Isoparametric Hypersurfaces in Products of Simply Connected Space Forms

Abstract

Let ${\mathbb{Q}}_{{\epsilon }_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature ${\epsilon }_i$. We prove that any connected isoparametric hypersurface of ${\mathbb{Q}}_{{\epsilon }_1}^{n_1}\times {\mathbb{Q}}_{{\epsilon }_2}^{n_2}$ has constant angle function. We then use this property to classify the isoparametric and homogeneous hypersurfaces of ${\mathbb{Q}}_{{\epsilon }_1}^{n_1}\times {\mathbb{Q}}_{{\epsilon }_2}^{n_2}$, $|{\epsilon }_1|+|{\epsilon }_2|\ne 0$, that satisfy a one-point condition.

39 pages, 1 figure. Comments are welcome