Invariant characterization of Liouville metrics and polynomial integrals

Abstract

A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric possess? The method is also applied to recognition of other polynomial integrals of geodesic flows.

References in corpus (3)

• Superintegrability in a two-dimensional space of non-constant curvature
• A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields
• Invariants of pseudogroup actions: Homological methods and Finiteness theorem

Cited by in corpus (24)

• Two-dimensional superintegrable metrics with one linear and one cubic integral
• Geodesically equivalent metrics in general relativity
• The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta
• Metrisability of three-dimensional projective structures
• Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors
• Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric
• Degree of mobility for metrics of lorentzian signature and parallel (0,2)-tensor fields on cone manifolds
• Differential invariants for cubic integrals of geodesic flows on surfaces
• A criterion for the existence of Killing vectors in 3D
• Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
• Integrable Cosmological Potentials
• 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
• Fubini Theorem for pseudo-Riemannian metrics
• When a $(1,1)$-tensor generates separation of variables of a certain metric
• All the three dimensional Lorentzian metrics admitting three Killing vectors
• Generalised Ricci Solitons
• Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method
• Two-dimensional metrics admitting precisely one projective vector field
• Differential invariants of the motion group actions
• Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures
• Differential Invariants and Symmetry: Riemannian metrics and beyond
• Hexagonal geodesic 3-webs
• Superintegrable metrics on surfaces admitting integrals of degrees 1 and 4
• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta