Two Problems in Sub-Riemannian Geometry
In this thesis we study two interesting problems in sub-Riemannian geometry. First, we pose and partially solve the Kepler Problem on the Heisenberg Group. Second, we present a formula for computing the Puiseux characteristic corresponding to a Goursat germ with prescribed small growth vector.
The Kepler Problem is among the oldest and most fundamental problems in mechanics. It has been studied in curved geometries, such as the sphere and hyperbolic plane. Here, we formulate the problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. We record many interesting properties of the system, prove the existence of periodic orbits, deduce a version of Kepler's third law, and reduce the integration of a fundamental integrable subsystem to the parametrization of a family of algebraic plane curves.
Germs of Goursat distributions can be classified according to a geometric coding called an RVT code. Jean and Mormul have shown that this coding carries precisely the same data as the small growth vector. Montgomery and Zhitomirskii have shown that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves: the Puiseux characteristic. Here we derive a simple formula for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector.