Invariant fibration of geodesic flows

TL;DR

This paper investigates the conditions under which geodesic flows on compact 3-manifolds are completely integrable, linking geometric properties with algebraic topology, and extends results to higher dimensions.

Contribution

It establishes new connections between integrable geodesic flows, the topology of 3-manifolds, and the structure of their fundamental groups, including higher-dimensional generalizations.

Findings

Complete integrability implies the fundamental group is almost polycyclic.

Existence of integrable geodesic flows with singular sets as real-analytic varieties.

Results extend to higher-dimensional manifolds.

Abstract

Let ({\Sigma}, g) be a compact $C^2$ finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then ${\pi}_1({\Sigma})$ is almost polycyclic. On the other hand, if {\Sigma} is a compact, irreducible 3-manifold and ${\pi}_1({\Sigma})$ is infinite polycyclic while ${\pi}_2({\Sigma})$ is trivial, then {\Sigma} admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven.