Geometry and Real-Analytic Integrability

TL;DR

This paper constructs a specific 4-dimensional real-analytic Riemannian manifold with integrable geodesic flow that has smooth but non-analytic integrals, revealing topological obstructions to real-analytic integrability.

Contribution

It provides a counterexample demonstrating that topological properties alone do not guarantee real-analytic integrability of geodesic flows.

Findings

Existence of a 4-manifold with smooth but not real-analytic integrals

Geodesic flow's limit set is dense on the universal cover

Obstructions to real-analytic integrability beyond topology

Abstract

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2 \times S^2$ ; and (3) the limit set of the geodesic flow on the universal cover is dense. This shows there are obstructions to realanalytic integrability beyond the topology of the configuration space.