A sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds

TL;DR

This paper extends the classical Gauss-Bonnet theorem to sub-Riemannian geometry, relating surface topology to geometric properties in contact manifolds using a limiting process with taming metrics.

Contribution

It introduces a sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds, linking topology to geometry via a limiting approach with taming metrics.

Findings

Derived a sub-Riemannian Gauss-Bonnet formula for surfaces

Revealed topological information from geometric data around characteristic points

Established results for surfaces with and without boundary

Abstract

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian result in the limit. In particular, we are able to recover topological information of the surface from the geometry around the characteristic set, i.e., the points where the tangent space to the surface and contact structure coincide. We both give a version for surfaces without boundary and surfaces with boundary.