f(R)-gravity and spacetimes with pseudo-projective curvature

One of the goals of the present article is to prove that a Lorentzian manifold is of constant curvature if and only if its pseudo-projective curvature tensor vanishes. It has been established that a conformally flat pseudo-projectively recurrent spacetime is a perfect fluid spacetime. As a consequence, it has been shown that such a spacetime represents a dust matter fluid, stiff matter and dark energy era under certain restrictions on the Ricci curvature. A conformally flat pseudo-projective recurrent spacetime has been analyzed in the perspective of \(f\left( R\right)\) $f\left(R\right)$ -gravity theory. For the model \(f\left( R\right) = \ln \left( aR\right) +a\exp \left( -\dfrac{R}{a}\right)\) $f\left(R\right)=ln\left(a,R\right)+aexp\left(-,{\displaystyle \frac{R}{a}}\right)$ , several energy conditions in terms of the Ricci curvature have been examined and, in addition, it has been shown that the Universe is in accelerating phase which satisfies the weak, null, dominant, and strong energy conditions.