Solution
= Solution
At fixed time and angles, proper radial distance is
$$
D=a(t)\int_0^{\Delta r}\frac{dr}{\sqrt{1-kr^2}}.
$$
Thus
$$
D=\begin{cases}
a\Delta r,&k=0,\\
\dfrac a{\sqrt k}\sin^{-1}(\sqrt k\,\Delta r),&k>0,\\
\dfrac a{\sqrt{-k}}\sinh^{-1}(\sqrt{-k}\,\Delta r),&k<0.
\end{cases}
$$
These are respectively flat Euclidean, positively curved spherical, and negatively curved hyperbolic spatial geometries.