Consider $k$ congruent copies of the geodesic compactification of $X:=\Bbb R\times \Bbb H^2.$ Denote the $k$ congruent copies as $X_k$ for $k\in \Bbb N.$ Let $p_k^-$ and $p_k^+$ represent the antipodal singular points of $X_k.$ Embed $X_k$ for all $k,$ into the interior of a sphere, (where all $p_k^+$ and $p_k^-$ lie on the surface of said sphere).
Q: What kind of space is this? And how does one rigorously treat it?
The real line in each copy of $X,$ acts as the diameter of the sphere, connecting the corresponding $p^+$ and $p^-.$ I am having trouble thinking about and defining this rigorously. It's like there are all these real line diameters that pass through the center of the sphere, reminding me of $\Bbb R P^2,$ but that's ignoring the hyperbolic directions of each copy of $X.$
Geodesic Compactification:
Product Compactification:
