Abstract
In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold (M, g) with potential function f, there exists a unique smooth function p on M that connects u to the gradient of potential function Vf. We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n -sphere Sn (c) and the Euclidean space P. We also show that the connecting function influences on a topology of the Riemannian manifold.