The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric
The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E1,1,gλ1,λ2, where λ1≥λ2>0. It provides a natural 2-parametric deformation family of the Riemannian homogeneous manifold Sol3=E1,1,g1,1 which is the model space to solve geometry in the eight mode...
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Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/1431082 |
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| author | Jianyun Guan Haiming Liu |
| author_facet | Jianyun Guan Haiming Liu |
| author_sort | Jianyun Guan |
| collection | DOAJ |
| description | The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E1,1,gλ1,λ2, where λ1≥λ2>0. It provides a natural 2-parametric deformation family of the Riemannian homogeneous manifold Sol3=E1,1,g1,1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in E1,1,gLλ1,λ2 away from characteristic points and signed geodesic curvature for the Euclidean C2-smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric. |
| format | Article |
| id | doaj-art-8cfb41d97d154c71ac920934d3b23875 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
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| series | Journal of Function Spaces |
| spelling | doaj-art-8cfb41d97d154c71ac920934d3b238752025-02-03T07:24:03ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/14310821431082The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant MetricJianyun Guan0Haiming Liu1School of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, ChinaSchool of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, ChinaThe group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E1,1,gλ1,λ2, where λ1≥λ2>0. It provides a natural 2-parametric deformation family of the Riemannian homogeneous manifold Sol3=E1,1,g1,1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in E1,1,gLλ1,λ2 away from characteristic points and signed geodesic curvature for the Euclidean C2-smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.http://dx.doi.org/10.1155/2021/1431082 |
| spellingShingle | Jianyun Guan Haiming Liu The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric Journal of Function Spaces |
| title | The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric |
| title_full | The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric |
| title_fullStr | The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric |
| title_full_unstemmed | The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric |
| title_short | The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric |
| title_sort | sub riemannian limit of curvatures for curves and surfaces and a gauss bonnet theorem in the group of rigid motions of minkowski plane with general left invariant metric |
| url | http://dx.doi.org/10.1155/2021/1431082 |
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