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.


1 Answer 1


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
Here 2 and here 3 you may find many related interesting results.


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