Hypersurfaces in space forms satisfying some curvature conditions

In Abdalla and Dillen (2002) an example of a non-semisymmetric Ricci-symmetric quasi-Einstein austere hypersurface M isometrically immersed in an Euclidean space was constructed. In this paper we state that, at every point of the hypersurface M, the following generalized Einstein metric curvature condition is satisfied: (∗) the difference tensor R⋅C−C⋅R and the Tachibana tensor Q(S,C) are linearly dependent. Precisely, (n−2)(R⋅C−C⋅R)=Q(S,C) on M. We also prove that non-conformally flat and non-Einstein hypersurfaces with vanishing scalar curvature having at every point two distinct principal curvatures, as well as some hypersurfaces having at every point three distinct principal curvatures, satisfy (∗). We present examples of hypersurfaces satisfying (∗).