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

TL;DR

This paper studies hypersurfaces in constant curvature spaces where the shape operator satisfies a specific algebraic relation, leading to the Riemann curvature tensor fulfilling a Roter type equation under certain conditions.

Contribution

It characterizes hypersurfaces with a shape operator satisfying a cubic relation, showing their curvature tensor obeys a Roter type equation on a significant subset.

Findings

The Riemann curvature tensor is a linear combination of Kulkarni-Nomizu products on a subset.

The tensor R.S can be expressed as a combination of Tachibana tensors.

Hypersurfaces with three distinct principal curvatures satisfy the Roter type equation.

Abstract

We investigate hypersurfaces M isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n > 3, such that the operator A^3, where A is the shape operator of M, is a linear combination of the operators A^2 and A and the identity operator Id. The main result states that on the set U of all points of M at which the square of the Ricci operator of M is not a linear combination of the Ricci operator and the identity operator, 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.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 with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.