On Dupin hypersurfaces in \(\mathbb R^{5}\) parametrized by lines of curvature (Q346589)

A hypersurface immersed in a space form is said to be Dupin if each principal curvature is constant along its corresponding line or surface of curvature. In this paper, the authors provide a complete local characterization of Dupin hypersurfaces in the five-dimensional Euclidean space \(\mathbb R^5\) with four distinct principal curvatures parametrized by lines of curvatures in terms of the principal curvatures and vector valued functions of one variable in \(\mathbb R^5\) which describe plane curves. They also provide explicit examples of such Dupin hypersurfaces which are irreducible and have nonconstant Lie curvature.