This article explores properties of conformal vector fields on a Riemannian manifold, focusing on conditions that lead to the manifold being isometric to the Euclidean space. Given a conformal vector ζ with conformal factor σ on a Riemannian manifold N,g, there is naturally associated a skew-symmetric tensor χ to ζ called the essential tensor of ζ. It is shown that the essential tensor χ plays a vital role in our study. We intend to analyze when a conformal vector field becomes a Killing vector field. In a first result of this article, we obtain a necessary and sufficient geometric condition on a complete and connected Riemannian manifold N,g admitting a conformal vector field ζ so that ζ is a Killing vector field. In the rest of the article, we obtain characterizations of a Euclidean space using conformal vector fields. In the first such result, it is shown that an n-dimensional complete and connected Riemannian manifold N,g, n>2 admits a conformal vector field ζ with conformal factor σ≠0 and essential tensor χ such that the affinity tensor of ∇σ is zero, the function ζσ is a constant, and ζ annihilates χ if and only if N,g is isometric to the Euclidean space En. Similarly, in a second characterization of the Euclidean space En using a conformal vector field ζ, we use the following conditions: ζ annihilates the Ricci operator S, ∇σ annihilates χ, and the vector field χζ is incompressible. Finally, we consider a conformal vector field ζ with conformal factor σ≠0 and essential tensor χ on a complete and connected Riemannian manifold N,g such that the Hessian operator Hσ is invariant under the local flow of ζ so that the function ζσ−σ2 is a subharmonic function and ζ annihilates χ, and show that N,g is isometric to the Euclidean space En. The converse holds as well.
Alohali et al. (Tue,) studied this question.