Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures

Abstract

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. In this work, we elaborate the algebroid spinor differential geometry and formulate the (scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids. The paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exact solutions with generic off-diagonal metrics and generalized symmetries in modern gravity define nonholonomic spacetime manifolds with uncompactified extra dimensions.

The manuscript was substantially modified following recommendations of JMP referee. The former Chapter 2 and Appendix were elliminated. The Introduction and Conclusion sections were modified

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• Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces
• Critical Remarks on Finsler Modifications of Gravity and Cosmology by Zhe Chang and Xin Li
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• Einstein Gravity, Lagrange-Finsler Geometry, and Nonsymmetric Metrics
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• Killing sections and sigma models with Lie algebroid targets
• A Kähler Structure on Cartan Spaces
• General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations
• Dynamical Equations and Lagrange--Ricci Flow Evolution on Prolongation Lie Algebroids
• Algebraic Constructions in the Category of Lie Algebroids
• Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci Flows, and Mathematical Relativity