Einstein Gravity, Lagrange-Finsler Geometry, and Nonsymmetric Metrics

TL;DR

This paper develops a geometric framework for nonsymmetric metrics and nonholonomic structures in Riemann-Cartan spaces, aiming to extend gravity theories and geometric mechanics with new compatible connection formulations.

Contribution

It introduces a method to model nonsymmetric metrics with nonholonomic frames and defines compatible connections, advancing geometric approaches to generalized gravity theories.

Findings

Formulated a geometric approach for nonsymmetric metrics and nonholonomic distributions.

Defined classes of compatible distinguished linear connections.

Constructed nonholonomic deformations linking Einstein and Finsler-Lagrange spaces.

Abstract

We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatible to a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrary distinguished connections, we define the class of distinguished linear connections which are compatible with the nonlinear connection and general nonsymmetric metric structures. The nonsymmetric gravity theory is formulated in terms of metric compatible connections. Finally, there are constructed such nonholonomic deformations of geometric structures when the Einstein and/or Lagrange-Finsler manifolds are transformed equivalently into spaces with generic local anisotropy induced by nonsymmetric metrics and generalized connections. We speculate on possible applications of such geometric methods in Einstein and generalized theories of gravity, analogous gravity and geometric mechanics.