Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry

Details Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry

2004-08-09

arxiv.org/abs/math/0408121v1

Chapter 15 in: "Clifford and Riemann Finsler Structures in Geometric Mechanics and Gravity, Selected Works by S. Vacaru, P. St

Mathematics

Differential Geometry

55 pages, latex 2e, Outline of a series of Lectures and Seminars

Scientific paper

We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in particular, related to geometrization of classical and quantum mechanical and field theories, even if we restrict our considerations only to commutative and noncommutative Riemannian spaces. We elaborate a geometric approach to the Clifford modules adapted to nonlinear connections, to the theory of spinors and the Dirac operators on nonholonomic spaces and consider possible generalizations to noncommutative geometry. We argue that any commutative Riemann--Finsler geometry and generalizations my be derived from noncommutative geometry by applying certain methods elaborated for Riemannian spaces but extended to nonholonomic frame transforms and manifolds provided with nonlinear connection structure.

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