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.

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• Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
• When a $(1,1)$-tensor generates separation of variables of a certain metric
• Generalised Ricci Solitons
• Two-dimensional metrics admitting precisely one projective vector field
• Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures
• Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method
• Differential invariants of the motion group actions
• Differential Invariants and Symmetry: Riemannian metrics and beyond