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Is the following assertion true : If $M$ is a simply-connected manifold with $\operatorname{Ric}<0$ (or $\operatorname{Ric}\leq -k$ for $k$ positive) then $M$ is diffeomorphic to $\mathbb{R}^n$? (i.e. I am trying to generalize Cartan-Hadamard theorem for manifold with negative Ricci curvature.)

Remark : It is not true if we assume $\operatorname{Ric}\leq 0$ as for example there is Ricci flat Schwartzchild metric on $S^2\times \mathbb{R}^2$.

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Every manifold of dimension $\ge 3$ admits a complete Ricci-negative metric, this is Lohkamp's theorem, see here. See also his paper in Annals of Math, 1994 for the detailed proof.

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