The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group | Zendy

Haiming Liu | Zendy; Jiajing Miao | Zendy; Wanzhen Li | Zendy; Jianyun Guan | Zendy

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The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group

Author(s) -

Haiming Liu,

Jiajing Miao,

Wanzhen Li,

Jianyun Guan

Publication year - 2021

Publication title -

journal of mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.252

H-Index - 13

eISSN - 2314-4785

pISSN - 2314-4629

DOI - 10.1155/2021/9981442

Subject(s) - mathematics , gaussian curvature , geodesic , lie group , euclidean geometry , group (periodic table) , limit (mathematics) , exponential map (riemannian geometry) , combinatorics , mathematical analysis , pure mathematics , curvature , sectional curvature , geometry , scalar curvature , quantum mechanics , physics

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.

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