A GEOMETRIC STUDY OF $f(r)$-GRAVITY SPACETIMES THROUGH PROJECTIVE CURVATURE TENSOR

This research investigates spacetime with projective curvature tensor in $f(r)$ gravity. It explores projectively flat and projectively flat perfect fluid spacetime, determining isotropic pressure ($p$) and energy density ($\sigma$). The study extends to energy conditions and reveals that spacetimes with divergence-free projective curvature tensor may represent a dark energy dominated era or have constant isotropic pressure and energy density, especially when the energy-momentum tensor is recurrent or bi-recurrent.