Finsler-Lagrange Geometries and Standard Theories in Physics: New Methods in Einstein and String Gravity

TL;DR

This paper reviews Finsler-Lagrange geometry's role in modern physics, showing how anisotropic structures can be modeled within Einstein and string gravity frameworks using nonholonomic deformations.

Contribution

It introduces a canonical scheme for modeling Finsler and Lagrange structures on Riemannian spaces, bridging geometric methods with Einstein and string theories.

Findings

Finsler-Lagrange structures can be modeled on Riemann-Cartan spaces.

Canonical transforms generate generalized Lagrange and Finsler configurations.

Criteria are provided for equivalent formulations using Levi Civita connection.

Abstract

In this article, we review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community ''orthodox'' physicists. Although the bulk of former models of Finsler-Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modelled equivalently on Riemann-Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold or on a corresponding tangent bundle. Such canonical transforms are defined by the coefficients of a prime metric (it can be a solution of the Einstein equations) and generate target spaces as generalized Lagrange structures, their models of almost Hermitian/ Kahler, or nonholonomic Riemann spaces with constant curvature, for some Finsler like connections. There are formulated the criteria when such constructions can be redefined equivalently in terms of the Levi Civita connection.