Dupin hypersurfaces and a Lie invariant
The author first shows that the cross ratio of four distinct principal curvatures of a hypersurface in a space form is invariant under Lie transformations. Let us call this invariant Lie curvature. Consequently, a necessary condition for a Dupin hypersurface (i.e., a hypersurface each of which principal curvatures has constant multiplicity and vanishing derivative in corresponding principal direction) with exactly four principal curvatures to be Lie equivalent to an isoparametric hypersurface is that its Lie curvature is constant. The author shows that this is also a sufficient condition if one adds a certain condition on how leaves of curvature distributions can intersect. The method of proof is similar in spirit, although much more difficult, to the paper of the author in [Math. Z. 187, 433-452 (1984; Zbl 0529.53045)]. The Lie curvature introduced in this paper has been used to prove that there are Dupin hypersurfaces that are not Lie equivalent to isoparametric hypersurfaces, see the author and \textit{T. Ozawa} [Proc. Symp. Differ. Geom. 1988, Perspect Math. 8, 181-189 (1989)] and \textit{U. Pinkall} and the reviewer [Proc. Am. Math. Soc. 107, No.4, 1037-1043 (1989)].
scientific article; zbMATH DE number 4126243