4 problems

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$, with $g$ distinct principal curvatures. The hypersurface has constant Lie curvatures when its Lie curvatures are…

Let $M$ be a closed embedded Dupin hypersurface, meaning that each principal curvature has constant multiplicity and is constant along its curvature direction. A Lie image is the i…

Let $M$ be an irreducible connected proper Dupin hypersurface in $S^n$ with four principal curvatures. Suppose their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and let $\Psi$…

Let $M\subset S^n$ be a compact proper Dupin hypersurface, meaning that it has constant multiplicities of its principal curvatures along each curvature surface. Two hypersurfaces a…