  The (4+1) dimensional conformally flat Eisenhart geometry is investigated in
this work, stressing the contribution of the stress tensor generating its
curvature. The energy-momentum tensor $T^{a}_{~b}$ is traceless and has only
one nonzero component. It could be written as an anisotropic fluid with null
transversal pressures and nonzero energy fluxes. The null and timelike
geodesics are computed in the pure cosmological case when the Eisenhart
potential energy is $V(r) = -m\omega^{2}r^{2}/2$, where $\omega$ is related to
the cosmological constant $\Lambda$. Although the metric is curved, the radial
null geodesics $R(T)$ and $Y(T)$ are straight lines, with finite $R_{max}$ and
$Y_{max}$, $Y$ being the 5th coordinate. In contrast, for a radial timelike
geodesic, $Y_{max} \rightarrow \infty$ if $T \rightarrow T_{max} = 1/\omega$.
