[Paper Review] Toric integrable geodesic flows

Eugene Lerman, Nadya Shirokova|ArXiv.org|Nov 20, 2000

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper proves that any toric integrable geodesic flow on a torus must arise from a flat metric, confirming a conjecture by Toth and Zelditch. By analyzing symplectic and contact toric structures on the punctured cotangent bundle, the authors show that non-flat metrics cannot support such integrable systems due to topological obstructions in the moment map image and isotropy group actions.

ABSTRACT

By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.

Motivation & Objective

• To resolve the conjecture that toric integrable geodesic flows on tori must correspond to flat metrics.
• To investigate the geometric and topological obstructions preventing non-flat metrics from supporting toric integrability.
• To apply symplectic and contact geometry techniques to classify toric integrable systems on tori.
• To establish that the absence of fixed points and the structure of isotropy groups in 3-dimensional contact toric manifolds lead to lens space topology, ruling out the co-sphere bundle of a torus.
• To demonstrate that the moment map image and Chern class obstruction in symplectic toric manifolds uniquely determine the underlying geometry, leading to flatness.

Proposed method

• Utilize the symplectization of contact manifolds to lift the geodesic flow on a torus to a symplectic cone structure on $T^*\mathbb{T}^n \setminus \mathbb{T}^n$.
• Apply the theory of symplectic toric manifolds and Delzant's classification to analyze the moment map image of the lifted action.
• Use the fact that toric integrability implies a Hamiltonian torus action commuting with dilations, preserving the symplectic form and energy function.
• Analyze isotropy groups and orbit types in 3-dimensional contact toric manifolds, showing that non-free actions lead to lens space topology.
• Leverage results from Lerman, Tolman, and Woodward on symplectic toric manifolds to show that the moment map image determines the topology up to homotopy.
• Apply the Haefliger–Salem theorem to classify 3-dimensional contact toric manifolds with non-free actions, concluding they are lens spaces, not the co-sphere bundle of a 2-torus.

Experimental results

Research questions

• RQ1 Can a non-flat metric on a torus admit a toric integrable geodesic flow?
• RQ2 What topological obstructions prevent a non-flat metric from supporting a toric integrable system?
• RQ3 How do isotropy group structures in 3-dimensional contact toric manifolds affect the global topology of the manifold?
• RQ4 Is the co-sphere bundle of a torus the only possible contact manifold supporting a toric integrable geodesic flow?
• RQ5 To what extent does the moment map image and Chern class obstruction determine the geometry of symplectic toric manifolds arising from geodesic flows?

Key findings

• Toric integrable geodesic flows on a torus $\mathbb{T}^n$ imply that the metric $g$ is flat, confirming the conjecture of Toth and Zelditch.
• The symplectization of the punctured cotangent bundle of a torus with a toric integrable metric is a symplectic toric manifold whose moment map image determines the topology.
• In dimension 3, if the torus action on a contact manifold is not free, the manifold must be a lens space, which is not diffeomorphic to the co-sphere bundle of a 2-torus.
• The absence of fixed points in the action on the contact manifold and the structure of isotropy groups (trivial or circle) lead to a quotient space that is an interval, forcing the global topology to be that of a lens space.
• The Chern class obstruction and the moment map image in symplectic toric geometry uniquely determine the symplectic type of the manifold, leading to the conclusion that only flat metrics support such systems.
• The result holds for finite covers: if a metric on a torus is toric integrable, then its pullback to a finite cover is also toric integrable, and hence flat.