A new structural approach to isoparametric hypersurfaces in spheres

Abstract

In this paper we show that the long-standing problem of classifying all isoparametric hypersurfaces in spheres with six different principal curvatures is still not complete. Moreover, we develop a structural approach that may be helpful for such a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented 2-planes in {mathbb {R}}^{n+2}. We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold.