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THE CONTACT NUMBER OF A EUCLIDEAN SUBMANIFOLD

Published online by Cambridge University Press:  27 May 2004

Bang-Yen Chen  and
Shi-Jie Li
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824–1027, USA (bychen@math.msu.edu)
Shi-Jie Li
Affiliation:
Department of Mathematics, South China Normal University, Guangzhou 510631, People's Republic of China (lisj@scnu.edu.cn)
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Abstract

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We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-$2$ submanifolds with contact number ${\geq}3$. We also study surfaces in $\mathbb{E}^6$ with contact number ${\geq}4$. As an immediate consequence, we obtain the first explicit examples of non-spherical pseudo-umbilical surfaces in Euclidean spaces.

AMS 2000 Mathematics subject classification: Primary 53C40; 53A10. Secondary 53B25; 53C42

Keywords

contact numberEuclidean submanifoldisotropyconstant isotropynormal sectionsgeodesic
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 1 , February 2004 , pp. 69 - 100
Copyright
Copyright © Edinburgh Mathematical Society 2004