  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.
