A note on conformal vector fields on a riemannian manifold

Abstract

We consider an n-dimensional compact Riemannian manifold (M; g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector elds with eigenvalue  λ > 0, together with an upper bound on the energy of the vector eld λ, implies that M is isometric to the n-sphere Sn(λ). We also introduce the notion of ϕ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres using these vector fields.