Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms

Abstract

The classification of curvature homogeneous hypersurfaces in real space forms was established by Tsukada in 1988, with the remaining rank-two cases in $\mathbb{S}^4$ and $\mathbb{H}^4$ settled by Bryant-Florit-Ziller in 2025. It is obvious that curvature homogeneity implies constant Ricci eigenvalues. In this paper, we prove that for hypersurfaces in real space forms, the converse also holds: a connected hypersurface immersed in real space forms has constant Ricci eigenvalues if and only if it is curvature homogeneous. Hence, hypersurfaces with constant Ricci eigenvalues in real space forms are also classified, which, in particular, generalizes the classification of Einstein hypersurfaces obtained by Lawson for minimal hypersurfaces and by Ryan for general cases in 1969. Moreover, as a byproduct, curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues can not be isometrically immersed in any real space form of codimension one. Finally, we show that a hypersurface with constant Ricci eigenvalues is isoparametric if it is a complete hypersurface either in $\mathbb{S}^{n+1}$ or nonflat in $\mathbb{R}^{n+1}$ for $n \geq 3$; or if it is not of constant sectional curvature $-1$ in $\mathbb{H}^{n+1}$ for $n \geq 5$.