Ricci curvature version of Cartan-Hadamard theorem?

Question

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 $S^2\times \mathbb{R}^2$.

Answer 1

Every manifold of dimension $\ge 3$ admits a complete Ricci-negative metric, this is Lohkamp's theorem, see the research announcement here. See also

Lohkamp, Joachim, Metrics of negative Ricci curvature, Ann. Math. (2) 140, No. 3, 655-683 (1994). ZBL0824.53033.

for a detailed proof.