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Metric with ergodic geodesic flow is completely determined by unparameterized geodesics
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## Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

### Vladimir S. Matveev and Petar J. Topalov

**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.

*Copyright 2000 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**06** (2000), pp. 98-104
- Publisher Identifier: S 1079-6762(00)00086-X
- 2000
*Mathematics Subject Classification*. Primary 53C20; Secondary 37J35, 37C40, 53A20, 53C22, 53B10
*Key words and phrases*. Projectively equivalent metrics, ergodic geodesic flows
- Received by the editors June 16, 2000
- Posted on December 7, 2000
- Communicated by Dmitri Burago
- Comments (When Available)

**Vladimir S. Matveev**

Isaac Newton Institute, Cambridge CB3 0EH, UK

*E-mail address:* `v.matveev@newton.cam.ac.uk`

**Petar J. Topalov**

Department of Differential Equations, Institute of Mathematics and Informatics, BAS, Acad. G. Bonchev Street, Bl. 8, Sofia 1113, Bulgaria

*E-mail address:* `topalov@math.bas.bg`

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