Application of classification theory for integrable Hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mapping near singular points (Q1917799)

The authors give a survey of recent results on integrable geodesic flows for Riemannian metrics on the 2-sphere \(S^2\) and the 2-torus \(T^2\). The known examples for such flows on \(T^2\) (resp. \(S^2\)) have a second integral whose restriction to the fibres of the fibration \(TT^2\to T^2\) (resp. \(TS^2\to S^2\)) is polynomial of degree 1 or 2 (resp. 1, 2, 3, 4). To the class of such a flow can be associated a numerical invariant with respect to equivalence under topological conjugacy, the so called complexity. A main conjecture discussed in the paper states that the complexity of any such system coincides with the complexity of a system with a fibrewise linear or quadratic integral. Some evidence supporting the conjecture is discussed. The paper consists mainly of a succession of definitions and theorems and does not contain any proof. It is not very informative for non-specialists.