Nonholonomic Ricci Flows: I. Riemann Metrics and Lagrange–Finsler Geometry
نویسنده
چکیده
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 nonhlonomic 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 nonholonomic flows in modern geometry, geometric mechanics and physics.
منابع مشابه
The Entropy of Lagrange–Finsler Spaces and Ricci Flows
We formulate a statistical analogy of regular Lagrange mechanics and Finsler geometry derived from Grisha Perelman’s functionals and generalized for nonholonomic Ricci flows. Explicit constructions are elaborated when nonholonomically constrained flows of Riemann metrics result in Finsler like configurations, and inversely, when geometric mechanics is modelled on Riemann spaces with a preferred...
متن کاملNonholonomic Ricci Flows: II. Evolution Equations and Dynamics
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 Ricc...
متن کاملNonholonomic Ricci Flows: III. Curve Flows and Solitonic Hierarchies
The geometric constructions are performed on (semi) Riemannian manifolds and vector bundles provided with nonintegrable distributions defining nonlinear connection structures induced canonically by metric tensors. Such spaces are called nonholonomic manifolds and described by two equivalent linear connections also induced unique forms by a metric tensor (the Levi Civita and the canonical distin...
متن کاملEinstein Gravity , Lagrange – Finsler Geometry , and Nonsymmetric Metrics
We formulate an approach to the geometry of Riemann–Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart–Moffat and Finsler–Lag...
متن کاملNonholonomic Ricci Flows and Parametric Deformations of the Solitonic pp–Waves and Schwarzschild Solutions
We study Ricci flows of some classes of physically valuable solutions in Einstein and string gravity. The anholonomic frame method is applied for generic off–diagonal metric ansatz when the field/ evolution equations are transformed into exactly integrable systems of partial differential equations. The integral varieties of such solutions, in four and five dimensional gravity, depend on arbitra...
متن کامل