Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg   group
  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. 
Zolt\'an Balogh, Jeremy T. Tyson, Eugenio Vecchi
math.DG