Nonholonomic Ricci Flows: III. Curve Flows and Solitonic Hierarchies

Abstract

The geometric constructions are elaborated 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 in unique forms by a metric tensor (the Levi Civita and the canonical distinguished connection, d-connection). The lifts of geometric objects on tangent bundles are performed for certain classes of d-connections and frame transforms when the Riemann tensor is parametrized by constant matrix coefficients. For such configurations, the flows of non-stretching curves and corresponding bi-Hamilton and solitonic hierarchies encode information about Ricci flow evolution, Einstein spaces and exact solutions in gravity and geometric mechanics. The applied methods were elaborated formally in Finsler geometry and allows us to develop the formalism for generalized Riemann-Finsler and Lagrange spaces. Nevertheless, all geometric constructions can be equivalently re-defined for the Levi Civita connections and holonomic frames on (semi) Riemannian manifolds.

47 pages, latex2e [12pt], version 2 with minor grammar modifications

References in corpus (6)

• Ricci flow with surgery on three-manifolds
• Ricci Flow and the Poincare Conjecture
• Nonholonomic Ricci Flows: II. Evolution Equations and Dynamics
• Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons
• Curve Flows and Solitonic Hierarchies Generated by (Semi) Riemannian Metrics
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Cited by in corpus (2)

• Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons
• Generalized Lagrange Transforms: Finsler Geometry Methods and Deformation Quantization of Gravity