Hypersurfaces in space forms satisfying some generalized Einstein metric condition

TL;DR

This paper studies hypersurfaces in space forms that satisfy a specific curvature condition involving the difference tensor of Einstein-like manifolds, establishing when this condition holds based on the hypersurface's properties.

Contribution

It characterizes hypersurfaces in space forms satisfying a generalized Einstein curvature condition, linking the condition to the hypersurface's Einstein or quasi-Einstein nature.

Findings

If the difference tensor is a linear combination of specific curvature tensors, condition (A) holds.

Condition (A) is satisfied on quasi-Einstein hypersurfaces under additional assumptions.

The results extend understanding of curvature conditions in Einstein and quasi-Einstein hypersurfaces.

Abstract

The difference tensor C.R - R.C of Einstein manifolds, some quasi-Einstein manifolds and Roter type manifolds, of dimension n > 3, satisfy the following curvature condition: (A) C.R - R.C = Q(S,C) - (k /(n-1)) Q(g,C). We investigate hypersurfaces M in space forms N satisfying (A). The main result states that if the tensor C.R - R.C of a non-quasi-Einstein hypersurface M in N is a linear combination of the tensors Q(g,C) and Q(S,C) then (A) holds on M. In the case when M is a quasi-Einstein hypersurface in N and some additional assumptions are satisfied then (A) also holds on M.