Abstract:Dupin hypersurfaces in five dimensional Euclidean space parametrized by lines of curvature, with four distinct principal curvatures, are considered. A generic family of such hypersurfaces is locally characterized in terms of the principal curvatures and four vector valued functions of one variable. These functions are invariant by inversions and homotheties.
“…From Remark 2.2 in [16], follows that for n ≥ 3, the higher-dimensional Laplace invariants do not change under inversions in spheres centered at the origin and homotheties.…”
Section: Preliminariesmentioning
confidence: 95%
“…Introduction. Dupin surfaces were first studied by Dupin in 1822 and more recently by many authors [1]- [6], [9]- [14] and [16], [17], which studied several aspects of Dupin hypersurfaces. The class of Dupin hypersurfaces is invariant under Lie transformations [11].…”
In this paper we study hypersurfaces in R 4 parametrized by lines of curvature with three distinct principal curvatures and with Laplace invariants m ji = m ki = 0, m jik = 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.
“…From Remark 2.2 in [16], follows that for n ≥ 3, the higher-dimensional Laplace invariants do not change under inversions in spheres centered at the origin and homotheties.…”
Section: Preliminariesmentioning
confidence: 95%
“…Introduction. Dupin surfaces were first studied by Dupin in 1822 and more recently by many authors [1]- [6], [9]- [14] and [16], [17], which studied several aspects of Dupin hypersurfaces. The class of Dupin hypersurfaces is invariant under Lie transformations [11].…”
In this paper we study hypersurfaces in R 4 parametrized by lines of curvature with three distinct principal curvatures and with Laplace invariants m ji = m ki = 0, m jik = 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.
“…The results of this paper were announced in [15]. Higher dimensional generalizations of Theorem 3.1 and the non generic case will appear elsewhere.…”
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.
“…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].…”
In this paper we study Dupin hypersurfaces in R 5 parametrized by lines of curvature, with four distinct principal curvatures and T ij kl = 0. We characterize locally a family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. Moreover, we show that these vector valued functions are invariant under inversions and homotheties.
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