A new structural approach to isoparametric hypersurfaces in spheres

The classification of isoparametric hypersurfaces in spheres with four or six different principal curvatures is still not complete. In this paper we develop a structural approach that may be helpful for 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.