ISOPARAMETRIC HYPERSURFACES WITH FOUR PRINCIPAL CURVATURES REVISITED

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 all isoparametric hypersurface of the type constructed by Ferus, Karcher and Munzner. The proof of the characterization in [2] is all extremely long calculation by exterior derivatives with remarkable cancellations. which is motivated by the idea that all 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. Inspite of its elementary, nature. the magnitude of the understand the surprisingly pleasant cancellations make, it desirable to the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter. proof of the characterization based oil Ozeki and Takeuchi's expansion formula for the Cartan-Munzner polynomial. Along the way the geometric meaning of these four set's of equations also becomes clear.