Impact of m-projective curvature tensor in f (, G)-gravity and f (,  m )-gravity

This article deals with the characterization of a spacetime and modified gravity, such as f  , G -gravity and f  ,  m -gravity endowed with the m -projective curvature tensor. At first, we show that a m -projectively flat perfect fluid spacetime satisfying Einstein’s field equations represents a dark energy era. Also, we prove that such a spacetime obeying the strong energy condition is locally isometric to Minkowski spacetime. Moreover, it is established that a m -projectively flat perfect fluid spacetime is either a de-Sitter spacetime or a Minkowski spacetime. In addition, we show that a perfect fluid spacetime admitting harmonic m -projective curvature tensor is a Robertson-Walker spacetime. Next, we study m -projectively flat perfect fluid spacetime as a solution of f  , G -gravity and f  ,  m -gravity theories and explain their physical meaning. Several energy conditions in terms of Ricci scalar are investigated with the models f  , G = exp (  ) + α ln ( 6 G ) and f  ,  m = λ +  2 2 +  m . For these models, the weak, null and dominant energy conditions are satisfied, while the strong energy condition is violated, which is good agreement with the recent observational studies and this reveals that the current Universe is in accelerating phase.