Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics

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 Schrödinger map equation. All geometric constructions can be re–defined for the Levi–Civita connection but with ”noholonomic mixing” of solitonic interactions.