2005/12/05 by Cecil, Thomas, Chi, Quo-Shin, Jensen, Gary · 2 citations
#53C40 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0512090
If M is an isoparametric hypersurface in a sphere Sn with four distrinct 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 Sn) with four distinct principal curvatures with multiplicities m1=m2 ≥ 1 and m3=m4=1, and constant Lie curvature r=-1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.