On the rigidity of Finslerian conformal circle-preserving transformations

Abstract

We prove that if a forward or backward complete Berwaldian or reversible Finslerian manifold $(M,F)$ admits a non-trivial (non-homothety) conformal concircular transformation ({\bf c}ircle-{\bf p}reserving {\bf t}ransformation or {\cpt} for short), $e^σ F$, where $σ$ has at least one critical point, then, $(M,F)$ is Riemannian. Consequently, $(M,g)$ is conformally diffeomorphic to either 1) the standard sphere, 2) the Euclidean space, or 3) the hyperbolic space. In particular, a compact Berwaldian or reversible Finslerian manifold does not admit any non-trivial conformal {\cpt}s, unless it is conformally diffeomorphic to the standard sphere.