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Variational characterization of integrable Riemannian metrics on the 2-torus

Determine whether integrable Riemannian metrics on the 2-torus, such as rotationally invariant or Liouville metrics, can be distinguished in variational terms analogous to Hopf’s no-conjugate-points characterization of flat metrics.

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Background

Hopf’s theorem implies that a Riemannian 2-torus with no conjugate points is flat, yielding a variational characterization for flat integrable geodesic flows.

The authors highlight that extending such a variational characterization to other integrable metrics (e.g., Liouville metrics) on the 2-torus is currently unknown.

References

It is unknown if other integrable Riemannian metrics on the 2-torus, e.g., rotationally invariant metrics or Liouville metrics, can be distinguished in variational terms.

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 2 (Birkhoff billiards), discussion of Hopf rigidity and variational methods