Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:42:17.771Z Has data issue: false hasContentIssue false

Umbilical points on surfaces in RN

Published online by Cambridge University Press:  22 January 2016

Kazuyuki Enomoto*
Affiliation:
1-17-13 Jiyugaoka Meguro-ku, Tokyo 152, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ϕ: MRN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Chen, B. Y. and Ludden, G. D., Surfaces with mean curvature vector parallel in the normal bundle, Nagoya Math. J., 47 (1972), 161167.CrossRefGoogle Scholar
[ 2 ] Chen, B. Y., On the surface with parallel mean curvature vector, Indiana Univ. Math. J., 22 (1973), 655666.CrossRefGoogle Scholar
[ 3 ] Chen, B. Y., Geometry of Submanifolds, Marcel Dekker, 1973.Google Scholar
[ 4 ] Erbacher, J., Isometric immersions of constant mean curvature and triviality of the normal connection, Nagoya Math. J., 45 (1971), 139165.CrossRefGoogle Scholar
[ 5 ] Henke, W., Riemannsche Mannigfaltigkeiten konstanter positiver Krümmung in euklidischen Raumen der Kodimension 2, Math. Ann., 193 (1971), 265278.CrossRefGoogle Scholar
[ 6 ] Hoffmann, D. A., Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geom., 8 (1973), 161176.CrossRefGoogle Scholar
[ 7 ] O’Neill, B. 0., Umbilics of constant curvature immersions, Duke Math. J., 32 (1965), 149159.CrossRefGoogle Scholar
[ 8 ] Yau, S. T., Submanifolds with constant mean curvature I, Araer. J. Math., 96 (1974), 346366.CrossRefGoogle Scholar