Compact Einstein manifolds with $\operatorname{Ric}(g)=\lambda g$ with $\lambda<0$ and sectional curvatures $\geq0$

Question

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

Answer 1

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.

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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.