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The Schwarzian derivative on Finsler manifolds of constant curvature

Abstract

Lagrange introduced the notion of Schwarzian derivative and Thurston discovered its mysterious properties playing a role similar to that of curvature on Riemannian manifolds. Here we continue our studies on the development of the Schwarzian derivative on Finsler manifolds. First, we obtain an integrability condition for the Möbius equations. Then we obtain a rigidity result as follows; Let (MF) be a connected complete Finsler manifold of positive constant Ricci curvature. If it admits non-trivial Möbius mapping, then M is homeomorphic to the n-sphere. Finally, we reconfirm Thurston’s hypothesis for complete Finsler manifolds and show that the Schwarzian derivative of a projective parameter plays the same role as the Ricci curvature on theses manifolds and could characterize a Bonnet–Mayer-type theorem.

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Acknowledgements

The first author would like to thank the “Institut de Mathématiques de Toulouse” (ITM) at the Paul Sabatier University of Toulouse, where this article is partially written.

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Correspondence to B. Bidabad.

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Bidabad, B., Sedighi, F. The Schwarzian derivative on Finsler manifolds of constant curvature. Period Math Hung 84, 346–357 (2022). https://doi.org/10.1007/s10998-021-00411-z

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  • DOI: https://doi.org/10.1007/s10998-021-00411-z

Keywords

  • Finsler
  • Schwarzian
  • Möbius
  • Constant curvature
  • Conformal
  • Completely integrable

Mathematics Subject Classification

  • 53C60
  • 58B20