Una clase especial de Hipersuperficies parametrizadas por líneas de curvatura en R4

• C. Riveros, Carlos M. ^[1]
  1. [1] Universidade de Brasília

     Universidade de Brasília

     Brasil

     
• Localización: Selecciones Matemáticas, ISSN-e 2411-1783, Vol. 5, Nº. 1 (Enero - Julio), 2018, págs. 48-57
• Idioma: español
• DOI: 10.17268/sel.mat.2018.01.07
• Títulos paralelos:
  • A Special class of Hypersurfaces parametrized by lines of curvature in R4
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En este artículo estudiamos hipersuperficies en R4 parametrizadas por líneas de curvatura con tres curvaturas principales distintas y con invariantes de Laplace mji = mki = 0; mjik 6= 0 para índices fijos i; j; k distintos. Caracterizamos localmente una familia genérica de tales hipersuperficies en términos de las curvaturas principales y tres funciones vectoriales de una variable, esta familia incluye una clase de hipersuperficies de Dupin. Ademas, mostramos que estas funciones vectoriales son invariantes por inversiones y dilataciones.

  • English

    In this paper we study hypersurfaces in R4 parametrized by lines of curvature with three distinct principal curvatures and with Laplace invariants mji = mki = 0; mjik 6= 0 for i; j; k distinct fixed indices. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable, this family includes a classe of Dupin hypersurfaces. Moreover, weshow that these vector valued functions are invariant under inversions and homotheties.

• Referencias bibliográficas
  • Cecil, T.E. and Chern, S.S. Dupin submanifolds in Lie sphere geometry, Differential geometry and topology 1 - 48, Lecture Notes in Math. vol....
  • Cecil, T.E. and Ryan, P.J. Conformal geometry and the cyclides of Dupin , Can. J. Math. 1980; (32): 767 - 782.
  • Cecil, T.E. and Ryan, P.J. Tight and taut immersions of manifolds , Pitman, London, 1985.
  • Chern, S.S. An introduction to Dupin submanifolds, Differential Geometry, 95 - 102, Pitman Monographs Surveys Pure Appl. Math. 52, 1991.
  • Cecil, T.E. and Jensen, G. Dupin hypersurfaces with three principal curvatures, Invent. Math. 1998; (132): 121 - 178.
  • Cecil, T.E. and Jensen, G. Dupin hypersurfaces with four principal curvatures, Geom. Dedicata, 2000; (79): 1 - 49.
  • Kamran, N. and Tenenblat, K. Laplace transformation in higher dimensions, Duke Math. Journal 1996; (84): 237 - 266.
  • Kamran, N. and Tenenblat, K. Periodic systems for the higher-dimensional Laplace transformation, Discrete and continuous dynamical systems,...
  • Miyaoka, R. Compact Dupin hypersurfaces with three principal curvatures, Math. Z. 1984; (187): 433 - 452.
  • Niebergall, R. Dupin hypersurfaces in R5, Geom. Dedicata 1991; (40): 1 - 22, and 1992; (41): 5 - 38.
  • Pinkall, U. Dupinsche Hyperflachen , Dissertation, Univ. Freiburg, 1981.
  • Pinkall, U. Dupinsche Hyperflachen in E4, Manuscripta Math. 1985; (51): 89 - 119.
  • Pinkall, U. Dupin hypersurfaces, Math. Ann. 1985; (270): 427 - 440.
  • Riveros, C.M.C. A Characterization of Dupin hypersurfaces in R4, Bull. Belg. Math. Soc. Simon Stevin 2013; (20): 145 - 154.
  • Riveros, C.M.C. Hipersuperfícies de Dupin em espacos Euclidianos, Dissertation, Univ. Brasília, 2001.
  • Riveros, C.M.C. and Tenenblat, K. On four dimensional Dupin hypersurfaces in Euclidean space An. Acad. Bras. Cienc. 2003; 75(1): 1 - 7.
  • Riveros, C.M.C. and Tenenblat, K. Dupin hypersurfaces in R5: Canad. J. Math. Vol. 2005; 57(6): 1291 - 1313.
  • Stolz,S. Multiplicities of Dupin hypersurfaces, Invent.Math. 1999; (138): 253 - 279.
  • Thorbergsson,G. Dupin hypersurfaces, Bull. London Math. Soc. 1983; (15): 493 - 498.