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

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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Published 28 April 2017