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A general rigidity theorem for complete submanifolds

Published online by Cambridge University Press:  22 January 2016

Katsuhiro Shiohama  and
Hongwei Xu
Katsuhiro Shiohama
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan
Hongwei Xu
Affiliation:
Institute of mathematics, Zhejiang University, Hangzhou, 310027-China
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Abstract.

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Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature normal in a complete and simply connected Riemannian (n+p) -manifold Nn+p with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number T(n,p) ∈ (0,1) with the property that if the sectional curvature of N is pinched in [T(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then Nn+p is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 150 , June 1998 , pp. 105 - 134
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

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