Hypersurfaces of the Space form for Which the Covariant Derivative of Their Ricci Tensors has Vanishing Minimal Norm Tensor

In this paper, we first calculate for a Riemannian manifold the minimal norm tensor of the covariant derivative of the Ricci curvature. Then we show that, for an n-dimensional ( \(n\geqslant 4\) $n⩾4$ ) umbilic-free hypersurface of the space form, if the covariant derivative of the Ricci curvature has vanishing minimal norm tensor, it has parallel Ricci curvature or it is a special rotational hypersurface \(M^n_{c,\sigma }\) $M_{c,\sigma }^n$ .