Conformal vector fields and conformal transformations on a Riemannian manifold

In this paper first it is proved that if ξ is a nontrivial closed conformal vector field on an n-dimensional compact Riemannian manifold (M, g) with constant scalar curvature S satisfying S ≤ λ1(n − 1), λ1 being first nonzero eigenvalue of the Laplacian operator ∆ on M and Ricci curvature in direction of a certain vector field is non-negative, then M is isometric to the n-sphere S(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 ∆′h = −λh whenever ∆h = −μh, for constants λ, μ, (g′ = F ∗g and ∆′ and ∆ are Laplacian operators on (M, g′) and (M, g) respectively), then F is a homothety. M.S.C. 2010: 53C20,53A50.