Characterizing spheres and Euclidean spaces by conformal vector fields

Abstract

It is well known that the Euclidean space \( (R^{n},\left\langle ,\right\rangle )\), the n-sphere \(S^{n}(c)\) of constant curvature c are examples of spaces admitting many conformal vector fields, and therefore conformal vector fields are used in obtaining characterizations of these spaces. In this paper, we use nontrivial conformal vector fields on a compact and connected Riemannian manifold to characterize the sphere \(S^{n}(c)\). Also, we use a nontrivial conformal vector field on a complete and connected Riemannian manifold and find characterizations for a Euclidean space \((R^{n},\left\langle ,\right\rangle )\) and the sphere \(S^{n}(c)\).