Characterizations of Spheres and Euclidean Spaces by Conformal Vector Fields

Abstract

A nontrivial conformal vector field ω on an m-dimensional connected Riemannian manifold Mm,g has naturally associated with it the conformal potential θ, 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 Tω, called the orthogonal reflection field, and in this article, we study the impact of the condition that commutator ω,Tω=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 Mm such that ∇θ=ρω; this function ρ 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 ω with a commuting orthogonal reflection Tω and constant proportionality function ρ 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.

Keywords

• conformal field
• conformal potential
• isometric to sphere
• isometric to Euclidean space