Hypersurfaces in space forms satisfying some generalized Einstein metric condition

Abstract

The difference tensor C⋅R−R⋅C of Einstein manifolds, some quasi-Einstein manifolds and Roter type manifolds, of dimension n≥4, satisfy the following curvature condition: (∗)C⋅R−R⋅C=Q(S,C)−(κ∕(n−1))Q(g,C). We investigate hypersurfaces M in space forms N satisfying (∗). 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 (∗) holds on M. In the case when M is a quasi-Einstein hypersurface in N and some additional assumptions are satisfied then (∗) also holds on M.