Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures

TL;DR

This paper develops a geometric framework on nonholonomic manifolds with Clifford-Lie algebroid symmetry, enabling new models in gravity and field theories with off-diagonal metrics and generalized symmetries.

Contribution

It introduces Clifford algebroids and spinor geometry on nonholonomic manifolds, extending physical models to include complex geometric structures and generalized field equations.

Findings

Formulated field equations on Lie algebroids.

Constructed explicit off-diagonal metric parametrizations.

Developed a geometric approach for nonholonomic gravity models.

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.