Abstract
In this paper first it is proved that if xi is a nontrivial closed conformal vector field on an n-dimensional compact Riemannian manifold (M, g) with constant scalar curvature S satisfying S <= lambda(1) (n - 1), lambda(1) being first nonzero eigenvalue of the Laplacian operator Delta on M and Ricci curvature in direction of a certain vector field is non-negative, then M is isometric to the n-sphere S-n(c), where S = n(n - 1)c. Finally we show that a conformal transformation F : M -> M of a Riemannian manifold (M, g) that preserves the eigenfunctions that is Delta'h = -lambda h whenever Delta h = -mu h, for constants lambda, mu, (g' = F*g and Delta' and Delta are Laplacian operators on (M, g') and (M, g) respectively), then F is a homothety.