Solitons in  \(f(\mathcal {R},T)\) -Gravity

The goal of this book chapter is to examine the perfect fluid spacetimes fulfilling \(f(\mathcal {R},T)\) -gravity when Ricci and gradient Ricci solitons, Yamabe and gradient Yamabe solitons, \(\eta \) -Ricci and gradient \(\eta \) -Ricci solitons are its metrics. We establish criteria for which Ricci solitons are steady, expanding, or shrinking. Moreover, we study gradient Ricci solitons and prove that either the perfect fluid spacetime represents the dark energy era, or the spacetime has zero vorticity under certain conditions. Moreover, we investigate perfect fluid spacetimes fulfilling \(f(\mathcal {R},T)\) -gravity permitting Yamabe solitons and gradient Yamabe solitons and show that the integral curves of the velocity vector field \(\rho \) are geodesics and energy density and isotropic pressure remain invariant under \(\rho \) and for the gradient case the Ricci scalar is constant. Also, we study \(\eta \) -Ricci and gradient \(\eta \) -Ricci solitons in \(f(\mathcal {R},T)\) -gravity, respectively. Specifically, we establish criteria in which \(\eta \) -Ricci solitons are shrinking, expanding, or steady and for gradient \(\eta \) -Ricci solitons, either the spacetime represents the equation of state \(p+\sigma =\) constant, or the perfect fluid has vanishing vorticity.