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) ∈ R2n × R exponential coordinates on G, we find constants C1, C2 > 0 and R1,R2 such that the number (Formula Presented) of geodesics joining the origin with a generic point p = (x, z) satisfies: (Formula Presented) We give conditions for p to be joined by a unique geodesic and we specialize our computations to standard Heisenberg groups, where (Formula Presented) The set of geodesics joining the origin with p ≠ p0, parametrized with their initial covector, is a topological space Γ(p), that naturally splits as the disjoint union (Formula Presented) where Γ0(p) is a finite set of isolated geodesics, while Γ∞(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 Γ(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 p0, we prove that the number ν(p) of geodesics in M joining p0 with p can be estimated from below with (Formula Presented) (p). The number ν(p) estimates indeed geodesics whose image is contained in a coordinate chart around p0 (we call these “local” geodesics). As a corollary we prove the existence of a sequence (Formula Presented) such that: (Formula Presented)i.e. the number of “local” geodesics between two points can be arbitrarily large, in sharp contrast with the Riemannian case.