On magnetic fields and sub-Riemannian geodesics.

The objective of this thesis is to study the effect of magnetic fields on Riemannian and Sub-Riemannian geodesics. We use the language of Differential Geometry and notions from Symplectic, Riemannian and sub-Riemannian Geometry. The scheme of our investigation is organized into five chapters as follows. The goal of the first chapter is to relate the dynamics of charged particles in a Riemannian surface under the presence of a magnetic field to the sub-Riemannian geodesics of a line bundle with base the Riemannian surface itself. In Chapter 2 we analyze the specific case in which the starting Riemannian space is the Euclidean plane. In the next chapter we generalize the previous description to a Riemannian manifold of any finite dimension and subsequently we consider the case in which the starting space is a sub-Riemannian manifold. In Chapter 4 we address the particular case in which the starting sub-Riemannian manifold is the flat Heisenberg group. Finally we give a description of abnormal curves arising from the structures previously constructed. We finally added two appendices to recall some basic facts about principal and affine connections that we use throughout the thesis.