Nonholonomic Ricci Flows of Riemannian Metrics and Lagrange-Finsler Geometry

Alexiou M,Stavrinos Pc

doi:10.4172/2090-0902.1000162

Alexiou M, Stavrinos Pc

Open Access

https://doi.org/10.4172/2090-0902.1000162

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Abstract

In this paper, the theory of the Ricci flows for manifolds is elaborated with nonintegrable (nonholonomic) distributions defining nonlinear connection structures. Such manifolds provide a unified geometrical arena for nonholonomic Riemannian spaces, Lagrange mechanics, Finsler geometry, and various models of gravity (the Einstein theory and string, or gauge, generalizations). Nonhlonomic frames are considered with associated nonlinear connection structure and certain defined classes of nonholonomic constraints on Riemann manifolds for which various types of generalized Finsler geometries can be modelled by Ricci flows. We speculate upon possible applications of the nonholonomic flows in modern geometrical mechanics and physics.

Highlights

A series of the most remarkable results in mathematics are related to Grisha Perelman’s proof of the Poincare Conjecture [1,2,3] built on geometrization (Thurston) conjecture [4,5] for three dimensional Riemannian manifolds, and R
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 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]

Summary

Introduction

A series of the most remarkable results in mathematics are related to Grisha Perelman’s proof of the Poincare Conjecture [1,2,3] built on geometrization (Thurston) conjecture [4,5] for three dimensional Riemannian manifolds, and R. Our strategy will be different: We shall formulate the criteria to determine when certain types of Finsler like geometries can be "extracted" (by imposing the corresponding nonholonomic constraints) from "well defined" Ricci flows of Riemannian metrics This is possible because such geometries can be equivalently described in terms of the Levi Civita connections or by metric configurations with nontrivial torsion induced by nonholonomic frames. By nonholonomic transforms of geometric structures, we shall be able to generate certain classes of nonmetric geometries and/or generalized torsion configurations.The aim of this paper (the first one in a series of works) is to formulate the Ricci flow equations on nonholonomic manifolds and prove the conditions under which such configurations (of Finsler–Lagrange type and in modern gravity) can be extracted from well defined flows of Riemannian metrics and evolution of preferred frame structures. Using the inverse d–tensor gαβ for both cases, we compute the corresponding scalar curvatures sR(∇) and sR(D ), see formulas (13) by contracting, respectively, with the Ricci tensor and Ricci d–tensor

Metrization procedure and preferred linear connections

Conclusion

There is a canonical metric structure

Result