A Special class of Hypersurfaces parametrized by lines of curvature in R4
DOI:
https://doi.org/10.17268/sel.mat.2018.01.07Keywords:
Dupin hypersurfaces, Laplace invariants, lines of curvatureAbstract
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, we
show that these vector valued functions are invariant under inversions and homotheties.
References
Cecil, T.E. and Chern, S.S. Dupin submanifolds in Lie sphere geometry, Differential geometry and topology 1 - 48, Lecture Notes in Math. vol. 1369, Springer, Berlin, Heidelberg, 1989.
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, 1998; 359 - 378.
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.
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