Rigidity of almost Ricci solitons on compact Riemannian manifolds

Mohammed Guediri; Norah Alshehri; Mohammed Guediri; Norah Alshehri

doi:10.3934/math.2025608

AIMS Mathematics

2025, Volume 10, Issue 6: 13524-13539. doi: 10.3934/math.2025608

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Research article

Rigidity of almost Ricci solitons on compact Riemannian manifolds

Mohammed Guediri ,
Norah Alshehri ^,

Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia

Received: 18 March 2025 Revised: 02 May 2025 Accepted: 06 May 2025 Published: 12 June 2025
MSC : 53C24, 53C25, 53C42, 53E20, 53Z05

Considering an almost Ricci soliton (ARS) $ \left(N, g, \eta, \kappa \right) $ on a compact Riemannian manifold $ (N, g) $, we use the Ricci curvature in the direction of the potential vector field $ \eta $ to derive necessary and sufficient conditions for $ (N, g) $ to be isometric to a sphere. This expands on several recent results regarding Ricci solitons and almost Ricci solitons by applying specific integral inequalities involving the Ricci curvature evaluated in the direction $ \eta $. Furthermore, we present conditions under which $ \eta $ is either Killing or parallel; in particular, the ARS is trivial.
- almost Ricci solitons,
- compact Riemannian manifolds,
- Ricci and scalar curvatures
Citation: Mohammed Guediri, Norah Alshehri. Rigidity of almost Ricci solitons on compact Riemannian manifolds[J]. AIMS Mathematics, 2025, 10(6): 13524-13539. doi: 10.3934/math.2025608

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Abstract

Considering an almost Ricci soliton (ARS) $ \left(N, g, \eta, \kappa \right) $ on a compact Riemannian manifold $ (N, g) $, we use the Ricci curvature in the direction of the potential vector field $ \eta $ to derive necessary and sufficient conditions for $ (N, g) $ to be isometric to a sphere. This expands on several recent results regarding Ricci solitons and almost Ricci solitons by applying specific integral inequalities involving the Ricci curvature evaluated in the direction $ \eta $. Furthermore, we present conditions under which $ \eta $ is either Killing or parallel; in particular, the ARS is trivial.

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