Moreover, it would be good that, if $M$ is such an example, its universal cover $\tilde M$ is "highly" symmetric, i.e. the group G of isometries of $\tilde M$ is "big" (for example, transitive, or a lie group with high dimension, or at the very least infinite!!).
2026-03-28 02:24:27.1774664667
Are there "interesting" examples of complete finite-volume non-compact Riemannian manifolds that are not non-positively curved?
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Take the product of your favorite locally symmetric finite volume nonpositively curved complete Riemannian manifold with a sphere. For instance, take the product of $S^n$, $n\ge 2$, and a complete noncompact hyperbolic surface of finite area. The universal cover is a symmetric space. (In the specific example, it is $S^n\times H^2$.) Maybe you also want manifolds which are K(G,1)? Then the examples you are after (assuming homogeneity of the universal cover) do not exist.