Lanczos-Lovelock-Cartan Gravity from Clifford Space Geometry Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia. 30314, perelmanc@hotmail.com June 2012 Abstract A rigorous construction of Clifford-space Gravity is presented which is compatible with the Clifford algebraic structure and permits the deriva- tion of the expressions for the connections with torsion in Clifford spaces ( C-spaces). The C-space generalized gravitational field equations are de- rived from a variational principle based on the extension of the Einstein- Hilbert-Cartan action. We continue by arguing how Lanczos-Lovelock- Cartan higher curvature gravity with torsion can be embedded into grav- ity in Clifford spaces and suggest how this might also occur for extended gravitational theories based on f (R),f (Rμν ), ... actions, for polynomial- valued functions. In essence, the Lanczcos-Lovelock-Cartan curvature ten- sors appear as Ricci-like traces of certain components of the C-space cur- vatures. Torsional gravity is related to higher-order corrections of the bosonic string-effective action. In the torsionless case, black-strings and black-brane metric solutions in higher dimensions D> 4 play an im- portant role in finding specific examples of solutions to Lanczos-Lovelock gravity. 1 Introduction In the past years, the Extended Relativity Theory in C-spaces (Clifford spaces) and Clifford-Phase spaces were developed [1], [2]. This extended relativity in Clifford spaces theory should not be confused with the extended relativity theory (ER) proposed by Erasmo Recami and collaborators [3] many years ago which was based on the Special Relativity theory extended to Antimatter and Super- luminal motions. Since the beginning of the seventies, an “Extended special Relativity” (ER) exists, which on the basis of the ordinary postulates of Special Relativity (chosen “com grano salis”) describes also superluminal motions in a 1