Untwisted Gromov-Witten invariants of Riemann-Finsler manifolds

TL;DR

This paper introduces a new deformation invariant for Riemann-Finsler manifolds, generalizes classical theorems on curvature, and provides novel estimates on closed geodesics, with implications for dynamical systems and geometric topology.

Contribution

It defines a rational-valued deformation invariant for Riemann-Finsler manifolds and applies it to generalize Preissman's theorem and analyze geodesic behavior.

Findings

Every rational number can be realized as an invariant value for some compact Riemannian manifold.

Certain non-compact product manifolds admit negative sectional curvature metrics under specific conditions.

Sky catastrophes in dynamical systems are not geodesible by particular classes of Riemann-Finsler metrics.

Abstract

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.