  Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian
structures and related mKdV hierarchies of soliton equations derived
geometrically from regular Lagrangians and flows of non-stretching curves in
tangent bundles. The total space geometry and nonholonomic flows of curves are
defined by Lagrangian semisprays inducing canonical nonlinear connections
(N-connections), Sasaki type metrics and linear connections. The simplest
examples of such geometries are given by tangent bundles on Riemannian
symmetric spaces $G/SO(n)$ provided with an N-connection structure and an
adapted metric, for which we elaborate a complete classification, and by
generalized Lagrange spaces with constant Hessian. In this approach,
bi-Hamiltonian structures are derived for geometric mechanical models and
(pseudo) Riemannian metrics in gravity. The results yield horizontal/ vertical
pairs of vector sine-Gordon equations and vector mKdV equations, with the
corresponding geometric curve flows in the hierarchies described in an explicit
form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrodinger
maps on a tangent bundle.
