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

Abstract

We use a Riemannnian approximation scheme to define a notion of $\text{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^2$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\text{sub-Riemannian signed geodesic curvature}$ 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-Carathéodory distance in $\mathbb{H}$ is provided.