Calculus of Variations and Geometric Measure Theory

A. Lerario - L. Rizzi

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

created by rizzi1 on 20 May 2014
modified on 15 May 2017


Published Paper

Inserted: 20 may 2014
Last Updated: 15 may 2017

Journal: Journal of symplectic geometry
Volume: Volume 15
Number: 1
Pages: 247 – 305
Year: 2017
Doi: 10.4310/JSG.2017.v15.n1.a7

ArXiv: 1405.4294 PDF
Links: arXiv preprint


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 $\nu(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{
^2} + R_1 \leq \nu(q) \leq C_2\frac{
^2} + R_2\] for some constants $C_1,C_2$ and $R_1,R_2$. We recover exact values for Heisenberg groups, where $C_1=C_2 = \frac{8}{\pi}$. 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 $p$).

Keywords: Carnot groups, Geodesics, sub-Riemannian, contact