Strongly and weakly affine vector fields on Finsler manifolds

In this paper, we investigate the affine vector fields on both compact and forward complete Finsler manifolds. We first give definitions of the affine transformation and the affine vector field. Unexpectedly, we find two kinds of affine fields, which are named as the strongly and weakly affine vector fields. Based on these definitions, we prove some rigidity theorems of affine fields on compact and forward complete Finsler manifolds.