2008/01/31 by Sergiu I. Vacaru, SERGIU I. VACARU · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Mathematical Theories and Applications #Einstein #General relativity #Geometric Analysis and Curvature Flows #Minkowski space #Nonholonomic system #Riemann hypothesis #Riemannian geometry #Spacetime #String (physics) #Tangent space #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1142/s0219887808002898
published as Int.J.Geom.Meth.Mod.Phys.05:473-511,2008 · latex 2e, 11pt, 44 pages; accepted to IJGMMP (2008) as a short variant of arXiv:0707.1524v3, on 86 pages
arxiv created 2008/01/31 · openalex publication_date 2008/06/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 modeled 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 modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.