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.



What are the manifolds whose Curvature tensor has a globally vanishing $k$th order covariant derivative
In fact, the result is true for any complete Riemannian manifold as I remember. The result was proved by Katsumi Nomizu and Hideki Ozeki [Here](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC220758/)  

Here [2](https://link.springer.com/content/pdf/10.1007/BF02414842.pdf) and here [3](https://projecteuclid.org/euclid.pjm/1102976647) you may find many related interesting results.


