The sub-Riemannian limit of curvatures for curves and surfaces and a Gauss-Bonnet theorem in the rototranslation group. (English) Zbl 1477.53055
Summary: The rototranslation group \(\mathscr{RT}\) is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean \(C^2\)-smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean \(C^2\)-smooth curves on surfaces. Based on these results, we obtain a Gauss-Bonnet theorem in the rototranslation group.
MSC:
| 53C17 | Sub-Riemannian geometry |
References:
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