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A hypersurface M in a standard sphere Sn 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 Sn by reducing that case to a situation where Mίmzner's argument can be applied. This also implied that the rank of the Z 2-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 Sn 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].

In this note, we show that some global hypotheses are necessary to reach Miyaoka's conclusion by exhibiting a non-compact proper Dupin hypersurface in Sn on which Ψ—l/2 which is not Lie equivalent to an open subset of an isoparametric hypersurface in Sn . We also produce examples on which Ψ has a constant value c, 0<c


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Originally published in Kodai Math. J. Volume 13, Number 1 (1990), 143-153.