If the fundamental group of a smooth closed aspherical manifold is a hyperbolic group, does that manifold admit a metric with non-positive sectional curvature?

If not, what's the obstruction to admitting a metric of non-positive curvature?

Here we are all talking about smooth closed manifold and the curvature are sectional curvature. Aspherical manifold are the maniflod which's universal covering is contractable.

We know that any manifold with a metric of non-positive curvature is aspherical by the Cartan-Hadamard theorem, and when in addition the manifold is negatively curved, then the fundamental group is a hyperbolic group in the context of Gromov. I am curious about the converse.

By the way, does there exist an aspherical manifold which admits a metric with non-negative sectional curvature?

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