The Schwarzian derivative and conformal transformation on Finsler manifolds
J. Korean Math. Soc. 2020 Vol. 57, No. 4, 873-892
https://doi.org/10.4134/JKMS.j190436
Published online July 1, 2020
Behroz Bidabad, Faranak Sedighi
Amirkabir University of Technology; Payame Noor University of Tehran
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 $ S^{n-1}$ in $ \mathbb{R}^n $. 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 : Finsler, Schwarzian, Mobius, conformal, projective, concircular
MSC numbers : Primary 53C60, 58B20
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