Table of Contents
ISRN Geometry
Volume 2012 (2012), Article ID 591296, 23 pages
http://dx.doi.org/10.5402/2012/591296
Research Article

On Some 𝐿 𝑘 -Finite-Type Euclidean Hypersurfaces

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran

Received 24 May 2012; Accepted 26 July 2012

Academic Editors: I. Biswas, A. Ferrandez, and G. Martin

Copyright © 2012 Akram Mohammadpouri and S. M. B. Kashani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, World Scientific Publishing, Singapore, 1984.
  2. B. Y. Chen, “A report on submanifolds of finite type,” Soochow Journal of Mathematics, vol. 22, no. 2, pp. 117–337, 1996. View at Google Scholar · View at Zentralblatt MATH
  3. R. C. Reilly, “Variational properties of functions of the mean curvatures for hypersurfaces in space forms,” Journal of Differential Geometry, vol. 8, no. 3, pp. 465–477, 1973. View at Google Scholar · View at Zentralblatt MATH
  4. H. Rosenberg, “Hypersurfaces of constant curvature in space forms,” Bulletin des Sciences Mathématiques, vol. 117, no. 2, pp. 211–239, 1993. View at Google Scholar · View at Zentralblatt MATH
  5. S. M. B. Kashani, “On some L1-fnite type (hyper)surfaces in n+1,” Bulletin of the Korean Mathematical Society, vol. 46, no. 1, pp. 35–43, 2009. View at Publisher · View at Google Scholar
  6. A. Ferrández and P. Lucas, “Null finite type hypersurfaces in space forms,” Kodai Mathematical Journal, vol. 14, no. 3, pp. 406–419, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. B. Y. Chen, “Null 2-type surfaces in E3 are circular cylinders,,” Kodai Mathematical Journal, vol. 11, no. 2, pp. 295–299, 1988. View at Publisher · View at Google Scholar
  8. Th. Hasanis and Th. Vlachos, “Surfaces of finite type with constant mean curvature,” Kodai Mathematical Journal, vol. 16, no. 2, pp. 244–252, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. L. J. Alías and N. Gürbüz, “An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures,” Geometriae Dedicata, vol. 121, pp. 113–127, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. G. Wei, “Complete hypersurfaces in a Euclidean space n+1 with constant th mean curvature,” Differential Geometry and Its Applications, vol. 26, no. 3, pp. 298–306, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. H. Alencar and M. Batista, “Hypersurfaces with null higher order mean curvature,” Bulletin of the Brazilian Mathematical Society, vol. 41, no. 4, pp. 481–493, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. B. G. Yang and X. M. Liu, “R-minimal hypersurfaces in space forms,” Journal of Geometry and Physics, vol. 59, no. 6, pp. 685–692, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. Y. Chen and S. Ishikawa, “On classification of some surfaces of revolution of finite type,” Tsukuba Journal of Mathematics, vol. 17, no. 1, pp. 287–298, 1993. View at Google Scholar · View at Zentralblatt MATH
  14. T. Takahashi, “Minimal immersions of Riemannian manifolds,” Journal of the Mathematical Society of Japan, vol. 18, pp. 380–385, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. B. Y. Chen and H. S. Lue, “Some 2-type submanifolds and applications,” Annales de la Faculté des Sciences de Toulouse, vol. 9, no. 1, pp. 121–131, 1988. View at Google Scholar
  16. B. Segre, “Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di dimensioni,” Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, vol. 27, pp. 203–207, 1938. View at Google Scholar
  17. R. Walter, “Compact hypersurfaces with a constant higher mean curvature function,” Mathematische Annalen, vol. 270, no. 1, pp. 125–145, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. B. O'Neill, “Immersion of manifolds of nonpositive curvature,” Proceedings of the American Mathematical Society, vol. 11, pp. 132–134, 1960. View at Google Scholar · View at Zentralblatt MATH
  19. T. Ôtsuki, “Minimal hypersurfaces in a Riemannian manifold of constant curvature,” American Journal of Mathematics, vol. 92, no. 1, pp. 145–173, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, Wiley-Interscience, New York, NY, USA, 1963, Vol. 2, 1969.
  21. S. Nishikawa and Y. Maeda, “Conformally flat hypersurfaces in a conformally flat Riemannian manifold,” The Tohoku Mathematical Journal, vol. 26, pp. 159–168, 1974. View at Google Scholar · View at Zentralblatt MATH
  22. A. Ferrandez, O. J. Garay, and P. Lucas, “On a certain class of conformally at Euclidean hypersurfaces,” in Proceedings of the Conference on Global Analysis and Global Differential Geometry, pp. 48–54, Berlin, Germany, 1990.