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Normal curvature of minimal submanifolds in a sphere

Published online by Cambridge University Press:  18 May 2009

Sharief Deshmukh
Sharief Deshmukh
Affiliation:
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh-11451, Saudi Arabia
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Simons [5] has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki [4] improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfies

then either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen [6] but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 1 , January 1997 , pp. 29 - 33
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

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