Sub-Riemannian Geometry of Curves and Surfaces in Roto-Translation Group Associated with Canonical Connection

The aim of this paper is to obtain the sub-Riemannian properties of the roto-translation group RT. At the same time, we compute the sub-Riemannian limits of Gaussian curvature associated with two kinds of canonical connections for a C2-smooth surface in the roto-translation group away from characteristic points and signed geodesic curvature associated with two kinds of canonical connections for C2-smooth curves on surfaces. Based on these results, we obtain a Gauss-Bonnet theorem in the RT.