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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.
Zhang et al. (Tue,) studied this question.