Geodesic equivalence of metrics as a particular case of integrability of geodesic flows (Q1592124)

Let \(M\) be smooth n-manifold with two metrics \(g\) and \(\overline{g}\). These are geodesically equivalent if they have the same geodesics, considered as unparametrized curves. The authors solve the following problem, on closed manifolds: If the metrics \(g\) and \(\overline{g}\) are geodesically equivalent, then the geodesic flows of the metrics are completely integrable [see also \textit{S. Yu. Dobrokhotov} and \textit{A. I. Shafarevich}, Funct. Anal. Appl. 34, No. 2, 133-134 (2000; Zbl 1019.58009], by using the \(n\) functions: \[ I_k (\xi) = \left(\frac{\det (g)}{\det (\overline{g}}\right) g (S_k \xi,\xi), \] where \(S_k\) is the linear operator given by \[ S_k \underset{\text{def}} = \sum_{i=0}^k c_i G^{k-i+1} \] and \(g(v,\xi)\) denotes the dot product of the tangent vectors \(v\) and \(\xi\) in the metric \(g\). The authors point out that: ``\dots{} the integral \(I_0\) was considered by Painlevé [\textit{T. Levi-Civita}, Ann. Mat. (2), 24, 255-300 (1896; JFM 27.0603.04)]''. They give various nontrivial examples on the \(n\)-sphere, on the \(n\)-torus and on the Poisson sphere, and analyse on real analytic manifolds admissibility of non proportional geodesically equivalent metrics. They also give an approach to the question: ``How many metrics are geodesically equivalent to a given one?'' by calculating the dimension of the manifold of the metrics that are geodesically equivalent to a given one. The second author has worked out more details of one of the examples [see \textit{P. Topalov}, Families of metrics geodesically equivalent to the analogs of the Poisson sphere, J. Math. Phys. 41, No. 11, 7510-7520 (2000; Zbl 1001.53005]. This is a very beautiful and readable article.