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On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

Published online by Cambridge University Press:  17 April 2009

Kang-Hai Tan  and
Xiao-Ping Yang
Kang-Hai Tan
Affiliation:
Department of Applied Mathematics, Nanjing University of Science and Technology, 210094, Nanjing, Jiangsu, Peoples Republic of China e-mail: tankanghai2000@yahoo.com.cn, yangxp@mail.njust.edu.cn
Xiao-Ping Yang
Affiliation:
Department of Applied Mathematics, Nanjing University of Science and Technology, 210094, Nanjing, Jiangsu, Peoples Republic of China e-mail: tankanghai2000@yahoo.com.cn, yangxp@mail.njust.edu.cn
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We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 177 - 198
Copyright
Copyright © Australian Mathematical Society 2004

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