Nonholonomic Ricci Flows: I. Riemann Metrics and Lagrange-Finsler Geometry

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