Spherical rigidities of submanifolds in Euclidean spaces
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概要
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In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space \bm{E}<SUP>n+p</SUP>. We prove that if M<SUP>n</SUP> is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in \bm{E}<SUP>n+p</SUP> and satisfies either:(1) s\displaystyle ≤\frac{n<SUP>2</SUP>H<SUP>2</SUP>}{n-1}, or(2) n<SUP>2</SUP>H<SUP>2</SUP>\displaystyle ≤\frac{(n-1)R}{n-2}, then M<SUP>n</SUP> is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of M<SUP>n</SUP> and the scalar curvature of M<SUP>n</SUP> respectively.On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [{11}] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere S<SUP>n</SUP>(c), the totally geodesic Euclidean space \bm{E}<SUP>n</SUP>, and the generalized cylinder S<SUP>n-1</SUP>(c)× \bm{E}<SUP>1</SUP> are only n-dimensional (n>2) complete connected submanifolds M<SUP>n</SUP> with constant mean curvature H in \bm{E}<SUP>n+p</SUP> if S≤ n<SUP>2</SUP>H<SUP>2</SUP>/(n-1) holds.
- 社団法人 日本数学会の論文
- 2004-04-01
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