In this work, we obtain the Raychaudhuri equations for various types of Finsler spaces as the Finsler-Randers (FR) space-time and in a generalized geometrical structure of the space-time manifold which contains two fibres that represent two scalar fields φ (1) , φ (2) .We also derive the Klein-Gordon equation for this model. In addition, the energy-conditions are studied in a FR cosmology and are correlated with FRW model. Finally, we apply the Raychaudhuri equation for the model M × {φ (1) } × {φ (2) }, where M is a FRW-spacetime.

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.

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.