Two characteristic properties of the sphere

Koutroufiotis, Dimitri

doi:10.1090/S0002-9939-1974-0339025-8

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Two characteristic properties of the sphere
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by Dimitri Koutroufiotis PDF

Proc. Amer. Math. Soc. 44 (1974), 176-178 Request permission

Abstract:

Let $S$ be a closed, convex surface in ${E^3}$ with positive Gaussian curvature $K$ and let ${K_{{\text {II}}}}$ be the curvature of its second fundamental form. It is shown that $S$ is a sphere if ${K_{{\text {II}}}} \equiv cK$ for some constant $c$ or if ${K_{{\text {II}}}} \equiv \surd K$.

References

Rolf Schneider, Closed convex hypersurfaces with second fundamental form of constant curvature, Proc. Amer. Math. Soc. 35 (1972), 230–233. MR 307133, DOI 10.1090/S0002-9939-1972-0307133-1