Abstract
Consider a hypersurface M n in R n+1 with n distinct princi- pal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hyper- surfaces for n � 4, and for n = 3, they are, up to Mobius transformations, Dupin hypersurfaces with constant Mobius curvature. (2) If the principal curvatures are given by a sum of functions of sepa- rated variables, there are no such hypersurfaces for n � 4, and for n = 3, they are, up to Mobius transformations, Dupin hypersurfaces with con- stant Mobius curvature.
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