A non-trivial example of a Riemannian manifold with universal covering $N^2\times \Bbb R$?

Question

I looking for an non-trivial example of complete Riemannian $3$-manifolds such that its universal covering isometric to a Riemann product $N^2\times \Bbb R$ where $N^2$ is a complete $2$-manifold with non-negative sectional curvature.

means of "non-trivial": $M$ not be $N^2\times \Bbb R$.