Characterizations of Spheres and Euclidean Spaces by Conformal Vector Fields

A nontrivial conformal vector field omega on an m-dimensional connected Riemannian manifold Mm,g has naturally associated with it the conformal potential theta, a smooth function on Mm, and a skew-symmetric tensor T of type (1,1) called the associated tensor. There is a third entity, namely the vector field T omega, called the orthogonal reflection field, and in this article, we study the impact of the condition that commutator omega,T omega=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 rho on Mm such that del theta=rho omega; this function rho is called the proportionality function. First, we show that an m-dimensional compact and connected Riemannian manifold Mm,g admits a nontrivial conformal vector field omega with a commuting orthogonal reflection T omega and constant proportionality function rho if and only if Mm,g is isometric to the sphere Sm(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.