  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.
