Untwisted Gromov-Witten invariants of Riemann-Finsler manifolds

We define a $\mathbb{Q}$-valued deformation invariant of certain complete Riemann-Finsler manifolds, in particular of complete Riemannian manifolds with non positive sectional curvature. It is proved that every rational number is the value of this invariant for some compact Riemannian manifold. We use this to find the first and mostly sharp generalizations, to non-compact products and fibrations, of Preissman's theorem on non-existence of negative sectional curvature metrics on compact products. For example, $\Sigma \times T ^{n}$ admits a metric of negative sectional curvature, where $\Sigma$ is a non-compact possibly infinite type surface, if and only if $\Sigma$ has genus zero. We also give novel estimates on counts of closed geodesics with restrictions on multiplicity. Along the way, we also prove that sky catastrophes of smooth dynamical systems are not geodesible by a certain class of forward complete Riemann-Finsler metrics, in particular by complete Riemannian metrics with non-positive sectional curvature. This partially answers a question of Fuller and gives important examples for our theory here.