Eigenvectors of the De-Rham Operator

Nasser Bin Turki, Sharief Deshmukh and Gabriel-Eduard Vîlcu ()
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Nasser Bin Turki: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Gabriel-Eduard Vîlcu: Department of Mathematics and Informatics, National University of Science and Technology Politehnica Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania

Mathematics, 2023, vol. 11, issue 24, 1-15

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 ( 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.

Keywords: de-Rham operator; eigenvector; k -sphere S c k; Ricci curvature; manifold; harmonic vector field (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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