Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs

Author:

Sergey I. Agafonov, Thaís G. P. Alves

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Dynamical Systems (math.DS)

journal:

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date:

2023-06-29 16:00:00

Abstract

We prove that if the geodesic flow on a surface has an integral, fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for existence of fractional-linear integrals: such integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.