  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.
