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Integrable geodesic flows on surfaces. (English) Zbl 1203.37092

In Section 2 of this paper, the author summarizes the facts which are needed on minimal geodesics and Lagrangian projections of invariant tori. In Section 3, it is proved that no minimal rays can be trapped “inside” compressible invariant tori. Section 4 contains the proofs of main Theorems 1.4 and 1.6. In Section 5, Theorems 1.8 and 1.10 are proved. The last Section 6 contains facts on critical points of \(F\).
This paper contains two parts. In the first part, Kozlov’s theorem on non-integrability on surfaces of higher genus is strengthed. In the second part, integrable geodesic flows on a 2-torus are studied. The main result for a 2-torus describes the phase portraits of integrable flows. It is proved that they are essentially standard outside what is called separatrix chains.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J50 Action-minimizing orbits and measures (MSC2010)
53D25 Geodesic flows in symplectic geometry and contact geometry
70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
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