Metric Compatible or Non-Compatible Finsler–ricci Flows

Abstract

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric noncompatible) connections, the Ricci flag curvature etc. In a series of works, we studied (non)commutative metric compatible . The goal of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived nonholonomic Hamilton evolution equations, when metric noncompatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric noncompatible connections.

Black holes with MDRs and Bekenstein–Hawking and Perelman entropies for Finsler–Lagrange–Hamilton Spaces

“…In a series of works [53,54,55,24,56,57,58,59,25,22,61,62,23], we developed a new statistical and geometric thermodynamics approach which allows us to characterize physical properties of generic off-diagonal configurations in GR and MGTs, see recent results in [26,63,64,27,65]. Such gravitational and matter field geometric flow theories and generalized Ricci solitons can be elaborated following G. Perelman's definitions of W-and F-entropies [19].…”

“…Such higher symmetry generalized metrics are with conventional horizons when a corresponding hypersurface area can be computed (for instance, black elipsoid/torus, holographic and/or emergent configurations in phase spaces) for physical objects imbedded into certain phase space backgrounds with high symmetry and flat space asymptotic structure. In our previous works [53,54,55,24,56,57,58,59,25,22,61,62,23,26,63,64,27,65] (see also a review of directions of research 10 and 17 in Appendix B4 of [29]), we proved that for generic off-diagonal exact solutions in GR and MGTs with nonholonomic, noncommutative, supersymmetric, fractional and Finsler like variables, we can elaborate a more general approach to the geometric and statistical thermodynamics of gravitational fields using G. Perelman's concepts of Wand F-entropy [19,20,21]. On mathematics of Ricci flows and certain physical applications, we cite [66,67,68,69,70,71,72,73,74,75]).…”

Nonholonomic Ricci Flows of Riemannian Metrics and Lagrange-Finsler Geometry

“…Recently, there were proposed various models of " analogous gravity", a review [53], which do not apply the methods of Finsler geometry and the formalism of nonlinear connections.…”

Nonholonomic ricci flows and Finsler–Lagrange f(R,F,L)–modified gravity and dark matter effects

“…We argue that selfconsistent and physically motivated "minimal" Finsler modifications of the standard Ricci flow and gravitational field equations can be elaborated using the so-called Cartan and canonical distinguished connections (d-connection) structures. The approaches with metric noncompatible Finsler connections, or without linear connections, do not have limits to standard theories of particle physics and do not allow formulations of certain analogs of the axiomatic formalism as GR 2,3 . The second goal of our works is to analyze possible implications of the geometric unifications of nonholonomic Ricci flow evolution and modified gravity theories, MGTs, in modern acceleration and study of dark energy and dark matter problems 4,6 .…”

“…b The symbols L and F is taken respectively from the Lagrange and Finsler nonlinear quadratic elements, ds 2 = L(x, dx) and = F 2 (x, dx), which generalize the quadratic element in (pseudo) Riemannian geometry, ds 2 , (g, N,D), variables. For instance, in the first case, we keep an explicit analogy between the Lagrange and Finsler geometry; in the second case, we introduce almost symplectic variables with allow to perform a rigorous deformation quantization of such geometries; in the third case, it is possible to decouple certain generalize Einstein-Finsler equations forD and solve such equations in very general forms with generic off-diagonal metrics determined by generating and integration functions, correspondingly depending on all spacetime coordinates (such variables can be introduced also in GR); see discussion, examples and references in [2][3][4]7,8 Finally, we consider generalized Grisha Perelman's functionals c ,…”