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On a class of submanifolds carrying an extrinsic totally umbilical foliation

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Abstract

In this paper we give a conformal classification of all euclidean submanifolds carrying a parallel principal curvature normal under the intrinsic additional assumption that the associated conformal conullity is involutive and the leaves are extrinsic spheres in the submanifold in the sense of Nomizu. We also provide several applications of this result.

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References

  1. A. Asperti and M. Dajczer,N-dimensional submanifolds of R N+1 and SN+2, Illinois Journal of Mathematics28 (1984), 621–645.

    MATH  MathSciNet  Google Scholar 

  2. O. Bonnet,Mémoire sur la théorie des surfaces applicables, Journal de l’École PolytechniqueXLII (1867).

  3. E. Cartan,La déformation des hypersurfaces dans l’espace conforme réel a n ≥ 5 dimensions, Bulletin de la Société Mathématique de France45 (1917), 57–121.

    MathSciNet  Google Scholar 

  4. T. E. Cecil,Lie Sphere Geometry, Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  5. T. E. Cecil,Lie sphere geometry and Dupin submanifolds, inGeometry and Topology of Submanifolds, III (L. Verstraelen and A. West, eds.), World Scientific, Singapore, 1991, pp. 90–107.

    Google Scholar 

  6. B. Y. Chen,Some conformal invariants of submanifolds and their applications, Unione Matematica Italiana. Bollettino10 (1974), 380–385.

    MATH  Google Scholar 

  7. M. do Carmo and M. Dajczer,Riemannian metrics induced by two immersions, Proceedings of the American Mathematical Society86 (1982), 115–119.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Darboux,Leçons sur la théorie des surfaces, Gauthier-Villars, Paris, 1914. (Reprinted by Chelsea Publ. Co., 1972.)

    MATH  Google Scholar 

  9. M. Dajczer and L. Florit,On Chen’s basic equality, Illinois Journal of Mathematics42 (1998), 97–106.

    MATH  MathSciNet  Google Scholar 

  10. M. Dajczer and R. Tojeiro,On Cartan’s conformally deformable hypersurfaces, Preprint.

  11. R. Fukuoka,On conformally flat hypersurfaces of euclidean space with constant mean curvature, Preprint.

  12. U. Hertrich-Jeromin,On conformally flat hypersurfaces and Guichard’s Nets, Beiträge zur Algebra und Geometrie35 (1994), 315–331.

    MATH  MathSciNet  Google Scholar 

  13. J. Lafontaine,Conformal geometry from the Riemannian viewpoint, Aspects of Mathematics, E 12, Vieweg, Braunschweig, 1988.

    Google Scholar 

  14. J. D. Moore,Conformally flat submanifolds in Euclidean space, Mathematische Annalen225 (1977), 89–97.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Nolker,Isometric immersions of warped products, Differential Geometry and its Applications6 (1996), 1–30.

    Article  MathSciNet  Google Scholar 

  16. K. Nomizu,On circles and spheres in Riemannian Geometry, Mathematische Annalen210 (1974), 163–170.

    Article  MATH  MathSciNet  Google Scholar 

  17. U. Pinkall,Dupin hypersurfaces, Mathematische Annalen270 (1985), 427–440.

    Article  MATH  MathSciNet  Google Scholar 

  18. H. Reckziegel,Krummungsflachen von isometrischen immersionen in raume konstanter krummung, Mathematische Annalen223 (1976), 169–181.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Ribaucour,Sur la théorie des surfaces, Bulletin de la Société Philomathique (1870).

  20. C.-L. Terng,Submanifolds with flat normal bundle, Mathematische Annalen277 (1987), 95–111.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática Pure e Aplicada, IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil

    M. Dajczer & L. A. Florit

  2. Departamento de Matemática, Universidade Federal de São Carlos, Via Washington Luiz km 235 CP676, 13565-905, São Carlos, SP, Brazil

    R. Tojeiro

Authors
  1. M. Dajczer
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  2. L. A. Florit
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  3. R. Tojeiro
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Correspondence to M. Dajczer.

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Dajczer, M., Florit, L.A. & Tojeiro, R. On a class of submanifolds carrying an extrinsic totally umbilical foliation. Isr. J. Math. 125, 203–220 (2001). https://doi.org/10.1007/BF02773380

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  • DOI: https://doi.org/10.1007/BF02773380

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