Isoparametric hypersurfaces with four principal curvatures revisited

TL;DR

This paper provides a shorter, more conceptual proof for the classification of isoparametric hypersurfaces with four principal curvatures in spheres, clarifying the geometric principles behind previous lengthy calculations.

Contribution

It introduces a concise proof based on Ozeki and Takeuchi's expansion formula, enhancing understanding of the geometric structure of these hypersurfaces.

Findings

Shorter proof of the characterization

Clarification of geometric meaning of equations

Improved understanding of hypersurface classification

Abstract

The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and M\"{u}nzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-M\"{u}nzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.