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 $\nu(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{|z|}{\|x\|^2} + R_1 \leq \nu(q) \leq C_2\frac{|z|}{\|x\|^2} + R_2\] for some constants $C_1,C_2$ and $R_1,R_2$. We recover exact values for Heisenberg groups, where $C_1=C_2 = \frac{8}{\pi}$. 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 $p$).
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https://hal.archives-ouvertes.fr/hal-01096718
Contributor : Luca Rizzi <>
Submitted on : Thursday, December 18, 2014 - 9:51:00 AM
Last modification on : Tuesday, October 23, 2018 - 3:24:19 PM

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Antonio Lerario, Luca Rizzi. How many geodesics join two points on a contact sub-Riemannian manifold?. Journal of Symplectic Geometry, International Press, 2017, ⟨10.4310/JSG.2017.v15.n1.a7⟩. ⟨hal-01096718⟩

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