Abstract

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.

Keywords and Phrases

Spacetime, Perfect Fluid Spacetime, $f(r)$-Gravity, Projective Curvature Tensor, Energy-momentum tensor.

A.M.S. subject classification

53A20, 53C25, 53C50, 83C05, 83C15.

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