Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE SCHWARZIAN DERIVATIVE AND CONFORMAL TRANSFORMATION ON FINSLER MANIFOLDS

  • Bidabad, Behroz (Faculty of Mathematics and Computer Science Amirkabir University of Technology) ;
  • Sedighi, Faranak (Faculty of Mathematics Payame Noor University of Tehran)
  • Received : 2019.06.25
  • Accepted : 2020.01.02
  • Published : 2020.07.01
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Abstract

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere Sn-1 in ℝn. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.

Keywords

Acknowledgement

The first author would like to thank the Institut de Mathématiques de Toulouse (IMT) in which this article is partially written.

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