Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

Abstract

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.

References in corpus (8)

• A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields
• Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems
• Geometrical classification of Killing tensors on bidimensional flat manifolds
• Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems
• Complex variables for separation of Hamilton-Jacobi equation on real pseudo-Riemannian manifolds
• (1+1)-dimensional separation of variables
• Reciprocal transformations and flat metrics on Hurwitz spaces
• Appendix: Dini theorem for pseudo-Riemannian metrics

Cited by in corpus (13)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Geodesically equivalent metrics in general relativity
• Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
• C-projective geometry
• Differential invariants for cubic integrals of geodesic flows on surfaces
• Normal forms of two-dimensional metrics admitting exactly one essential projective vector field
• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry
• Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries
• Koenigs Theorem and Superintegrable Liouville Metrics
• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta
• Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures