Rigidity of almost Ricci solitons with affine vector fields on semi-Riemannian manifolds

Norah Alshehri; Mohammed Guediri; Norah Alshehri; Mohammed Guediri

doi:10.3934/math.20261096

AIMS Mathematics

2026, Volume 11, Issue 8: 27396-27410. doi: 10.3934/math.20261096

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Research article

Rigidity of almost Ricci solitons with affine vector fields on semi-Riemannian manifolds

Norah Alshehri ^,,
Mohammed Guediri

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

Received: 23 April 2026 Revised: 19 August 2026 Accepted: 21 August 2026 Published: 31 August 2026
MSC : 53C21, 53C25, 53C44, 53C50, 53C80

We investigated almost Ricci solitons on semi-Riemannian manifolds when the potential vector field was affine. Our central theorem showed that in every dimension $ n \geq 3 $, such almost Ricci solitons coincide with Ricci solitons, and the manifolds had constant scalar curvature. In contrast, in dimension 2, we constructed explicit examples of proper (but trivial) almost Ricci solitons, highlighting a genuine low dimensional distinction. In the Riemannian setting, this leads to rigidity: Under natural geometric assumptions, an almost Ricci soliton must be parallel, trivial, or Ricci-flat.
- semi-Riemannian manifolds,
- Ricci solitons,
- almost Ricci solitons,
- killing vector fields,
- parallel vector fields,
- affine vector fields,
- scalar curvature
Citation: Norah Alshehri, Mohammed Guediri. Rigidity of almost Ricci solitons with affine vector fields on semi-Riemannian manifolds[J]. AIMS Mathematics, 2026, 11(8): 27396-27410. doi: 10.3934/math.20261096

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Abstract

We investigated almost Ricci solitons on semi-Riemannian manifolds when the potential vector field was affine. Our central theorem showed that in every dimension $ n \geq 3 $, such almost Ricci solitons coincide with Ricci solitons, and the manifolds had constant scalar curvature. In contrast, in dimension 2, we constructed explicit examples of proper (but trivial) almost Ricci solitons, highlighting a genuine low dimensional distinction. In the Riemannian setting, this leads to rigidity: Under natural geometric assumptions, an almost Ricci soliton must be parallel, trivial, or Ricci-flat.

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