Dupin hypersurfaces with constant Laguerre curvatures (Q2408131)

A hypersurface \(M^n\) of a space form (\(\mathbb{R}^{n+1}\), \(\mathbb{S}^{n+1}\) or \(\mathbb{H}^{n+1}\)) is called \textit{proper Dupin}, if the number of distinct principal curvatures is constant on \(M\) and each principal curvature is constant along its corresponding surface of curvature. This property is invariant under the group of Lie sphere transformations, a group, generated by the subgroups of Moebius transformations and Laguerre transformations. Using distinct principal curvatures \textit{Moebius curvature, Laguerre curvature and Lie curvature} can be defined. In the present paper the authors consider proper Dupin hypersurfaces in \(\mathbb{R}^{n+1}\) with \(n\) distinct nonvanishing principle curvatures and constant Laguerre curvature, that admit principal coordinate systems. They classify explicitly all such hypersurfaces with constant Laguerre curvatures. These hypersurfaces are determined explicitly by \(n\) constants, namely \(n-2\) Laguerre curvatures and two other constants, one of them being nonzero.