I will give a construction of certain Q-valued deformation invariants of (in particular) complete non-positively curved Riemannian manifolds. These are obtained as certain elliptic Gromov-Witten curve counts. As one immediate application we give the (possibly) first  generalization to non-compact fibrations, of Preissman's now classical theorem on non-existence of negative sectional curvature metrics on compact products. One additional goal of the talk is to use the above theory to motivate a very elementary but deep open problem in Riemannian geometry/dynamics concerning existence of Reebable and geodesible sky catastrophes. I will give a partial answer to this problem for surfaces.