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

Abstract

We review the theory of geometric flows on nonholonomic manifolds and tangent bundles and self-similar configurations resulting in generalized Ricci solitons and EinsteinFinsler equations. There are provided new classes of exact solutions on Finsler-Lagrange f(R,F,L)-modifications of general relativity and discussed possible implications in acceleration cosmology.Keywords: Nonholonomic Ricci flows; Finsler-Lagrange geometry and modified gravity; locally anisotropic Finsler cosmolgy.Current important and fascinating problems in modern accelerating cosmology and dark energy and dark matter physics involve the finding of canonical (optimal) metric and connection spacetime structures, search for possible topological configurations, and to find the relevant physical applications, see 5,6,8 and references therein. There are strong observational cosmological data and theoretical arguments (e.g. the fundamental unsolved problem of constructing a self-consistent model of quantum gravity) that the standard general relativity, GR, theory of gravity should be modified in a non-Riemannian geometric form and/or as modified gravity theory, * DAAD fellowship affiliations for two host institutions The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com by 54.202.163.228 on 05/07/18. For personal use only.

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].…”

“…In this subsection, we provide an introduction to the theory of nonholonomic relativistic flows on stationary phase spaces modelled as cotangent Lorentz bundles T * V of total dimension 8. Such geometric evolution models are certain dual analogs and Hamilton-Finsler-Ricci modifications of theories elaborated in [23,26,63,27,65], see also our previous works [53,54,55,24,56,57,58,59,25,22,61,62]. We generalized for phase spaces the G. Perelman's definitions of W-and F-entropies [19,20,21] (rigorous mathematic results on of Ricci flows of Riemanian and Kähhler metrics can be found in [69,70,71,72,73,74]).…”

“…Finally we note that the formulas for generalized Perelman's functionals (43) and (44) and derived thermodynamic values (46) are witten in terms of geometric values like D and R all defined by a total phase space, or projections on the base spacetime, metric structure. Such values can be defined and computed in pure geometric forms in various types of commutative and noncommutative/ supersymmetric Finsler geometric flow evolution models and MGTs [53,54,55,56,57,58,62,63,64,65]. There is also an indirect dependence on underlying theories of gravity because the metric and (non) linear connection structures are defined differently in MGTs.…”

“…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]).…”

Dynamical equations and Lagrange–Ricci flow evolution on prolongation Lie algebroids

“…We use τ = h τ = v τ for a couple of possible h-and v-flows parameters, τ = ( h τ, v τ ), and introduce a new function f instead of f. The scalar functions are re-defined in such a form that the "sub-integral" formula (34) under the distortion of Ricci tensor (29) is re-written in terms of geometric objects derived for the canonical d-connection,…”