Affine and fundamental vector fields

This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectively general affine, general linear and translation groups (locally) represented on the manifold via the described in this work left actions; in a case of the manifold Kn = Rn,Cn, the actions mentioned have the usual meaning of affine, linear and translation transformations. Bozhidar Z. Iliev: Affine and fundamental vector fields 1