Abstract
A well-known theorem by E. Hopf states that if a Riemannian 2-torus has no conjugate points, then its Gaussian curvature vanishes identically. This result is a generalization of a theorem by M. Morse and G. Hedlund who have proved that a 2-torus without focal points is flat. In the present paper it is shown that under some additional assumptions a 2-torus is flat if there are no focal points just on a single geodesic.
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Polterovich, I.V. On a characterization of flat metrics on 2-torus. Journal of Dynamical and Control Systems 2, 89–101 (1996). https://doi.org/10.1007/BF02259624
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DOI: https://doi.org/10.1007/BF02259624