Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. (English) Zbl 0979.53032
Summary: Let \(g\) be a Riemannian metric with ergodic geodesic flow. Then if some metric \(\bar g\) has the same geodesics (regarded as unparameterized curves) with \(g\), then the metrics are homothetic. If two metrics on a closed surface of genus greater than one have the same geodesics, then they are homothetic.
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 53C22 | Geodesics in global differential geometry |
| 53D25 | Geodesic flows in symplectic geometry and contact geometry |
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