Inhomogeneity of Isoparametric Hypersurfaces of OT-FKM-type in the Pseudo-Sphere

TL;DR

This paper explores the inhomogeneity of certain isoparametric hypersurfaces in pseudo-spheres, providing explicit Clifford system constructions and identifying conditions under which these hypersurfaces are inhomogeneous.

Contribution

It offers explicit Clifford system constructions for any signature and proves inhomogeneity of OT-FKM-type hypersurfaces under specific signature conditions.

Findings

Clifford systems of signature (m, r) are explicitly constructed for all (m, r).

Connected components of focal varieties are shown to be inhomogeneous under certain signature conditions.

The inhomogeneity depends on the signature (m, r) satisfying specific modular conditions.

Abstract

We study isoparametric hypersurfaces, whose principal curvatures are all constant, in the pseudo-Riemannian space forms. In this paper, we investigate two topics. Firstly, according to representations of Clifford algebras, we give a construction of Clifford systems of signature $(m, r)$ for any $(m, r)$ explicitly. Secondly, we show that a (connected) isoparametric hypersurface of OT-FKM-type whose focal variety is $M_{+}$ in the pseudo-sphere is inhomogeneous if the signature $(m, r)$ of its Clifford system on $\mathbb{R}^{2l}_{s}$ satisfies $m\equiv 0\pmod{4}$, $r\equiv 0\pmod{2}$ and $l>m$, showing that each connected component of $M_{+}$ is inhomogeneous.