Rigidity and structural constraints for $ \rho $-Einstein solitons on twisted warped product manifolds

Ayman Elsharkawy; Clemente Cesarano; Emad F. Wanas; Ayman Elsharkawy; Clemente Cesarano; Emad F. Wanas

doi:10.3934/math.2026669

AIMS Mathematics

2026, Volume 11, Issue 6: 16288-16304. doi: 10.3934/math.2026669

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Research article

Rigidity and structural constraints for $ \rho $-Einstein solitons on twisted warped product manifolds

1.
Department of Mathematics, Faculty of Science, Tanta University, Egypt; Email: ayman_ramadan@science.tanta.edu.eg
2.
Section of Mathematics, International Telematic University Uninettuno, Roma, Italy; Email: c.cesarano@uninettunouniversity.net
3.
Institute of Basic and Applied Science, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt; Email: efw@aast.edu

Received: 17 April 2026 Revised: 25 May 2026 Accepted: 02 June 2026 Published: 08 June 2026
MSC : 53C21, 53C25, 53C44, 53C50, 83C20

We investigate the geometric structure of $ \rho $-Einstein solitons on twisted warped product manifolds, establishing fundamental rigidity phenomena and structural constraints. Our main result proves that the existence of a $ \rho $-Einstein soliton on a twisted warped product $ M_1 \times_f M_2 $ with $ \dim(M_2) \geq 2 $ forces the warping function to be multiplicatively separable, $ f(x_1, x_2) = \phi_1(x_1)\cdot \phi_2(x_2) $, thereby excluding genuinely twisted structures. We derive complete decomposition formulas showing how the soliton equation separates into base and fiber components, with the mixed component imposing severe restrictions. For gradient solitons with separated potentials, we prove additional constraints linking the warping function to the geometry of factor manifolds. Applications to generalized Robertson–Walker and static space-times demonstrate the physical significance of these results. Our findings reveal an inherent geometric obstruction: non-separable warping functions are incompatible with $ \rho $-Einstein soliton structures, suggesting deep connections between soliton geometry and product manifold topology.
- $ \rho $-Einstein solitons,
- twisted warped products,
- Ricci-Bourguignon flow,
- geometric rigidity,
- separability constraints
Citation: Ayman Elsharkawy, Clemente Cesarano, Emad F. Wanas. Rigidity and structural constraints for $ \rho $-Einstein solitons on twisted warped product manifolds[J]. AIMS Mathematics, 2026, 11(6): 16288-16304. doi: 10.3934/math.2026669

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Abstract

We investigate the geometric structure of $ \rho $-Einstein solitons on twisted warped product manifolds, establishing fundamental rigidity phenomena and structural constraints. Our main result proves that the existence of a $ \rho $-Einstein soliton on a twisted warped product $ M_1 \times_f M_2 $ with $ \dim(M_2) \geq 2 $ forces the warping function to be multiplicatively separable, $ f(x_1, x_2) = \phi_1(x_1)\cdot \phi_2(x_2) $, thereby excluding genuinely twisted structures. We derive complete decomposition formulas showing how the soliton equation separates into base and fiber components, with the mixed component imposing severe restrictions. For gradient solitons with separated potentials, we prove additional constraints linking the warping function to the geometry of factor manifolds. Applications to generalized Robertson–Walker and static space-times demonstrate the physical significance of these results. Our findings reveal an inherent geometric obstruction: non-separable warping functions are incompatible with $ \rho $-Einstein soliton structures, suggesting deep connections between soliton geometry and product manifold topology.

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