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Characterizations of the sphere by the mean II-curvature

Published online by Cambridge University Press:  17 April 2009

George Stamou
George Stamou
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, Thessalonicki, Greece.
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Abstract

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The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.

In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 2 , April 1981 , pp. 249 - 253
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Glässner, Ekkehart, “Über die Minimalflächen der zweiten Fundamental-form”, Monatsh. Math. 78 (1974), 193–214.CrossRefGoogle Scholar
[2]Hopf, E., “Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus”, S.-ber. Preuss. Akad. Hiss. (1927), 147–152.Google Scholar
[3]Laugwitz, Detlef, Differentialgeometrie, Zweite, durchgesehene Auflage (Teubner, Stuttgart, 1968).Google Scholar
[4]Stamou, George, “Global characterizations of the sphere”, Proc. Amer. Math. Soc. 68 (1978), 328–330.CrossRefGoogle Scholar