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Metric with ergodic geodesic flow is completely determined by unparameterized geodesics


Authors: Vladimir S. Matveev and Petar J. Topalov
Journal: Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 98-104
MSC (2000): Primary 53C20; Secondary 37J35, 37C40, 53A20, 53C22, 53B10
DOI: https://doi.org/10.1090/S1079-6762-00-00086-X
Published electronically: December 7, 2000
MathSciNet review: 1796527
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Abstract | References | Similar Articles | Additional Information

Abstract:

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.


References [Enhancements On Off] (What's this?)

  • 1. Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR 1377265
  • 2. E. Beltrami, Resoluzione del problema: riportari i punti di una superficie sopra un piano in modo che le linee geodetische vengano rappresentante da linee rette, Ann. Mat. 1 (1865), no. 7.
  • 3. George D. Birkhoff, Dynamical systems, With an addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. MR 0209095
  • 4. U. Dini, Sopra un problema che si presenta nella theoria generale delle rappresetazioni geografice di una superficie su un'altra, Ann. Mat., ser.2, 3 (1869), 269-293.
  • 5. Kazuyoshi Kiyohara, Compact Liouville surfaces, J. Math. Soc. Japan 43 (1991), no. 3, 555–591. MR 1111603, https://doi.org/10.2969/jmsj/04330555
  • 6. Shoshichi Kobayashi, Transformation groups in differential geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR 1336823
  • 7. V. N. Kolokol′tsov, Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial with respect to velocities, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 994–1010, 1135 (Russian). MR 675528
  • 8. V. S. Matveev and P. Ĭ. Topalov, Trajectory equivalence and corresponding integrals, Regul. Chaotic Dyn. 3 (1998), no. 2, 30–45 (English, with English and Russian summaries). MR 1693470, https://doi.org/10.1070/rd1998v003n02ABEH000069
  • 9. P. I. Topalov, Tensor invariants of natural mechanical systems on compact surfaces, and their corresponding integrals, Mat. Sb. 188 (1997), no. 2, 137–157 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 2, 307–326. MR 1453263, https://doi.org/10.1070/SM1997v188n02ABEH000205
  • 10. T. Y. Thomas, On the projective theory of two dimensional Riemann spaces, Proc. Nat. Acad. Sci. U. S. A. 31 (1945), 259-261. MR 7:33g

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Additional Information

Vladimir S. Matveev
Affiliation: Isaac Newton Institute, Cambridge CB3 0EH, UK
Email: v.matveev@newton.cam.ac.uk

Petar J. Topalov
Affiliation: Department of Differential Equations, Institute of Mathematics and Informatics, BAS, Acad. G. Bonchev Street, Bl. 8, Sofia 1113, Bulgaria
Email: topalov@math.bas.bg

DOI: https://doi.org/10.1090/S1079-6762-00-00086-X
Keywords: Projectively equivalent metrics, ergodic geodesic flows
Received by editor(s): June 16, 2000
Published electronically: December 7, 2000
Communicated by: Dmitri Burago
Article copyright: © Copyright 2000 American Mathematical Society