Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group

TL;DR

This paper introduces a sub-Riemannian Gaussian curvature and signed geodesic curvature for smooth surfaces and curves in the Heisenberg group, leading to a Gauss-Bonnet theorem adaptation and applications to distance formulas.

Contribution

It defines new curvature notions in the Heisenberg group using Riemannian approximations and proves a Gauss-Bonnet theorem in this sub-Riemannian setting.

Findings

Defined sub-Riemannian Gaussian curvature for surfaces in $ ext{Heisenberg}$

Proved a Gauss-Bonnet theorem in the sub-Riemannian context

Applied results to Steiner's formula for Carnot-Carathéodory distance

Abstract

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carath\'eodory distance in $\mathbb{H}$ is provided.