Dupin Hypersurfaces in ℝ⁵

Abstract

Abstract. We study Dupin hypersurfaces in R 5 parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.

On Dupin hypersurfaces with constant Möbius curvature

“…In the last three decades, several aspects of Dupin hypersurfaces were studied by many authors [Cecil and Chern 1989;Cecil et al 2007;Cecil and Jensen 1998;2000;1980;Miyaoka 1984;1989;Niebergall 1991;1992;Pinkall 1981;1985b;1985a;Pinkall and Thorbergsson 1989;Riveros and Tenenblat 2005;Stolz 1999;Thorbergsson 1983]. A hypersurface is said to be Dupin if each principal curvature is constant along its corresponding surface of curvature.…”

“…We observe that, for Dupin hypersurfaces parametrized by lines of curvature, the condition of having constant Möbius curvature is equivalent to having all higher dimensional Laplace invariants equal to zero; see Lemma 3.2. These invariants were introduced by Kamran and Tenenblat [1996;1998], and they were used to study a class of Dupin hypersurfaces of ‫ޒ‬ 5 in [Riveros and Tenenblat 2005].…”

Dupin hypersurfaces with four principal curvatures in ℝ5 with principal coordinates

“…Dupin surfaces were first studied by Dupin in 1822 and more recently by many authors [1][2][3][4][5][6][9][10][11][12][13][14][15][16][17][18][19][20][21], which studied several aspects of Dupin hypersurfaces. The class of Dupin hypersurfaces is invariant under Lie transformations [12].…”

Classes of Hypersurfaces With Vanishing Laplace Invariants

“…In [10], Riveros and Tenenblat used the Laplace invariants to obtain a characterization of Dupin hypersurfaces in R 5 , parametrized by lines of curvature and with four distinct principal curvatures.…”