Gauss-Bonnet Theorem in Sub-Riemannian Heisenberg Space $\mathbb H^{1}$

Abstract

We prove a version of the Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $\mathbb H^{1}$. The sub-Riemannian distance makes $\mathbb H^{1}$ a metric space that consequently has a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and, for compact surfaces (which are topologically a torus), we obtain ${\int }_{S}K=0$.