  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.
