Matveev, V. S.; Topalov, P. Ĭ. Trajectory equivalence and corresponding integrals. (English) Zbl 0928.37003 Regul. Chaotic Dyn. 3, No. 2, 30-45 (1998). 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. Reviewer: L.Lerman (Berlin) Cited in 1 ReviewCited in 42 Documents MSC: 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 53C22 Geodesics in global differential geometry Keywords:integrability; Hamiltonian systems; geodesic equivalence; Riemannian metrics; geodesic flow; Liouville integrable PDFBibTeX XMLCite \textit{V. S. Matveev} and \textit{P. Ĭ. Topalov}, Regul. Chaotic Dyn. 3, No. 2, 30--45 (1998; Zbl 0928.37003)