Eigenvectors of the De-Rham Operator

We aim to examine the influence of the existence of a nonzero eigenvector ζ of the de-Rham operator Γ on a k -dimensional Riemannian manifold ( N k , g ) . If the vector ζ annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field ζ on ( N k , g ) , there are two operators T ζ and Ψ ζ associated with ζ , called the basic operator and the associated operator of ζ , respectively. We show that the existence of an eigenvector ζ of Γ on a compact manifold ( N k , g ) , such that the integral of Ric ( ζ , ζ ) admits a certain lower bound, forces ( N k , g ) to be isometric to a k -dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field ζ on a connected and complete Riemannian space ( N k , g ) , having div ζ ≠ 0 and annihilating the associated operator Ψ ζ , forces ( N k , g ) to be isometric to the k -dimensional Euclidean space, provided that the squared length of the covariant derivative of ζ possesses a certain lower bound.