Characterizations of Spheres and Euclidean Spaces by Conformal Vector Fields

A nontrivial conformal vector field ω on an m -dimensional connected Riemannian manifold M m , g has naturally associated with it the conformal potential θ , a smooth function on M m , and a skew-symmetric tensor T of type ( 1 , 1 ) called the associated tensor. There is a third entity, namely the vector field T ω , called the orthogonal reflection field, and in this article, we study the impact of the condition that commutator ω , T ω = 0 ; this condition that we refer to as the orthogonal reflection field is commutative. As a natural impact of this condition, we see the existence of a smooth function ρ on M m such that ∇ θ = ρ ω ; this function ρ is called the proportionality function. First, we show that an m -dimensional compact and connected Riemannian manifold M m , g admits a nontrivial conformal vector field ω with a commuting orthogonal reflection T ω and constant proportionality function ρ if and only if M m , g is isometric to the sphere S m ( c ) of constant curvature c . Secondly, we find three more characterizations of the sphere and two characterizations of a Euclidean space using these ideas. Finally, we provide a condition for a conformal vector field on a compact Riemannian manifold to be closed.