According to Gromoll and Meyer:

- Let M be a complete non-compact Riemannian manifold of positive sectional curvature. Then M is diffeomorphic to $\mathbb{R}^n$.

Thus, I think to classify non-compact Riemannian manifolds up to isometry, it suffices to seek for metrics of positive curvature on euclidean spaces. I did some search to find some results related to this problem, but I found few ones.

Therefore, I would be very thankful if give me some information, or/and introduce me some references about metrics of positive curvature on $\mathbb{R}^n$.