linear connections that are compatible with G, thus, permitting the construction of Riemann-Cartan geometries. Many results of genuine mathematical interest are obtained. Moreover, such results are fundamental for building a consistent formulation of a theory of the gravitational field in flat spacetime. Some important examples of applications of our theory are worked in details.

Multivector and Extensor Fields on Smooth Manifolds

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.

Lagrangian Formalism for Multiform Fields on Minkowski Spacetime

We present the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus. The formulation gives a unified mathematical description for the relativistic field theories including the gravitational field. We work out examples including the Dirac—Hestenes field on the gravitational background.

Metric tensor vs. metric extensor

In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms oftensors andextensors. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors donot have inverses. We relate the definition of determinant of a metric extensor with the classical determinant of the corresponding matrix associated to the metric tensor in a given vector basis. Previous identifications of these concepts are equivocated. The use of metric extensor permits sophisticated calculations without the introduction of matrix representations.

GEOMETRIC AND EXTENSOR ALGEBRAS AND THE DIFFERENTIAL GEOMETRY OF ARBITRARY MANIFOLDS

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U ⊂ M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field γ defining a parallelism structure on U ⊂ M, which represents in a well-defined way the action on U of the restriction there of some given connection ∇ defined on M. Also we give a novel and intrinsic presentation (i.e. one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartans theory. Two kinds of Cartans connection operator fields are identified, and both appear in the intrinsic Cartans structure equations satisfied by the Cartans torsion and curvature extensor fields. We introduce moreover a metrical extensor g in U corresponding to the restriction there of given metric tensor g defined on M and also introduce the concept of a geometric structure(U, γ ,g) for U ⊂ M and study metric compatibility of covariant derivatives induced by the connection extensor γ. This permits the presentation of the concept of gauge (deformed) derivatives which satisfy noticeable properties useful in differential geometry and geometrical theories of the gravitational field. Several derivatives of operators in metric and geometrical structures, like ordinary and covariant Hodge co-derivatives and some duality identities are exhibited.