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Proceedings of the American Mathematical Society

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

 

 

A characteristic property of the sphere


Authors: Themis Koufogiorgos and Thomas Hasanis
Journal: Proc. Amer. Math. Soc. 67 (1977), 303-305
MSC: Primary 53C45
DOI: https://doi.org/10.1090/S0002-9939-1977-0487927-7
MathSciNet review: 0487927
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Abstract: On an ovaloid S with Gaussian curvature $ K > 0$ in Euclidean three-space $ {E^3}$, the second fundamental form defines a nondegenerate Riemannian metric with curvature $ {K_{{\text{II}}}}$. It is shown that S is a sphere if $ {K_{{\text{II}}}} = c{H^s}{K^r}$, where c, s and r are constants, H is the mean curvature of S and $ 0 \leqslant s \leqslant 1$.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0487927-7
Keywords: Ovaloid, curvature of the second fundamental form
Article copyright: © Copyright 1977 American Mathematical Society