A Clifford Bundle Approach to the Differential Geometry of Branes

Abstract

We first recall using the Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on ${\mathcal{C}\ell}$(M, g) (the Clifford bundle of differential forms) the formulation of the intrinsic geometry of a differential manifold M equipped with a metric field g of signature (p, q) and an arbitrary metric compatible connection ${\nabla}$ introducing the torsion (2−1)-extensor field ${\tau}$, the curvature (2 − 2) extensor field ${\Re}$ and (once fixing a gauge) the connection (1−2)-extensor ${\omega}$ and the Ricci operator ${\partial \bigwedge \partial}$ (where ${\partial}$ 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 the Riemann or the Lorentzian geometry of an orientable submanifold M (dim M =  m) living in a manifold M̊ (such that M̊ ${\simeq \mathbb{R}^n}$ is equipped with a semi- Riemannian metric g̊ with signature (p̊, q̊) and p̊+q̊ = n and its Levi- Civita connection D̊) and where there is defined a metric g =  i*g̊, where ${i : M \rightarrow}$ M̊ is the inclusion map. We prove several equivalent forms for the curvature operator ${\Re}$ 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 ∂̊ of ${\mathcal{C}\ell}$(M̊, g̊) to the projection operator P). Also we disclose the relationship between the (1−2)-extensor ${\omega}$ and the shape biform ${\mathcal{S}}$ (an object related to S). The results obtained are used to give a mathematical formulation to Clifford’s theory of matter. It is hoped that our presentation will be useful for differential geometers and theoretical physicists interested, e.g., in string and brane theories and relativity theory by divulging, improving and expanding very important and so far unfortunately largely ignored results appearing in reference [13].