A new structural approach to isoparametric hypersurfaces in spheres

TL;DR

This paper introduces a novel structural method for studying isoparametric hypersurfaces in spheres by analyzing associated Lagrangian submanifolds, offering new geometric insights that could aid classification efforts.

Contribution

It develops a structural approach using Lagrangian submanifolds in Grassmannians to better understand isoparametric hypersurfaces, advancing classification techniques.

Findings

New geometric invariants derived from Lagrangian submanifold analysis

Identities relating classical invariants to Grassmannian geometry

Potential framework for complete classification in future work

Abstract

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.