Integrability in the theory of geodesically equivalent metrics (Q2716908)

In the paper under review the property of two metrics on a manifold having the same geodesics is considered. This property is equivalent to a special kind of integrability of the geodesic flow of these metrics (both in the classical and in the quantum sense). This gives nontrivial restrictions on the topology of the manifold, and allows the authors to construct new examples of such pairs of metrics on the sphere and to give a local description of geodesically equivalent metrics near the points where the eigenvalues of one metric with respect to the other bifurcate.