Well, as a first stab, I think with the normalising factor the map $\mathbb{R}^{n,1} \to S^n\times S^1 \subset \mathbb{R}^{n,1}\times\mathbb{R}^{1,1}$ looks like

$\frac{1}{\sqrt{q(x,t)^2 + 2(|x|^2+t^2)+1}}(2x,2t,1-q(x,t),-1-q(x,t))$in the notation of conformal compactification.

]]>I added to conformal compactification the actual formula for the embedding $\mathbb{R}^{n,1}\hookrightarrow \mathbb{R}^{n+1,2}\setminus \{0\}$. In fact this is a diffeomorphism with the intersection of the real points of the Klein quadric with a particular hyperplane (the original Minkowski space times one of the null planes in the extra $\mathbb{R}^{1,1}$ factor – there’s another disjoint embedding using the other null plane). I need to calculate the correct normalisation factor that means this will land in $S^n\times S^1$ instead, but not now.

]]>Added link to Klein quadric to published version of Klein’s thesis (is Springer hiding results from Google? I could only get it by searching the journal website itself, once I discovered where it was published) in German, and a link to a pdf of the English translation, not sure where *that* pdf is from, searching its title gives only that one result.

Whoops, thanks.

]]>Thanks! (we should say that your edit is at *Klein quadric*)

I added the equation, and the reference to Klein’s thesis.

]]>I have added the brief remark that the example of complexified Minkowski spacetimes conformally compactifying to the Klein quadric is key in the *twistor correspondence*.

And thereby this discussion connects to the solitonic M5-dicussion here. Is that the reason why you are looking into conformal compactification at the moment? Still building that explicit String 2-bundle for Christian?

]]>I added that the Klein quadric conformally compactifies complexified $\mathbb{R}^{3,1}$, with a sketchy reference as really I have to go offline now. (Lecturing in 8.5 hours and need to sleep first…)

]]>Thanks. It seems that’s a better place to engage Willie than here, but I may copy over any insights if they are warranted.

]]>edits and edit discussion on the entry *conformal compactification* is going on here