The Schwarzian derivative and conformal transformation on Finsler manifolds

TL;DR

This paper extends the concept of the Schwarzian derivative to Finsler manifolds, introduces Mobius mappings, and explores their properties, including circle preservation and conditions for conformal equivalence to Euclidean spheres or Riemannian spaces.

Contribution

It defines a tensor field generalizing the Schwarzian derivative on Finsler manifolds and characterizes Mobius mappings, linking them to conformal geometry and curvature properties.

Findings

Mobius mappings preserve circles on Finsler manifolds.

If such a mapping exists, the indicatrix is conformally equivalent to a Euclidean sphere.

Certain Finsler manifolds with scalar flag curvature admit non-trivial Mobius mappings only if they are Riemannian with constant curvature.

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.