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.

Two-dimensional metrics admitting precisely one projective vector field

“…We will also use the following two statements: first is due to Knebelman [18], another proof could also be found in [5,9,21,25,34,45], one more proof easily follows from the theory of invariant operators, see for example [3]. The second is combination of the formula (11) and the connection between projectively equivalent metrics and integrable systems due to [25,26], see also Darboux [11, §608], see also of [9, §2.4].…”

“…Assume the metrics g andḡ have the Liouville form (21). Then, the condition (18) is equivalent to a system of 6 PDE's on the unknown functions v 1 (x, y), v 2 (x, y), X (x), and Y (y).…”

“…Recently Kruglikov [21] and, independently, Bryant et al [10] used Mathematica and Maple (and also quite advanced theory) to construct curvature invariants such that if they do not vanish the dimension of A is at most 2. But their invariants are still too complicated, and there is no hope to calculate them for our metrics without using a computer (though one can easily do it with computer).…”

“…4.5, in the cases 3a-3d we know the space of quadratic integrals of the metrics 3a-3d, so it it sufficient to check that no quadratic integral is degenerate at every point, which is an easy exercise. Moreover, in the case 3d the space of quadratic integrals is precisely 3-dimensional implying the metric admits no Killing vector field, see [21,Section 5]. We will use the following approach which was known to Darboux [11, § §688,689] and Eisenhart [15, pp.…”

Degree of mobility for metrics of Lorentzian signature and parallel (0,2)-tensor fields on cone manifolds

“…⇒ Let ( M ,ĝ) be locally the cone over (M, g). Then there exist coordinates (r, x), such thatĝ has the form (13). By direct calculations we see that the function v = 1 2 r 2 satisfies (19).…”