A NOTE ON CONFORMAL VECTOR FIELDS ON A RIEMANNIAN MANIFOLD

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