Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-05T02:34:45.389Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterisation of Helices and Cornu spirals in real space forms

Published online by Cambridge University Press:  17 April 2009

J. Arroyo ,
M. Barros  and
O.J. Garay
J. Arroyo
Affiliation:
Departamento de Geometris y TopologiaUniversidad de Granada18071 GranadaSpain e-mail: [email protected]
M. Barros
Affiliation:
Departamento de MatématicasFacultad de CienciasUniversidad del Pail Vasco/Euskal Herriko Unibertsitatea48080 BilbaoSpain e-mail: [email protected]
O.J. Garay
Affiliation:
Departamento de MatématicasFacultad de CienciasUniversidad del Pail Vasco/Euskal Herriko Unibertsitatea48080 BilbaoSpain e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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.

We classify unit speed curves contained in a real space form of arbitrary dimension Nm(c), whose mean curvature vector is proper for the Laplacian. Then we use these results to classify Hopf cylinders of S3 and semi-Riemannian Hopf cylinders of with proper mean curvature function.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 1 , August 1997 , pp. 37 - 49
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Barros, M., ‘General helices and a Theorem of Lancret’, Procc. Amer. Math. Soc. (to appear).Google Scholar
[2]Barros, M., Ferrández, A., Lucas, P. and Meroño, M.A., ‘Hopf cylinders, B-Scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti De Sitter space’, Comptes Rendus Acad. Sci. Paris, (to appear).Google Scholar
[3]Barros, M. and Garay, O.J., ‘On submanifolds with harmonic mean curvature’, Proc. Amer. Math. Soc. 123 (1995), 2545–2549.CrossRefGoogle Scholar
[4]Do Carmo, M.P., Riemannian geometry, Mathematics Theory and Applications (Birkhausser, 1992).CrossRefGoogle Scholar
[5]Chen, B-Y, Total mean curvature and submanifolds of Finite Type (World Scientific, Singapore, 1984).Google Scholar
[6]Chen, B-Y, ‘Some open problems and conjectures on submanifolds of finite type’, Soochow J. Math. 17 (1991), 169–188.Google Scholar
[7]Chen, B-Y, ‘Null-2 type surfaces in Euclidean spaces’, Algebra, Analysis and Geometry (1988), 1–18.Google Scholar
[8]Chen, B-Y, ‘Submanifolds in De Sitter space-time satisfying ΔH = λH’ Israel J. Math. 91 (1995), 373–391.Google Scholar
[9]Chen, B-Y, ‘A report on submanifolds of Finite Type’, Soochow Math. J. 22 (1996), 117–337.Google Scholar
[10]Chen, B-Y and Ishikawa, S., ‘Biharmonic surfaces in pseudo-Euclidean spaces’, Mem. Fac. Sci. Kyushu Univ. Ser. A. Math. 45 (1991), 323–347.Google Scholar
[11]Dimitric, I., ‘Submanifolds of E m with harmonic mean curvature vector’, Bull. Inst. Math. Acad. Sinica. 20 (1992), 53–65.Google Scholar
[12]Erbacher, J., ‘Reduction of codimension of a isometric immersion’, J. Differential Geom. 5 (1971), 333–340.CrossRefGoogle Scholar
[13]Hassanis, Th. and Vlachos, Th., ‘Hypersurfaces in E 4 with harmonic mean curvature vector field’, Math. Nachr. 172 (1995), 145–169.Google Scholar
[14]Pinkall, U., ‘Hopf Tori in S 3’, Invent. Math. 81 (1985), 379–386.Google Scholar
[15]Simons, J., ‘Minimal varieties in Riemannian manifolds’, Ann. Math. 88 (1968), 62–105.CrossRefGoogle Scholar
[16]Takahashi, T., ‘Minimal immersions of Riemannian manifolds’, J. Math. Soc. Japan. 18 (1966), 380–385.CrossRefGoogle Scholar