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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Characterizations of the sphere by the curvature of the second fundamental form


Author: Udo Simon
Journal: Proc. Amer. Math. Soc. 55 (1976), 382-384
MSC: Primary 53C45
DOI: https://doi.org/10.1090/S0002-9939-1976-0405301-5
MathSciNet review: 0405301
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Abstract: On an ovaloid $ S$ with Gaussian curvature $ K({\text{I)}} > 0$ in Euclidean three-space $ {E^3}$ the second fundamental form defines a nondegenerate Riemannian metric with curvature $ K({\text{II}})$. R. Schneider [7] proved that the spheres in Euclidean space $ {E^{n + 1}}$ are the only closed hypersurfaces on which the second fundamental form defines a nondegenerate Riemannian metric of constant curvature. For surfaces in $ {E^3}$ we give a common generalization of Schneider's theorem and the classical theorem of Liebmann [6] (which states that any ovaloid in $ {E^3}$ with constant Gaussian curvature is a sphere).


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DOI: https://doi.org/10.1090/S0002-9939-1976-0405301-5
Article copyright: © Copyright 1976 American Mathematical Society