  For (2+2)-dimensional nonholonomic distributions, the physical information
contained into a spacetime (pseudo) Riemannian metric can be encoded
equivalently into new types of geometric structures and linear connections
constructed as nonholonomic deformations of the Levi-Civita connection. Such
deformations and induced geometric/physical objects are completely determined
by a prescribed metric tensor. Reformulation of the Einstein equations in
nonholonomic variables (tetrads and new connections, for instance, with
constant coefficient curvatures and/or Yang-Mills like potentials) reveals
hidden geometric and rich quantum structures. It is shown how the Einstein
gravity theory can be re-defined equivalently as certain gauge models on
nonholonomic affine and/or de Sitter frame bundles. We speculate on possible
applications of the geometry of nonholonomic distributions with associated
nonlinear connections in classical and quantum gravity.
