Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus

In this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of Riemannian metrics on the two-dimensional torus. Also our method gives a new proof of Birkhoff's theorem for twist maps of the cylinder. We prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the Aubry-Mather set with this rotation number.