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

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.

latex 2e, 11pt, 86 pages, v3 with a up-dated references; a short variant of this review was accepted for IJGMMP (2008), see also arXiv: 0801.4958 [gr-qc]

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Cited by in corpus (6)

• Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces
• Deformation Quantization of Nonholonomic Almost Kahler Models and Einstein Gravity
• Critical Remarks on Finsler Modifications of Gravity and Cosmology by Zhe Chang and Xin Li
• Einstein Gravity in Almost Kahler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics
• Generalized Lagrange Transforms: Finsler Geometry Methods and Deformation Quantization of Gravity
• Modelling Cosmic Springs with Finsler and Generalised Finsler Geometries