Bulletin of the
Korean Mathematical Society BKMS

Consider a hypersurface $M^n$ in $ \mathbb{R}^{n+1}$ with $n$ distinct principal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hypersurfaces for $n\geq 4$, and for $n=3$, they are, up to M\"obius transformations, Dupin hypersurfaces with constant M\"obius curvature. (2) If the principal curvatures are given by a sum of functions of separated variables, there are no such hypersurfaces for $n\geq 4$, and for $n=3$, they are, up to M\"obius transformations, Dupin hypersurfaces with constant M\"obius curvature.

Keywords: lines of curvature, Laplace invariants, Dupin hypersurfaces

MSC numbers: Primary 53A07; Secondary 35N10