Study of η ‐Ricci–Yamabe Solitons and Ricci–Yamabe Solitonss in a Lorentzian Nearly Kähler Space‐Time Manifold

An η ‐Ricci–Yamabe solitonss is a notion of both Ricci and Yamabe solitons, defined by a geometric equation involving a tensor field, which has applications in general relativity and cosmology. The objective of the present research is to examine η ‐Ricci–Yamabe solitonss and Ricci–Yamabe solitonss on covariant projectively flat and concircularly flat Lorentzian nearly Kähler space‐time manifolds that satisfy the Einstein field equations, both with and without a cosmological constant. Next, the necessary and sufficient conditions under which these solitons exhibit expanding, shrinking, or steady behaviour and identify the parameter restrictions that determine their dynamics are established. Furthermore, the analysis is extended to η ‐Ricci–Yamabe solitonss and Ricci–Yamabe solitons corresponding to various cosmological fluids, including dark fluid, dust fluid, stiff matter and radiation fluid. Finally, the existence of η ‐Ricci–Yamabe solitons on projectively flat Lorentzian nearly Kähler space‐time manifolds with nontrivial examples using differential equations is proved.