Topological rigidity for negatively curved manifolds?

Question

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.

Answer 1

I'm assuming that your intention is that negatively curved means having negative sectional curvature.

Your question, with regard to uniqueness up to homeomorphism, is a special case of the Borel Conjecture (topological uniqueness of aspherical manifolds) which is still unsolved despite much progress. In the special case of negative curvature, there is a monumental series of works of Farrell and Jones suggesting the correctness of the Borel conjecture. I would look on Mathscinet for these papers, including several detailed surveys.

The smooth version of the Borel conjecture does not hold; one can change the smooth structure of a negatively curved manifold, and still have a negatively curved metric. See [Farrell, F. T.; Jones, L. E. Negatively curved manifolds with exotic smooth structures. J. Amer. Math. Soc. 2 (1989), no. 4, 899–908.]