The rigity of spaces with cyclic parallel Ricci tensor

The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \begin{equation} D_{ijk}=\nabla_{i}R_{jk}+\nabla_{j}R_{ki}+\nabla_{k}R_{ij}=0\nonumber \end{equation} These structures include non-compact gradient shrinking Ricci soliton, compact $(m>1)$-quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.