How many geodesics join two points on a contact sub-Riemannian manifold?

Abstract

We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordinates $(x,z)\in {\mathbb{R}}^{2n}\times \mathbb{R}$ centered at $p$. In this case we prove that for the generic $q$ the number of geodesics $\hat{I}½(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{|z|}{\|x\|^2} + R_1 \leq ν(q) \leq C_2\frac{|z|}{\|x\|^2} + R_2\] for some constants $C_1,C_2$ and $R_1,R_2$. We recover exact values for Heisenberg groups, where $C_1=C_2=\frac{8}{\ddot{I}}€$. Removing the genericity condition for $q$, geodesics might appear in families and we prove a similar statement for their topology. We study these families, and in particular we focus on the unexpected appearance of isometrically non-equivalent geodesics: families on which the action of isometries is not transitive. We apply the previous study to contact sub-Riemannian manifolds: we prove that for any given point $p\in M$ there is a sequence of points $p_n$ such that $p_n\to p$ and that the number of geodesics between $p$ and $p_n$ grows unbounded (moreover these geodesics have the property of being contained in a small neighborhood of $p$).

37 pages, 10 figures, final version to appear on "Journal of Symplectic Geometry". Minor corrections prior to publication