Abstract
In this paper we are interested in obtaining characterizations of the Euclidean complex space form (C-n, J, <,>) using specific conformal vector fields on a Kaehler manifold. On the Euclidean complex space form there exist a conformal vector field, whose expression for its covariant derivative motivates the definition of a specific vector field on a Kaehler manifold, which we call a special conformal vector field. We show that a complete simply connected complex space form M(c) (a Kaehler manifold of constant holomorphic sectional curvature c) admits a special conformal vector field if and only if it is isometric to the Euclidean complex space form. We also show that a complete simply connected Kaehler manifold (M, J, g) that admits a non-parallel harmonic special conformal vector field, is isometric to the Euclidean complex space form.