Eigenvectors of the De-Rham Operator

Abstract

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 (Nk,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 (Nk,g), there are two operators Tζ and Ψζ 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 (Nk,g), such that the integral of Ric(ζ,ζ) admits a certain lower bound, forces (Nk,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 (Nk,g), having divζ≠0 and annihilating the associated operator Ψζ, forces (Nk,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.