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Finsler and Lagrange geometries in Einstein and string gravity. (English) Zbl 1152.53016
In this survey paper the author presents the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers well informed of general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. He also proposes a canonical scheme when geometrical objects on a (pseudo) Riemannian space are non-holonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generated target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or non-holonomic Riemann spaces. Finally, the author considers some classes of exact solutions in string and Einstein gravity modeling Lagrange-Finsler structures with solitonic pp-waves and speculates on their physical meaning.
MSC:
53B40Finsler spaces and generalizations (areal metrics)
53B50Applications of local differential geometry to physics
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
53C55Hermitian and Kählerian manifolds (global differential geometry)
83C15Closed form solutions of equations in general relativity
83E99Unified, higher-dimensional and super field theories