The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group
| Autor: | Wanzhen Li, Haiming Liu, Jiajing Miao, Jianyun Guan |
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| Rok vydání: | 2021 |
| Předmět: |
Article Subject
Group (mathematics) General Mathematics 010102 general mathematics Mathematical analysis Lie group 02 engineering and technology 01 natural sciences symbols.namesake Gauss–Bonnet theorem Scheme (mathematics) Euclidean geometry QA1-939 0202 electrical engineering electronic engineering information engineering Gaussian curvature symbols 020201 artificial intelligence & image processing Mathematics::Differential Geometry Limit (mathematics) 0101 mathematics Mathematics Geodesic curvature |
| Zdroj: | Journal of Mathematics, Vol 2021 (2021) |
| ISSN: | 2314-4785 2314-4629 |
| DOI: | 10.1155/2021/9981442 |
| Popis: | The rototranslation group ℛ T 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. |
| Databáze: | OpenAIRE |
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