Let $(M,g)$ be a boundaryless Riemannian manifold whose curvature tensor have the property that there exists $k\geq 2$ such that $\nabla^k R\equiv0$. What is known about such Riemannian manfiolds ? Is there a classification ?
I vaguely remember that I came (a long time ago) across a paper that claims that if $(M,g)$ is also known to be irreducible then we must have $\nabla R\equiv 0$. However, I cant find this paper anymore and I m starting to doubt myself so I decided to ask the question here.