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

¹ Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, via Trieste 63, Padova, Italy.
² CNRS, Laboratoire Jacques-Louis Lions, team Inria CAGE, Université de Paris, Sorbonne Université boîe courrier 187, 75252 Paris Cedex 05 Paris France.
³ Université de Paris and Sorbonne Université, CNRS, INRIA, IMJ-PRG, 75013 Paris, France.

^* Corresponding author: daniele.cannarsa@imj-prg.fr

Received: 9 November 2020
Accepted: 29 November 2021

Abstract

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 K̂ 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.