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Trajectory equivalence and corresponding integrals. (English) Zbl 0928.37003

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)

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
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