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∈−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 φ+c, where φ is the Golden Ratio. We also extend this result to Lorentzian ambient space, observing the existence of a non totally umbilical model.

Keywords

• Pythagorean triple
• Golden Ratio
• isoparametric hypersurface
• shape operator
• Gauss–Kronecker curvature