Nonholonomic Ricci Flows of Riemannian Metrics and Lagrange-Finsler Geometry

A series of the most remarkable results in mathematics are related to Grisha Perelman’s proof of the Poincare Conjecture built on geometrization (Thurston) conjecture for three dimensional Riemannian manifolds, and R. Hamilton’s Ricci flow theory see reviews and basic references explained by Kleiner. 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 flowtheory were proposed, by Vacaru.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.

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. Such spaces are characterized by three fundamental geometric objects: nonlinear connection (N–connection), linear connection and metric. There is an important geometrical problem to prove the existence of the ” best possible” metric and linear connection adapted to a N– connection structure. From the point of view of Riemannian geometry, the Thurston conjecture only asserts the existence of a best possible metric on an arbitrary closed three dimensional (3D) manifold. It is a very difficult task to define Ricci flows of mutually compatible fundamental geometric structures on non–Riemannian manifolds (for instance, on a Finsler manifold). For such purposes, we can also apply the Hamilton’s approach but correspondingly generalized in order to describe nonholonomic (constrained) configurations. The first attempts to construct exact solutions of the Ricci flow equations on nonholonomic Einstein and Riemann–Cartan (with nontrivial torsion) manifolds, generalizing well known classes of exact solutions in Einstein and string gravity, were performed and explanied by Vacaru.

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