Equifocal hypersurfaces in symmetric spaces (Q1591534)

A hypersurface \(M\) in a symmetric space is called equifocal if all geodesics which meet \(M\) orthogonally are closed and of same length. The author studies equifocal hypersurfaces in Riemannian symmetric spaces of compact type and rank 1, i.e. in \(\mathbb{F} P^n\), where \(\mathbb{F}=\mathbb{R}, \mathbb{C},\mathbb{H}, \mathbb{O}\). For \(\mathbb{F}=\mathbb{R}, \mathbb{C},\mathbb{H}\) one has the classical Hopf fibration \(\mathbb{S}^m \to\mathbb{F} P^n\), and \(M\) pulls back to an isoparametric hypersurface \(\widetilde M\subset\mathbb{S}^m\). Using \textit{S. Stolz}' result [Invent. Math. 138, No. 2, 253-279 (1999; Zbl 0944.53035)], the author obtains conditions on the number \(g\) of focal points on such a geodesic and their multiplicities, for \(\mathbb{F}=\mathbb{C}, \mathbb{H}\) (the case \(\mathbb{F}= \mathbb{R}\) is completely covered by Stolz' work). Next, he shows that certain of the Clifford hypersurfaces constructed by \textit{D. Ferus, H. Karcher} and \textit{H.-F. Münzner} [Math. Z. 177, 479-502 (1981; Zbl 0452.53032)] can be pushed down into \(\mathbb{F} P^n\) (here, \(\mathbb{F}\) depends on the division ring defined by the Clifford algebra). Finally, he studies the case of the Cayley plane \(\mathbb{O} P^2\), where the methods above break down. Instead, he uses a certain path space fibration and the Leray-Serre spectral sequence to obtain bounds on the number and the multiplicities of focal points.