%0 Journal Article
%T Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$
%A Jos¨¦ M. M. Veloso
%A Marcos M. Diniz
%J Mathematics
%D 2012
%I arXiv
%X We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with 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 $\int_S K = 0$.
%U http://arxiv.org/abs/1210.7110v1