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

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 $\hat{K}$ 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.

arXiv comment24 pages, 15 figures

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• Minimal surfaces in the Heisenberg group
• Structures de contact en dimension trois et bifurcations des feuilletages de surfaces
• Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds
• Heat content asymptotics for sub-Riemannian manifolds
• Limit of Gaussian and normal curvatures of surfaces in Riemannian approximation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet theorem