## 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].

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## References

L. Auslander and R. E. MacKenzie,

*Introduction to Differential Manifolds*.McGraw Hill Book Co. New York (1963).K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory. Cambridge Univ. Press, Cambridge (2007).

Y Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick,

*Analysis Manifold and Physics. Part 1: Basics*. (Revised edition), North Holland, Amsterdam, 1982.M. Duff,

*M-Theory (The theory Formely Known as Strings)*. Int. J. Mod. Phys. A**11**(1996), 5623–5642. [hep-th/9608117]Clarke C.J. S.: On the Global Isometric Embedding of Pseudo-Riemannian Manifolds. Proc. Roy. Soc. A

**314**, 417–428 (1970)W. K. Clifford,

*On the Space-Theory of Matter*. Proc. Cambridge Phil. Soc.**2**(1864–1876 – Printed 1876), 157–158.C. Doran and A. Lasenby,

*Geometric Algebra for Physicists*. Cambridge Univ. Press, Cambridge (2003).R. da Rocha, A. E. Bernardini and J. M. Hoff da Silva,

*Exotic Dark Spinor Fields*, JHEP**4**, article:110 [arXiv:1103.4759 ] [hep-th].A. S. Eddington,

*The Mathematical Theory of Relativity*. 3rd edn., Chelsea, New York, 1975.V. V. Fernández and W. A. Jr. Rodrigues,

*Gravitation as a Plastic Distortion of the Lorentz Vaccum*. Fundamental Theories of Physics**168**, Springer, Berlin, 2010. Errata at : http://www.ime.unicamp.br/~walrod/plastic04162013.Geroch R.: Spinor Structure of Space-Times in General Relativity I. J. Math. Phys.

**9**, 1739–1744 (1968)N. J. Hicks,

*Notes in Differential Geometry*, van Nostrand, Princeton, 1965.D. Hestenes and G. Sobczyk,

*Clifford Algebra to Geometric Calculus*. D. Reidel Publ. Co., Dordrecht, 1984.D. Hestenes,

*Curvature Calcualtions with Spacetime Algebra*. Int. J. Theor. Phys.**25**(1986), 581–588.P. Horava and E. Witten,

*Eleven-Dimensional Supergravity on a Manifold with Boundary*. Nucl. Phys B**457**(1996), 94–114. [hep-th/9603142]P. D. Mannhein,

*Brane Localized Gravity*. World Sci. Publ. Co., Singapore (2005).R. A Mosna and W. A. Rodrigues Jr.,

*The Bundles of Algebraic and Dirac-Hestenes Spinor Fields*, J. Math. Phys**45**, 2945–2966 (2004). [mathph/0212033].E. Notte-Cuello, W. A. Rodrigues Jr. and Q. A. G. de Souza,

*The Square of the Dirac and spin-Dirac Operators on a Riemann-Cartan Space(time)*. Rep. Math. Phys**60**(2007), 135–157 [arXiv:math-ph/0703052 ].W. A. Rodrigues Jr. and E. Capelas de Oliveira,

*The Many Faces of Maxwell Equations. A Clifford Bundle Approach*. Lecture Notes in Physics**722**, Springer, Heildeberg, 2007. Errata and preliminary version of a second edition at http://www.ime.unicamp.br/~walrod/recentes.htm.W. A. Rodrigues Jr.,

*Killing Vector Fields, Maxwell Equations and Lorentzian Spacetimes*. Adv. Appllied. Clifford Algebras**20**(2010), 871–884.G. Sobczyk,

*Conformal Mappings in Geometric Algebra*, Not. Am. Math. Soc**59**(2012), 264–273.

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Rodrigues, W.A., Wainer, S.A. A Clifford Bundle Approach to the Differential Geometry of Branes.
*Adv. Appl. Clifford Algebras* **24**, 817–847 (2014). https://doi.org/10.1007/s00006-014-0452-6

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DOI: https://doi.org/10.1007/s00006-014-0452-6