Abstract
In this paper, we consider an
n
-dimensional compact Riemannian manifold (
M
,
g
) of constant scalar curvature 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 fields with eigenvalue
λ
together with a condition on Ricci curvature of
M
, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (
n
− 1)
λ
, forces
M
to be isometric to the
n
-sphere
S
n
(
λ
).