Trajectory equivalence and corresponding integrals
The authors show that if on a smooth manifold two Riemannian metrics exist which are geodesically equivalent then the related geodesic flow of any of these metrics is Liouville integrable. They also construct a nontrivial one-parameter family of geodesically equivalent metrics if there are two such metrics. Finally, they present a nontrivial metric on the \(n\)-dimensional ellipsoid geodesically equivalent to the standard one.
scientific article; zbMATH DE number 1281656