The Schwarzian derivative and conformal transformation on Finsler manifolds
J. Korean Math. Soc.
Published online March 12, 2020
Behroz Bidabad and 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 : 53C60; 58B20.
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