$ \eta $-Ricci-Bourguignon solitons in an almost pseudo-$ W_8 $ flat and $ M $-projective flat symmetric Lorentzian Kähler space-time manifold

B. B. Chaturvedi; Prabhawati Bhagat; Mohammad Nazrul Islam Khan; B. B. Chaturvedi; Prabhawati Bhagat; Mohammad Nazrul Islam Khan

doi:10.3934/math.2025757

AIMS Mathematics

2025, Volume 10, Issue 7: 16837-16864. doi: 10.3934/math.2025757

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Research article

$ \eta $-Ricci-Bourguignon solitons in an almost pseudo-$ W_8 $ flat and $ M $-projective flat symmetric Lorentzian Kähler space-time manifold

1.
Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), 495009, Bilaspur (C.G.), India
2.
Department of Computer Engineering, College of Computer, Qassim University, Saudi Arabia

Received: 31 May 2025 Revised: 04 July 2025 Accepted: 17 July 2025 Published: 28 July 2025
MSC : 35C08, 53C50, 53C55

In this paper, we investigated $ \eta $-Ricci-Bourguignon solitons within the framework of almost pseudo-$ W_8 $ flat and $ M $-projective flat symmetric Lorentzian Kähler space-time manifolds that satisfy the Einstein field equation with and without a cosmological constant. We established necessary and sufficient conditions under which such solitons exhibit expanding, shrinking, or steady behavior. Specifically, we derived constraints on the parameters that govern the soliton dynamics. Furthermore, we extended the results to the case of an $ \eta $-Ricci-Bourguignon soliton for dark fluid, dust fluid, stiff matter, and radiational fluid. Our results contribute to the broader understanding of geometric flows and their interaction with the structure of Lorentzian Kähler geometry using partial differential equations.
Citation: B. B. Chaturvedi, Prabhawati Bhagat, Mohammad Nazrul Islam Khan. $ \eta $-Ricci-Bourguignon solitons in an almost pseudo-$ W_8 $ flat and $ M $-projective flat symmetric Lorentzian Kähler space-time manifold[J]. AIMS Mathematics, 2025, 10(7): 16837-16864. doi: 10.3934/math.2025757

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Abstract

In this paper, we investigated $ \eta $-Ricci-Bourguignon solitons within the framework of almost pseudo-$ W_8 $ flat and $ M $-projective flat symmetric Lorentzian Kähler space-time manifolds that satisfy the Einstein field equation with and without a cosmological constant. We established necessary and sufficient conditions under which such solitons exhibit expanding, shrinking, or steady behavior. Specifically, we derived constraints on the parameters that govern the soliton dynamics. Furthermore, we extended the results to the case of an $ \eta $-Ricci-Bourguignon soliton for dark fluid, dust fluid, stiff matter, and radiational fluid. Our results contribute to the broader understanding of geometric flows and their interaction with the structure of Lorentzian Kähler geometry using partial differential equations.

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