## Nagoya Mathematical Journal

- Nagoya Math. J.
- Volume 193 (2009), 129-154.

### Isoparametric hypersurfaces with four principal curvatures revisited

#### 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 Münzner. 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-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

#### Article information

**Source**

Nagoya Math. J., Volume 193 (2009), 129-154.

**Dates**

First available in Project Euclid: 3 March 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.nmj/1236089984

**Mathematical Reviews number (MathSciNet)**

MR2502911

**Zentralblatt MATH identifier**

1165.53032

**Subjects**

Primary: 53C40: Global submanifolds [See also 53B25]

**Keywords**

isoparametric hypersurface

#### Citation

Chi, Quo-Shin. Isoparametric hypersurfaces with four principal curvatures revisited. Nagoya Math. J. 193 (2009), 129--154. https://projecteuclid.org/euclid.nmj/1236089984