Radial kinetic nonholonomic trajectories are Riemannian geodesics!

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and timelike geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generaliza...

Hamiltonian Geodesics in Nonholonomic Differential Systems

Let a smooth manifold M be equipped with a smoothly varying positive definite quadratic form on a distribution-subbundle V of the tangent bundle TM. One assumes that V is completely nonintegrable, equivalently all sections of V together with all brackets between them generate TM at each point of M. Then by the well-known theorem of Rashevski-Chow [3] it is possible to connect any two points (in...

Closed Geodesics in Compact Riemannian Good Orbifolds and Horizontal Periodic Geodesics of Riemannian Foliations

In this paper we prove the existence of closed geodesics in certain types of compact Riemannian good orbifolds. This gives us an elementary alternative proof of a result due to Guruprasad and Haefliger. In addition, we prove some results about horizontal periodic geodesics of Riemannian foliations and stress the relation between them and closed geodesics in Riemannian orbifolds. In particular w...

The regularity problem for sub-Riemannian geodesics

One of the main open problems in sub-Riemannian geometry is the regularity of length minimizing curves, see [12, Problem 10.1]. All known examples of length minimizing curves are smooth. On the other hand, there is no regularity theory of a general character for sub-Riemannian geodesics. It was originally claimed by Strichartz in [15] that length minimizing curves are smooth, all of them being ...

Computing Geodesics on a Riemannian Manifold