A Clifford Bundle Approach to the Differential Geometry of Branes

Abstract

The Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on $\mathcal{C\ell}(M,g)$ is first used for a fomulation of the intrinsic geometry of a differential manifold $M$ equipped with a metric field $\boldsymbol{g}$ 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 $\mathfrak{R}$ and (once fixing a gauge) the connection $(1-2)$-extensor $\omega$ 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 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 $M$. It is shown that the Ricci operator of $M$ is the (negative) square of the shape operator $\mathbf{S}$ of $M$. Also we disclose the relationship between the connection (1-2%)-extensor $\omega$ and the shape biform $\mathcal{S}$ (an object related to $\mathbf{S}$). We hope that our presentation will be useful for differential geometers and theoretical physists interested, e.g, in string and brane theories and relativity theory.