Abstract:
The geodesic curves of a Riemannian metric on a surface are described by a Hamiltonian system with two degrees of freedom whose Hamiltonian is quadratic in the momenta. Because of the homogeneity, every integral of the geodesic problem is a function of integrals that are polynomial in the momenta. The geodesic flow on a surface of genus greater than one does not admit an additional nonconstant integral at all, but on the other hand there are numerous examples of metrics on a torus whose geodesic flows are completely integrable: there are polynomial integrals of degree $\leqslant2$ that are independent of the Hamiltonian. It appears that the degree of an additional 'irreducible' polynomial integral of a geodesic flow on a torus cannot exceed two. In the present paper this conjecture is proved for metrics which can arbitrarily closely approximate any metric on a two-dimensional torus.
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