It is easy to see that $S^2\times S^1$ as a Riemannian manifold is not Euclidean, hyperbolic or elliptic. Is it also easy to see that the topological manifold $S^2\times S^1$ does not admit one of these three geometric structures (in particular elliptic geometry)?
I could not come up with an argument why that should be the case.
If Euclidean geometry is Riemannian geometry whose curvature vanishes. It has been shown by Bieberbach that a compact Euclidean manifold is finitely covered by the torus and $S^2\times S^1$ is not finitely covered by the torus since its universal cover is $S^2\times \mathbb{R}$ and the universal cover of $T^3$ is $\mathbb{R}^3$.
For the hyperbollic geometric, the universal cover of an hyperbolic manifold i the $H^n$ which is contractible and the universal cover of $S^2\times S^1$ is not contractible.
For elliptic geometry a theorem of Myers implies that the fundamental group of an elliptic manifold is finite