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RIGIDITY OF HYPERSURFACES IN A EUCLIDEAN SPHERE

Published online by Cambridge University Press:  02 February 2006

Qiaoling Wang
Affiliation:
Departamento de Matemática-IE, Universidade de Brasília, Campus Universitário, 70910-900 Brasília-DF, Brazil (wang@mat.unb.br; xia@mat.unb.br)
Changyu Xia
Affiliation:
Departamento de Matemática-IE, Universidade de Brasília, Campus Universitário, 70910-900 Brasília-DF, Brazil (wang@mat.unb.br; xia@mat.unb.br)
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Abstract

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This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2006