In an online webinar, I heard (not directly) the statement that **(closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces**. Is this a valid theorem? Any reference that contains its proof? I am not sure that in the above statement whether "positive curvature" is a part of assumptions or not.


So by the above claim, it seems that $\mathcal{R}\geq 0\iff$ $M$ is symmetric space!



Closed manifolds of nonnegative curvature operator are symmetric spaces
As Igor Belegradek commented, the correct statement is as follows:



> 
> **Theorem (classification of closed simply connected manifold with nonnegative curvature operator):** A closed simply connected manifold with nonnegative curvature operator is isometric to a Riemannian product of
> 
> 
> 1. standard spheres with metrics of nonnegative curvature operator
> 2. closed Kahler manifolds biholomorphic to complex projective spaces whose
> Kahler metric has nonnegative curvature operator on real (1, 1)-forms
> 3. *compact irreducible Riemannian symmetric spaces* with their natural
> metrics of nonnegative curvature operator.
> 
> 
> 




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See for instance [Page 270, theorem 7.34](https://books.google.com/books?id=dQYPCgAAQBAJ&lpg=PP1&pg=PA270#v=onepage&q&f=false) of


*Chow, Bennett; Lu, Peng; Ni, Lei*, **Hamilton’s Ricci flow**, Graduate Studies in Mathematics 77. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4231-5/hbk). xxxvi, 608 p. (2006). [ZBL1118.53001](https://zbmath.org/?q=an:1118.53001).


So closed simply connected irreducible manifolds of nonnegative curvature operator are isometric to locally symmetric spaces.


