Clifford Bundle Approach to the Differential Geometry of Branes

Abstract

Using the Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on \(\mathcal{C}\ell(M,g)\) (introduced in Chap. 4) we first recall the formulation of the intrinsic geometry of a differential manifold M (a brane) equipped with a metric field \(\boldsymbol{g}\) of signature (p, q) and an arbitrary metric compatible connection ∇ introducing the torsion (2 − 1)-extensor field τ, the curvature (2 − 2) extensor field \(\mathfrak{R}\) and (once fixing a gauge) the connection (1 − 2)-extensor ω and the Ricci operator \(\boldsymbol{\partial } \wedge \boldsymbol{ \partial }\) (where \(\boldsymbol{\partial }\) is the Dirac operator acting on sections of \(\mathcal{C}\ell(M,g)\)) which plays an important role in this paper. Next, using the CBF we give a thoughtful presentation of the Riemann or the Lorentzian geometry of an orientable submanifold M (\(\dim M = m\)) living in a manifold \(\mathring{M}\) (such that \(\mathring{M} \simeq \mathbb{R}^{n}\) is equipped with a semi-Riemannian metric \(\boldsymbol{\mathring{g}}\) with signature \((\,\mathring{p},\mathring{q})\) and \(\mathring{p}+\mathring{q} = n\) and its Levi-Civita connection \(\mathring{D}\)) and where there is defined a metric \(\boldsymbol{g = i}^{{\ast}}\mathring{g}\), where \(\boldsymbol{i}:\) \(M \rightarrow\mathring{M}\) is the inclusion map. We prove several equivalent forms for the curvature operator \(\mathfrak{R}\) of M. Moreover we show a very important result, namely that the Ricci operator of M is the (negative) square of the shape operator S of M (object obtained by applying the restriction on M of the Dirac operator \(\boldsymbol{\mathring{\partial}}\) of \(\mathcal{C}\ell(\mathring{M},\mathring{g})\) to the projection operator P). Also we disclose the relationship between the (1 − 2)-extensor ω and the shape biform \(\mathcal{S}\) (an object related to S). The results obtained are used in Chap. 11 to give a mathematical formulation to Clifford’s theory of matter (Rodrigues and Wainer, Adv Appl Clifford Algebras 24:817–847, 2014).