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If M is an isoparametric hypersurface in a sphere Sn with four distinct principal curvatures, then the principal curvatures κ1, . . . , κ4 can be ordered so that their multiplicities satisfy m1 = m2 and m3 = m4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in Rn (or...
Using the method of moving frames, we prove that any irreducible Dupin hypersurface in S5 with four distinct principal curvatures and constant Lie curvature is equivalent by Lie sphere transformation to an isoparametric hypersurface in S5.
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