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Global characterizations of the sphere


Author: George Stamou
Journal: Proc. Amer. Math. Soc. 68 (1978), 328-330
MSC: Primary 53C45
DOI: https://doi.org/10.1090/S0002-9939-1978-0467620-8
MathSciNet review: 0467620
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Abstract: Let S be an ovaloid in Euclidean three-space $ {E^3}$ with Gaussian curvature $ K > 0$ and let $ {K_{{\text{II}}}}$ be the curvature of the second fundamental form II of S. We give some global characterizations of the sphere by the curvature $ {K_{{\text{II}}}}$ which generalize some results of R. Schneider [4], D. Koutroufiotis [2] and the well-known ``H-Satz'' theorem of H. Liebmann.


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  • [1] P. J. Erard, Über die zweite Fundamentalform von Flächen im Raum, Doctoral Thesis, Diss. No. 4234, Eidgenössische Technische Hochschule, Zürich, 1968. MR 44 #5866. MR 0288670 (44:5866)
  • [2] D. Koutroufiotis, Two characteristic properties of the sphere, Proc. Amer. Math. Soc. 44 (1974), 176-178. MR 0339025 (49:3788)
  • [3] D. Laugwitz, Differentialgeometrie, 2nd ed., Teubner, Stuttgart, 1968. MR 0243432 (39:4754)
  • [4] R. Schneider, Closed convex hypersurfaces with second fundamental form of constant curvature, Proc. Amer. Math. Soc. 35 (1972), 230-233. MR 46 #6254. MR 0307133 (46:6254)
  • [5] U. Simon, Characterizations of the sphere by the curvature of the second fundamental form, Proc. Amer. Math. Soc. 55 (1976), 382-384. MR 0405301 (53:9095)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0467620-8
Keywords: Ovaloid, curvature of the second fundamental form, sphere
Article copyright: © Copyright 1978 American Mathematical Society

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