Eigenvectors of the De-Rham Operator
We aim to examine the influence of the existence of a nonzero eigenvector z of the de-Rham
operator G on a k-dimensional Riemannian manifold (Nk, g). If the vector z 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 z on (Nk, g), there are two operators Tz and Yz associated with z, called the
basic operator and the associated operator of z, respectively. We show that the existence of an
eigenvector z of G on a compact manifold (Nk, g), such that the integral of Ric(z, z) 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 z on a connected and complete Riemannian space
(Nk, g), having div(z) 6= 0 and annihilating the associated operator Yz , forces (Nk, g) to be isometric
to the k-dimensional Euclidean space, provided that the squared length of the covariant derivative of
z possesses a certain lower bound.
We aim to examine the influence of the existence of a nonzero eigenvector z of the de-Rham
operator G on a k-dimensional Riemannian manifold (Nk, g). If the vector z annihilates the de-Rham…
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We use a nontrivial concircular vector field u on the unit sphere Sn+1 in studying
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naturally…