Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

Abstract

Let M be a hypersurface isometrically immersed in an $(n+1)$-dimensional semi-Riemannian space of constant curvature, $n>3$, such that its shape operator $A$ satisfies $A^3=\varphi A^2+\psi A+\rho Id$, where ϕ, ψ and ρ are some functions on M and Id is the identity operator. The main result of this paper states that on the set U of all points of M at which the square $S^2$ of the Ricci operator $S$ of M is not a linear combination of $S$ and Id, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor $S^2$ of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the $(0,4)$-tensor $R\cdot S$ is on U a linear combination of some Tachibana tensors formed by the tensors g, S and $S^2$. In particular, if M is a hypersurface isometrically immersed in the $(n+1)$-dimensional Riemannian space of constant curvature, $n>3$, with three distinct principal curvatures and the Ricci operator $S$ with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.