On the Lie curvature of Dupin hypersurfaces
The Lie curvature of a hypersurface with four principal curvatures is by definition their cross ratio. It turns out that the Lie curvature is invariant under Lie sphere transformations and that it is identically equal to \(\frac12\) for isoparametric hypersurfaces after ordering their principal curvatures appropriately, see the paper [ibid. 12, No. 2, 228-256 (1989; Zbl 0687.53053)] by \textit{R. Miyaoka}. In the paper under review the author gives an example of a noncompact Dupin hypersurface with four principal curvatures that is not Lie equivalent to an isoparametric hypersurface although its Lie curvature is identically \(\frac12\). The reason for this example seems rather to be that it is reducible than that it is noncompact.
scientific article; zbMATH DE number 4169613
scientific article