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

TL;DR

This paper studies the metric geometry induced on surfaces within 3D contact sub-Riemannian manifolds, focusing on characteristic points, foliation, and conditions for finite induced distances, especially for spherical surfaces.

Contribution

It introduces a coefficient to analyze characteristic points and establishes global conditions for the induced distance to be finite on certain surfaces.

Findings

The coefficient $ ilde{K}$ determines the characteristic foliation locally.

Finite induced distance is guaranteed for spherical surfaces in tight coorientable distributions.

Surfaces with isolated characteristic points can have finite induced distances under specific conditions.

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 $\widehat K$ 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.