摘要： We construct of the curvature-type differential invariants  for a wide class of geometric structures and control systems on manifolds, especially for  sub-Riemannian sructures on nonholonomic vector distributions with application to Hamiltonian Dynamics and Physics (for example, to magnetic fields)

 

The approach to the above problems is based on the study of differential geometry of curves in Lagrangian Grassmannians.
We construct the canonical bundle of moving frames and the complete system of symplectic invariants for parametrized curves in Lagrange Grassmannians satisfying very general  assumption.

 

We can then calculate the curvature maps for a class of sub-Riemannian structures on distributions  having additional transversal infinitesimal symmetry. Such structures are reduced to a base Riemannian manifold equipped with a magnetic field. We estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the Riemannian curvature tensor of the base manifold  and the magnetic field  in the case of a uniform magnetic field.

Finally we study the hyperbolic flows in sub-Riemannian structures appearing naturally on principal connections of principal bundles over Riemannian manifolds, when the structure group of the bundle is commutative.

 

The talk is mainly based on the joint works with Igor Zelenko.