Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

DOI10.1090/S1079-6762-00-00086-XzbMath0979.53032MaRDI QIDQ4523229

Peter J. Topalov, Vladimir S. Matveev

Publication date: 8 January 2001

Published in: Electronic Research Announcements of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/231946

zbMATH Keywords

projectively equivalent metricsergodic geodesic flow

Mathematics Subject Classification ID

Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Geodesics in global differential geometry (53C22) Global Riemannian geometry, including pinching (53C20) Smooth ergodic theory, invariant measures for smooth dynamical systems (37C40) Geodesic flows in symplectic geometry and contact geometry (53D25) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40)

Related Items (12)

Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures ⋮ Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic ⋮ Algebraic integrability conditions for Killing tensors on constant sectional curvature manifolds ⋮ On the rigidity of magnetic systems with the same magnetic geodesics ⋮ Geodesic rigidity of conformal connections on surfaces ⋮ A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields ⋮ Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two ⋮ Complete Einstein metrics are geodesically rigid ⋮ Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta ⋮ Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics ⋮ Convex projective surfaces with compatible Weyl connection are hyperbolic ⋮ Open problems and questions about geodesics

Cites Work

• Unnamed Item
• Compact Liouville surfaces
• Trajectory equivalence and corresponding integrals
• Transformation groups in differential geometry.
• Tensor invariants of natural mechanical systems on compact surfaces, and the corresponding integrals
• On the Projective Theory of Two Dimensional Riemann Spaces

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