Inhomogeneity of Isoparametric Hypersurfaces of OT-FKM Type in the Pseudo-sphere

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 $M$ 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.