On the Lie curvature of Dupin hypersurfaces

Abstract

A hypersurface M in a standard sphere S is said to be Dupin if each of its principal curvatures is constant along its corresponding curvature surfaces. If the number of distinct principal curvatures is constant, then M is called a proper Dupin hypersurface. There is a close relationship between the class of compact proper Dupin hypersurfaces and the class of isoparametric hypersurfaces. Miinzner [11] showed that the number g of distinct principal curvatures of an isoparametric hypersurface must be 1, 2, 3, 4 or 6. Thorbergsson [15] then showed that the same restriction holds for a compact proper Dupin hypersurface embedded in S by reducing that case to a situation where Mίmzner's argument can be applied. This also implied that the rank of the Z2-cohomology ring in both cases must be 2g. Later Grove and Halperin [6] found more topological similarities between these two classes of hypersurfaces. All of this led to the conjecture [5, p. 184] that every compact proper Dupin hypersurface in S is equivalent by a Lie sphere transformation to an isoparametric hypersurface. The conjecture was known to be true in the cases g=l (umbilic hypersurfaces), g=2[4] and g=3[7]. Recently, however, counterexamples to the conjecture for g=4 have been discovered by Miyaoka and Ozawa [10] and by Pinkall and Thorbergsson [14]. Miyaoka and Ozawa also produced counterexamples in the case £—6. In all cases, the proof that the counterexamples are not Lie equivalent to an isoparametric hypersurface uses the so-called Lie curvature ^introduced by Miyaoka [8]. For a proper Dupin hypersurface Mwith four principal curvatures, Ψ is the cross-ratio of these principal curvatures. Viewed in the context of Lie sphere geometry, Ψ is the cross-ratio of the four points on a projective line corresponding to the four curvature spheres of M. Hence, Ψ is a natural Lie invariant. From Mϋnzner's work, it is easy to compute that Ψ has a constant value 1/2 on a Dupin hypersurface which is Lie equivalent to an isoparametric hypersurface. In projective geometric terms, this means that the four curvature spheres at each point of M form a harmonic set. For the counterexamples to the conjecture above, Ψ does not have the constant value 1/2.