I was wondering if two compact oriented manifold carrying a negatively curved Riemannian metric, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or homeomorphic) ?

In the hyperbolic case, the ingredients in the proof of Mostow rigidity strongly rely on the hyperbolic structure, and it makes me think that such a result, if true, must have a proof essentially different. One the other hand, the topological differences between hyperbolic manifolds and only negatively curved manifold (which are known to differ when the dimension is greater than $4$) makes it hard to think of a counterexample for the non-specialist that I am.

I was wondering if two compact oriented manifold carrying a Riemannian metric with negative sectional curvature, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or homeomorphic) ?

In the hyperbolic case, the ingredients in the proof of Mostow rigidity strongly rely on the hyperbolic structure, and it makes me think that such a result, if true, must have a proof essentially different. One the other hand, the topological differences between hyperbolic manifolds and only negatively curved manifold (which are known to differ when the dimension is greater than $4$) makes it hard to think of a counterexample for the non-specialist that I am.