3 problems

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 $M^{n-1}\subset S^n$ be a compact, connected proper Dupin hypersurface. Let $g$ be the number of distinct principal curvatures, and call its Lie curvatures constant Lie curvatu…

Let $M^{n-1}subset S^n$ be a compact, connected proper Dupin hypersurface, and let $g$ denote its number of distinct principal curvatures. Dupin classification conjecture. Every su…