Finsler and Lagrange Geometries in Einstein and String Gravity

Abstract

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 Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "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. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/ Kähler, or nonholonomic Riemann spaces.Finally, we consider some classes of exact solutions in string and Einstein gravity modelling Lagrange-Finsler structures with solitonic pp-waves and speculate on their physical meaning.

Nonholonomic Ricci flows. II. Evolution equations and dynamics

“…On the geometry of N-anholonomic manifolds (i.e. manifolds enabled with nonholonomic distributions defining nonlinear connection, N-connection, structures), we follow the conventions from the first partner work [10] and [16,17]. We shall use boldface symbols for spaces/ geometric objects enabled/ adapted to N-connection structure.…”

“…[17,16,10,14]. Here we note that originally the Lagrange geometry was elaborated on the tangent bundle T M of a manifold M, i.e.…”

Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

“…Such constructions were developed [1,3,15] for curve flows in Riemannian manifolds of constant curvature [9,16]. These which gave rise, for instance, to a vector generalization of the mKdV equa-curvatures, using certain ideas and methods from the geometry of nonholonomic distributions, and modelling of Lagrange-Finsler geometries [18]. Such models with nontrivial nonlinear connection structure were elaborated on (co) tangent bundles (see [13] and references therein) and our proposal was to apply such methods on (pseudo) Riemannian and Einstein manifolds endowed with nonholonomic distributions.…”

Einstein Gravity, Lagrange-Finsler Geometry, and Nonsymmetric Metrics

“…This way, we shall provide a synthesis of the methods and ideas developed in directions two, three and four (mentioned above) in a general nonsymmetric metric compatible form, for various classes of linear and nonlinear connection, in strong relations to the fifth direction following the methods of geometry of nonholonomic manifolds. As general references on nonholonomic manifolds enabled with nonlinear connection structure, on the geometry of spaces with generic local anisotropy and applications to modern physics and mechanics, we cite the works [24,25].…”

“…In this work, one follows the conventions from [24]. The Einstein's convention on summing "up" and "low" indices will be applied if the contrary will not be stated.…”