Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics

TL;DR

This paper explores the geometric structures of nonholonomic manifolds with Einstein metrics, deriving solitonic hierarchies and integrable equations like sine-Gordon and mKdV, with implications for physics and geometry.

Contribution

It introduces a new class of N-adapted linear connections on nonholonomic manifolds that generate solitonic hierarchies and integrable equations related to Einstein metrics.

Findings

Generalized sine-Gordon equations derived from nonholonomic splitting.

Explicit form of solitonic hierarchies via wave map equations.

Potential applications in modeling gravitational interactions.

Abstract

We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are 'N-adapted', metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in solitonic hierarchies described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrodinger map equation. All geometric constructions can be re-defined for the Levi-Civita connection but with "noholonomic mixing" of solitonic interactions. Finally, we speculate why certain methods and results from the geometry of nonholonmic manifolds and solitonic equations have general importance in various directions of modern mathematics, geometric mechanics, fundamental theories in physics and applications, and briefly analyze possible nonlinear wave configurations for modeling gravitational interactions by effective continuous media effects.