Isoparametric Hypersurfaces in Sn+1: The Chern Conjecture

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookInstead of low dimensional manifolds, one can also considerthose with a certain number g of pairwise different principalcurvatures; g = 3 is the first non-trivial case, and one has thefollowing result:Theorem (Chang 1994)Let M ⊂ S n+1 be a closed hypersurface with constant meanand scalar curvatures which has exactly three pairwise distinctprincipal curvatures in every point. Then M is isoparametric.