Journal of Symplectic Geometry

Volume 15 (2017)

Number 1

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

Pages: 247 – 305



A. Lerario (Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)

L. Rizzi (Univ. Grenoble Alpes, CNRS, Institut Fourier, Grenoble, France)


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 $C_1,C_2 \gt 0$ 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:

\begin{equation}C_1 \frac{\lvert z \rvert}{{\lVert x \rVert}^2} + R_1 \leq \hat{\nu}(p) \leq C_2 \frac{\lvert z \rvert}{{\lVert x \rVert}^2} + R_2.\end{equation}

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\[\Gamma (p) = \Gamma_0(p) \cup \Gamma_{\infty} (p) \; \textrm{,}\]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 (1) 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 “local” geodesics).

As a corollary we prove the existence of a sequence ${\lbrace p_n \rbrace }_{n \in \mathbb{N}}$ in $M$ such that:\[\lim_{n \to \infty} p_n = p_0 \; \; \textrm{and} \; \; \lim_{n \to \infty} \nu (p_n) = \infty \: \textrm{,}\]

i.e. the number of “local” geodesics between two points can be arbitrarily large, in sharp contrast with the Riemannian case.

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