  This article concerns with the characterization of a spacetime and modified
gravity, such as $f\left(R,G\right)$, $f\left(R,T\right)$ and
$f\left(R,L_{m}\right)$-gravity equipped with the projective curvature tensor.
We establish that a projectively flat perfect fluid spacetime represents dark
energy era. Also, we prove that a projectively flat perfect fluid spacetime is
either locally isometric to Minkowski spacetime or a de-Sitter spacetime.
Furthermore, it is shown that a perfect fluid spacetime permitting harmonic
projective curvature tensor becomes a generalized Robertson-Walker spacetime
and is of Petrov type $I$, $D$ or $O$. Lastly, we investigate the effect of
projectively flat perfect fluid spacetime solutions in $f\left(R,G\right)$,
$f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity, respectively. We also
investigate the spacetime as a $f\left(R,G\right)$-gravity solution of and use
the flat Friedmann-Robertson-Walker metric to establish a relation among jerk,
snap, and deceleration parameters. Numerous energy conditions are studied in
terms of Ricci scalar with the model $f\left(R,G\right)=\exp(R)+\alpha
\left(6G\right)^{\beta}$. For this model, the strong energy condition is
violated but the weak, dominant and null energy conditions are fulfilled, which
is in excellent accordance with current observational investigations that show
the universe is now accelerating.
