Ricci Vector Fields Revisited

Hanan Alohali, Sharief Deshmukh and Gabriel-Eduard Vîlcu ()
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Hanan Alohali: 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, 2024, vol. 12, issue 1, 1-16

Abstract: We continue studying the σ -Ricci vector field u on a Riemannian manifold ( N m , g ) , which is not necessarily closed. A Riemannian manifold with Ricci operator T , a Coddazi-type tensor, is called a T - manifold . In the first result of this paper, we show that a complete and simply connected T - manifold ( N m , g ) , m > 1 , of positive scalar curvature τ , admits a closed σ -Ricci vector field u such that the vector u − ∇ σ is an eigenvector of T with eigenvalue τ m − 1 , if and only if it is isometric to the m -sphere S α m . In the second result, we show that if a compact and connected T - manifold ( N m , g ) , m > 2 , admits a σ -Ricci vector field u with σ ≠ 0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature R i c u , u that has a suitable lower bound, then ( N m , g ) is isometric to the m -sphere S α m , and the converse also holds. Finally, we show that a compact and connected Riemannian manifold ( N m , g ) admits a σ -Ricci vector field u with σ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature R i c u , u has a lower bound depending on a positive constant α , if and only if ( N m , g ) is isometric to the m -sphere S α m .

Keywords: Ricci vector field; m -sphere; Riemannian manifold; static perfect fluid equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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