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PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  01 February 2008

MAYUKO KON
MAYUKO KON*
Affiliation:
Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Sapporo 060-0810, Japan (email: mayuko_k13@math.sci.hokudai.ac.jp)
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Abstract

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We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

Keywords

minimal submanifoldcomplex projective spacesectional curvatureflat normal connectionsemi-flat normal connection

MSC classification

Secondary: 53B25: Local submanifolds 53C40: Global submanifolds 53B35: Hermitian and Kählerian structures 53C55: Hermitian and Kählerian manifolds
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 1 , February 2008 , pp. 99 - 114
Copyright
Copyright © Australian Mathematical Society 2008

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