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Topological rigidity for closed hypersurfaces of elliptic space forms

Part of: Global differential geometry

Published online by Cambridge University Press:  20 June 2019

Eduardo Rosinato Longa  and
Jaime Bruck Ripoll
Eduardo Rosinato Longa
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil (eduardo.longa@ufrgs.br; jaime.ripoll@ufrgs.br)
Jaime Bruck Ripoll
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil (eduardo.longa@ufrgs.br; jaime.ripoll@ufrgs.br)
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Abstract

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

Keywords

rigidityhypersurfacestopologyspherespace forms

MSC classification

Primary: 53C24: Rigidity results
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1063 - 1072
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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