Nonholonomic Ricci flows. II. Evolution equations and dynamics

Abstract

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler-Lagrange and Riemann geometries. We verify some assertions made in the first partner paper and develop a formal scheme in which the geometric constructions with Ricci flow evolution are elaborated for canonical nonlinear and linear connection structures. This scheme is applied to a study of Hamilton's Ricci flows on nonholonomic manifolds and related Einstein spaces and Ricci solitons. The nonholonomic evolution equations are derived from Perelman's functionals which are redefined in such a form that can be adapted to the nonlinear connection structure. Next, the statistical analogy for nonholonomic Ricci flows is formulated and the corresponding thermodynamical expressions are found for compact configurations. Finally, we analyze two physical applications: the nonholonomic Ricci flows associated to evolution models for solitonic pp-wave solutions of Einstein equations, and compute the Perelman's entropy for regular Lagrange and analogous gravitational systems.

Einstein Gravity, Lagrange-Finsler Geometry, and Nonsymmetric Metrics

“…Equivalently, we can say that such coefficients define a class of (linearly depending on N a i ) linear frames. A geometric/physical motivation for such constructions can be provided if, for instance, the N-coefficients are induced by a (regular) Lagrange structure (in geometric mechanics, see [25,15]), or by certain off-diagonal coefficients of symmetric and/or nonsymmetric metrics, see examples and details in [24,30,17,23]. A manifold is not N-anholonomic if it is enabled with a trivial N-connection structure when the corresponding nonholonomic coefficients vanish.…”

“…For applications in modern physics [24,30,17,23], the constructions with metric compatible d-connections which are defined in a unique way by a metric and certain prescribed torsion structures are considered to be more related to standard theories.…”

“…Our aim is to redefine the A. Kawaguchi metrization procedure [32,33,34] considered in Finsler geometry and developed for Lagrange spaces (including their noncommutative metric generalizations) [13,14,15] for nonholonomic manifolds enabled with nonsymmetric metric structures [23,16,17].…”

“…A set of additional arguments for geometric and physical models with nonsymmetric metrics follows from the theory of nonholonomic Ricci flows [16,17] (the fifth direction of research with nonsymmetric metrics). It was mentioned that nonholonomic distributions on manifolds result in nonholonomic deformations of geometric objects (local frames, metrics, connections, functional and effective thermodynamical effects, .…”

Nonholonomic Ricci Flows of Riemannian Metrics and Lagrange-Finsler Geometry

“…Much of the works on Ricci flows has been performed and validated by experts in the area of geometrical analysis and Riemannian geometry. Recently, a number of applications in physics of the Ricci flow theory were proposed, by Vacaru [12][13][14][15][16].Some geometrical approaches in modern gravity and string theory are connected to the method of moving frames and distributions of geometric objects on (semi) Riemannian manifolds and their generalizations to spaces provided with nontrivial torsion, nonmetricity and/or nonlinear connection structures [17,18]. The geometry of nonholonomic manifolds and non-Riemannian spaces is largely applied in modern mechanics, gravity, cosmology and classical/quantum field theory expained by Stavrinos [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”