On four dimensional Dupin hypersurfaces in Euclidean space

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.

A Special class of Hypersurfaces parametrized by lines of curvature in R4

“…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.…”

“…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].…”

Dupin Hypersurfaces in ℝ5

“…The results of this paper were announced in [15]. Higher dimensional generalizations of Theorem 3.1 and the non generic case will appear elsewhere.…”

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].…”