An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If X is a weakly conformal complete vector field on a connected Finsler space (M, F) of dimension n ≥ 2, then, at least one of the following statements holds: (a) There exists a Finsler metric F1 weakly conformally equivalent to F such that X is a Killing vector field of the Finsler metric, (b) M is diffeomorphic to the sphere Sn and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on Sn , and (c) M is diffeomorphic to the Euclidean space Rn and the Finsler metric F is weakly conformally equivalent to a Minkowski metric on Rn . The considerations invite further dynamics on Finsler manifolds.
