| Horizontal Connection and Horizontal Mean Curvature in Carnot Groups |
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| 作者姓名： | Kang Hai TAN Xiao Ping YANG |
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| 作者单位： | Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China |
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| 基金项目： | This work is supported by the National Natural Science Foundation of China （No. 10471063）,Acknowledgements The authors thank the anonymous referees for their careful reading and valuable suggestions. |
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| 摘 要： | In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.
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| 关 键 词： | 水平连接 水平均值曲率 最小界面 几何分析 黎曼几何 |
| 收稿时间： | 2003-09-10 |
| 修稿时间： | 2003-09-102005-01-19 |
Horizontal Connection and Horizontal Mean Curvature in Carnot Groups |
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| Kang Hai TAN Xiao Ping YANG.Horizontal Connection and Horizontal Mean Curvature in Carnot Groups[J].Acta Mathematica Sinica,2006,22(3):701-710. |
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| Authors: | Kang Hai Tan Xiao Ping Yang |
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| Institution: | (1) Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China |
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| Abstract: | In this paper we give a geometric interpretation of the notion of the horizontal mean
curvature which is introduced by Danielli–Garofalo–Nhieu and Pauls who recently introduced sub–Riemannian minimal surfaces
in Carnot groups. This will be done by introducing a natural nonholonomic
connection which is the restriction (projection) of the natural Riemannian connection on the
horizontal bundle. For this nonholonomic connection and (intrinsic) regular hypersurfaces we introduce
the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out
that the horizontal mean curvature is the trace of the horizontal shape operator.
This work is supported by the National Natural Science Foundation of China (No. 10471063) |
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| Keywords: | Carnot groups Nonholonomic connection Horizontal mean curvature Sub-Riemannian minimal surfaces |
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