spacetime - What is the curvature of this shape? Is it closed?

This is a follow up question to this issue. I have the following metric:$$\mathrm{d}s_s^2=[1-4\phi^2T^2]\mathrm{d}T^2-\phi^2T^4\mathrm{d}\Theta^2-\phi^2T^4\sin^2\Theta \mathrm{d}\Phi^2$$ Where $\phi$ is a constant in units of $km$ $s^{-2}$ (length per square time), $\Phi$ and $\Theta$ are 3D polar coordinates and $T$ is time. What is the curvature of this space at time, $T$?

If I travel in a straight line in the space defined by this metric and travel long enough, will I end up back where I started (is it closed)?