Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group

Abstract

We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean \(C^{2}\)-smooth surface in the Heisenberg group \(\mathbb {H}\) away from characteristic points, and a notion of intrinsic 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–Caratheodory distance in \(\mathbb {H}\) is provided.