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.

Keywords

• de-Rham operator
• eigenvector
• Sck%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <i>k</i>-sphere <named-content content-type="inline-formula"><inline-formula><mml:math display="block" id="mm900"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>S</mml:mi><mml:mi>c</mml:mi><mml:mi>k</mml:mi></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula></named-content>
• Ricci curvature
• manifold
• harmonic vector field