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Submanifolds with constant scalar curvature

Published online by Cambridge University Press:  12 July 2007

Qing-Ming Cheng
Qing-Ming Cheng
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan ([email protected])
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Abstract

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 132 , Issue 5 , October 2002 , pp. 1163 - 1183
Copyright
Copyright © Royal Society of Edinburgh 2002

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