Impact of pseudo projective curvature tensor on a space-time and $f(r,G)$-gravity

In this article, we classify pseudo projectively flat space-times and acquire that it represents either an anti- de-Sitter or de-Sitter space-time. Furthermore, we obtain that a pseudo projectively flat perfect fluid space-time represents dark matter era and Robertson-Walker space-time. Next, we establish that a perfect fluid pseudo projectively symmetric space-time represents either a dark matter era, or a static space-time. Also, we examine the effect of pseudo projectively symmetric space-time in [Formula: see text]-gravity by choosing a model and we observe that the null, dominant and weak energy conditions are satisfied, but the strong energy condition has failed, which is also consistent with present observational studies that reveal the universe is expanding. Finally, we apply the flat Friedmann-Robertson-Walker metric to derive a relation between deceleration, jerk and snap parameters.