Communications of the
Korean Mathematical Society CKMS

This paper is concerned with the study of spacetimes satisfying $\mathrm{div}\mathcal{M}=0$, where ``div" denotes the divergence and $\mathcal{M}$ is the $m$-projective curvature tensor. We establish that a perfect fluid spacetime with $\mathrm{div}\mathcal{M}=0$ is a generalized Robertson-Walker spacetime and vorticity free; whereas a four-dimensional perfect fluid spacetime becomes a Robertson-Walker spacetime. Moreover, we establish that a Ricci recurrent spacetime with $\mathrm{div}\mathcal{M}=0$ represents a generalized Robertson-Walker spacetime.

Keywords: Divergence free $m$-projective curvature tensor, vorticity, geodesic, perfect fluid spacetimes, generalized Robertson-Walker spacetimes

MSC numbers: Primary 53C22, 53C50, 53Z05, 83C05