$f(\mathcal{R},T)$-gravity is a generalization of Einstein’s field equations $\left({EFE}_s\right)$ and $f\left(\mathcal{R}\right)$-gravity. In this research article, we demonstrate the virtues of the $f(\mathcal{R},T)$-gravity model with Einstein solitons $\left(ES\right)$ and gradient Einstein solitons $\left(GES\right)$. We acquire the equation of state of $f(\mathcal{R},T)$-gravity, provided the matter of $f(\mathcal{R},T)$-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a $f(\mathcal{R},T)$-gravity filled with perfect fluid admits an Einstein soliton $(g,\rho ,\lambda )$ and the Einstein soliton vector field $\rho$ of $(g,\rho ,\lambda )$ is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the $f(\mathcal{R},T)$-gravity model. Next, we prove that if a $f(\mathcal{R},T)$-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the $f(\mathcal{R},T)$-gravity model together with gradient Einstein soliton.

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle was established. Some theorems on the curvature tensor and the projective curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection to its tangent bundle were proved. Finally, locally $\varphi$-symmetric Sasakian manifolds with respect to the quarter-symmetric metric connection to its tangent bundle were studied.

The purpose of this study is to examine the complete lifts from the symmetric and concircular symmetric n-dimensional Lorentzian para-Sasakian manifolds (briefly, ${\left(LPS\right)}_n$) to its tangent bundle $TM$ associated with a Riemannian connection $D^C$ and a quarter-symmetric metric connection (QSMC) ${\overline{D}}^C$.

The purpose of this study is to evaluate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection on the tangent bundle $TM$. Certain results on a semisymmetric P-Sasakian manifold, generalized recurrent P-Sasakian manifolds, and pseudo-symmetric P-Sasakian manifolds on $TM$ are proved.

The subject of the present paper is the investigation of a new type of solitons, called $\eta$-∗-Ricci solitons in $(k,\mu )$-almost co-Kähler manifold (briefly, $ackm$), which generalizes the notion of the $\eta$-Ricci soliton introduced by Cho and Kimura. First, the expression of the ∗-Ricci tensor on $ackm$ is obtained. Additionally, we classify the $\eta$-∗-Ricci solitons in $(k,\mu )$-$ackm$s. Next, we investigate $(k,\mu )$-$ackm$s admitting gradient $\eta$-∗-Ricci solitons. Finally, we construct two examples to illustrate our results.