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Topology of real algebraic curves and integrability of geodesic flows on algebraic surfaces. (English) Zbl 1158.53051
Funct. Anal. Appl. 42, No. 2, 98-102 (2008); translation from Funkts. Anal. Prilozh. 42, No. 2, 98-102 (2008).
Summary: The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
14H70 Relationships between algebraic curves and integrable systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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