Calculus of Variations and Geometric Measure Theory
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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 2015


Accepted Paper

Inserted: 20 may 2014
Last Updated: 15 may 2015

Journal: Journal of symplectic geometry
Year: 2015
Links: arXiv preprint


We investigate the structure and the topology of the set of geodesics (critical points for the energy functional) between two points on a contact Carnot group $G$ (or, more generally, corank-one Carnot groups). Denoting by $(x,z)\in \mathbb{R}^{2n}\times \mathbb{R}$ exponential coordinates on $G$, we find constants $C_1, C_2>0$ and $R_1, R_2$ such that the number $\hat{\nu}(p)$ of geodesics joining the origin with a generic point $p=(x,z)$ satisfies: \[ C_1\frac{\lvert z\rvert\phantom{^2}}{\lVert x\rVert^2}+R_1\leq \hat{\nu}(p)\leq C_2\frac{\lvert z\rvert\phantom{^2}}{\lVert x\rVert^2}+R_2. \] We give conditions for $p$ to be joined by a unique geodesic and we specialize our computations to standard Heisenberg groups, where $C_1=C_2=\frac{8}{\pi}$.

The set of geodesics joining the origin with $p\neq p_0$, parametrized with their initial covector, is a topological space $\Gamma(p)$, that naturally splits as the disjoint union \[ \Gamma(p) = \Gamma_0(p) \cup \Gamma_\infty(p), \] where $\Gamma_0(p)$ is a finite set of isolated geodesics, while $\Gamma_\infty(p)$ contains continuous families of non-isolated geodesics (critical manifolds for the energy). We prove an estimate similar to the one above for the ``topology'' (i.e. the total Betti number) of $\Gamma(p)$, with no restriction on $p$.

When $G$ is the Heisenberg group, families appear if and only if $p$ is a vertical nonzero point and each family is generated by the action of isometries on a given geodesic. Surprisingly, in more general cases, families of non-isometrically equivalent geodesics do appear.

If the Carnot group $G$ is the nilpotent approximation of a contact sub-Riemannian manifold $M$ at a point $p_0$, we prove that the number $\nu(p)$ of geodesics in $M$ joining $p_0$ with $p$ can be estimated from below with $\hat{\nu}(p)$. The number $\nu(p)$ estimates indeed geodesics whose image is contained in a coordinate chart around $p_0$ (we call these ``affine'' geodesics).

As a corollary we prove the existence of a sequence $\{p_n\}_{n\in \mathbb{N}}$ in $M$ such that: \[ \lim_{n\to \infty}p_n=p_0\qquad \text{and}\qquad \lim_{n\to \infty}\nu(p_n)=\infty, \] i.e. the number of ``affine'' geodesics between two points can be arbitrarily large, in sharp contrast with the Riemannian case.

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


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