Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry

Abstract

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.

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

References in corpus (9)

• Geometric Structures in Field Theory
• Lagrange-Fedosov Nonholonomic Manifolds
• Generalized Finsler Geometry in Einstein, String and Metric--Affine Gravity
• (Non)Commutative Finsler Geometry from String/M--theory
• A Method of Constructing Off-Diagonal Solutions in Metric-Affine and String Gravity
• Noncommutative Finsler Geometry, Gauge Fields and Gravity
• Noncommutative Nonlinear Supersymmetry
• Noncommutative Symmetries and Stability of Black Ellipsoids in Metric--Affine and String Gravity
• A brief introduction to the noncommutative geometry description of particle physics standard model

Cited by in corpus (4)

• Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures
• Nonholonomic Deformations of Disk Solutions and Algebroid Symmetries in Einstein and Extra Dimension Gravity
• Nonlinear Connections on Gerbes, Clifford-Finsler Modules, and the Index Theorems
• Clifford Algebroids and Nonholonomic Spinor Deformations of Taub-NUT Spacetimes