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Does there exist a compact Einstein manifold $(M,g)$ with $\operatorname{Ric}(g)=\lambda g$ and $\lambda<0$ and nonnegative sectional curvatures?

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    $\begingroup$ I suppose you mean sectional curvature nonnegative at some plane? $\endgroup$ – user99914 Nov 12 '15 at 3:48
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No, because the Ricci curvature is ($n-1$ times) the average of the sectional curvatures, so if they are all nonnegative it must be nonnegative.


More precisely, the result is false regardless of the compact Einstein setting.

We have, for every $p \in M$, for every unit vector $e_1 \in T_pM$, $$ \operatorname{Ric}(g)(e_1)=(n-1) \sum_{j=2}^n K(\operatorname{span}(e_1,e_j))=(n-1) \operatorname{Ave}_{\Pi \ni e_1}K(\Pi), $$ where $e_1, \dots, e_n$ is an orthonormal frame. Here $M$ is a $n$-dimensional Riemannian manifold.

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