Pythagorean Isoparametric Hypersurfaces in Riemannian and Lorentzian Space Forms

Abstract

We introduce the notion of a Pythagorean hypersurface immersed into an (n + 1)-dimensional pseudo-Riemannian space form of constant sectional curvature c is an element of {-1, 0, 1}. By using this definition, we prove in Riemannian setting that if an isoparametric hypersurface is Pythagorean, then it is totally umbilical with sectional curvature phi + c, where phi is the Golden Ratio. We also extend this result to Lorentzian ambient space, observing the existence of a non totally umbilical model.