On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Davide Barilari (1) , Ugo Boscain (2, 3) , Daniele Cannarsa (2, 4)

1 Dipartimento di Matematica "Tullio Levi-Civita"
2 CaGE - Control And GEometry
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.

Mots clés: Gaussian curvature; Heisenberg group; Riemannian approximation; Length space; Sub-Riemannian geometry; Contact geometry

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

Davide Barilari, Ugo Boscain, Daniele Cannarsa. On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, 2022, 28 (9). ⟨hal-03091917⟩