Deformations of Dupin hypersurfaces

Abstract

Abstract.Examples are given of compact proper-Dupin hypersurfaces that are not Lie equivalent to an isoparametric hypersurface. DUPIN HYPERSURFACESA question raised by many authors is whether a compact proper-Dupin hypersurface is equivalent to an isoparametric hypersurface under a Lie transformation. Here we say that a hypersurface or more generally a submanifold of S"4 is proper-Dupin if the principal curvatures have globally constant multiplicities and are constant along the corresponding curvature leaves which lie in the unit normal bundle if the codimension is greater than one. An evidence for a positive answer to the question is that the geometry and topology of proper-Dupin hypersurfaces and isoparametric hypersurfaces are very similar. E.g., for both classes the number g of principal curvatures can only be 1, 2, 3, 4 or 6 and the dimension of the Z-,-cohomology ring is 2g, see [Mii] and [Th]. See also [GH] for further similarities in the topology of these two classes of hypersurfaces.The question has been answered positively for the cases g = 1 (trivial), g -2 [CR], see also [Pi], and g = 3 [Mi 1]. It will be shown in the present paper that the answer is negative for g = 4. More precisely, we will find a oneparameter deformation M(t) of any isoparametric hypersurface M of Clifford

Isoparametric and Dupin Hypersurfaces

“…This is the Clifford-Stiefel manifold of Clifford orthogonal 2-frames of length 1/ √ 2 in R l (see Pinkall and Thorbergsson [117]), where vectors u and v in R l are said to be Clifford orthogonal…”

“…However, in 1988 Pinkall and Thorbergsson [117], and Miyaoka and Ozawa [90] gave two different methods for producing counterexamples to this conjecture with four principal curvatures. The method of Miyaoka and Ozawa also yields counterexamples to the conjecture in the case of six principal curvatures.…”

“…The counterexamples of Pinkall and Thorbergsson [117] (see also [26, pp. 112-117]) to the conjecture are obtained by modifying the isoparametric hypersurfaces of FKM-type discussed in the previous section.…”

Dupin hypersurfaces with four principal Curvatures, II

“…At one time it was thought that perhaps every compact proper Dupin is Lie equivalent to an isoparametric hypersurface. However, by two separate constructions, Pinkall and Thorbergsson [18] (g = 4) and Miyaoka and Ozawa [13] (g = 4 or 6) produced compact proper Dupin hypersurfaces which do not have constant Lie curvatures and therefore cannot be Lie equivalent to an isoparametric hypersurface.…”

Laplace invariants in hypersurfaces parametrized by lines of curvature

“…Introducción. Superficies de Dupin fueron estudiadas inicialmente por Dupin en 1822 y mas recientemente por otros autores por ejemplo [1]- [3], [6]- [11] y [13]- [16], los cuales estudiaron varios aspectos de las hipersuperficies de Dupin. la clase de hipersuperficies de Dupin es invariante por el grupo de transformaciones de Lie (ver [10]).…”