A sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds

Abstract We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of three-dimensional contact sub-Riemannian manifolds. Using a family of taming Riemannian metrics on the surface which approach a sub-Riemannian metric, we carefully study the usual Gauss-Bonnet formula under this limit. We are then able to recover the Euler characteristic of the surface from the geometry around the surface’s characteristic set, i.e., the points where the tangent space to the surface and contact structure coincide. For the case of surfaces with boundary, we also give a Gauss-Bonnet type result which includes limits related to the boundary’s curvature.