Inclusion mapping in conformal compactifications

The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the null-cone $\mathcal N$ given by the null-set of the obvious quadratic form $Q$ on $\mathbb R^{n+1,1}$, with the origin removed in order to have a proper hypersurface. The final step is to define the conformal compactification as the quotient $$\mathcal N/\sim,$$ where $\sim$ is the equivalence relation on rays, i.e. $x,y\in\mathcal N$ are equivalent (and we write $x\sim y$) if there exists $\lambda\in\mathbb R\smallsetminus\{0\}$ such that $x = \lambda y$.

As a concrete example we can take the real line $\mathbb R$. The null-cone $\mathcal N$, in this case, is just a double cone without vertex in $\mathbb R^3$, and the equivalence relation $\sim$ gives $\mathcal N/\sim\equiv S^1$, i.e. the unit circle. Therefore the circle $S^1$ is the conformal compactification of the real line $\mathbb R$, as one'd expect.

Similarly one obtains that the conformal compactification of $\mathbb R^2$ is the sphere $S^2$, and it turns out that the relation between the two objects is the stereographic projection.

My question is: how can I explicitly and systematically derive the inclusion of, say, $\mathbb R$ into $S^1$, or that of $\mathbb R^2$ into $S^2$, according to the above outlined construction? In the example of the sphere, for instance, such correspondence is, like I've said before, the stereographic projection.

Further hints towards the systematic determination of such inclusion for generic (pseudo-)Riemannian manifolds (e.g. Minkowski 4-spacetime) into their conformal compactification would be greatly appreciated.

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Not sure if it's what you want, but: take a $0\neq v\in N$, let $V=\{u;u\cdot v=0\}$ (a hyperplane tangent to $N$ along $v$), let $W=V+u$ be a parallel hyperplane (for some $u\notin V$), then you can identify $W\cap N$ with $V/\mathbb R v$ and $W\cap N$ maps injectively to $N/\equiv$; that's what corresponds to the stereographic projection – user8268 Aug 26 '13 at 17:33
Well these are basically equivalent steps to reproduce what I have outlined above. In particular your intersection $W\cap N$ should be a paraboloid, and the quotient $V/\mathbb Rv$ is (isometrically? how?) isomorphic to $\mathbb R^n$. What I don't really see is: 1. how to explicitly write the correspondence between $\mathbb R^n$ and $V/\mathbb Rv$ (I can surely write one, and probably make it isometric, but how can I derive it out of the construction directly?); 2. how to express the injection from $W\cap N$ to $N/\sim$. – Phoenix87 Aug 26 '13 at 18:11

1 Answer

Let us first consider the example of the compactification of the real line $\mathbb R$. The first step is to embed $\mathbb R$ into $\mathbb R^{2,1}$ as, say $x\mapsto (x,0,0)$. The next step is to map $\mathbb R$ onto the parabola given by the intersection of the light-cone $\mathcal N$ with the time-like plane $\mathcal T$ given by $$\mathcal T =\{(x,t,y)\in\mathbb R^{2,1}\ |\ t = 1 + y\},$$ which leads to the locus parametrised by $$\mathcal P:\begin{cases}x = u\\t=\frac{u^2+1}2\\y=\frac{u^2-1}2\end{cases},\qquad u\in\mathbb R.$$ The final step is to map $\mathcal P$ onto the space-like plane $\Sigma=\{(x,t,y)\in\mathbb R^{2,1}\ |\ t=1\}$. This is accomplished by projecting the point of the parabola $\mathcal P$ on the intersection between $\Sigma$ and $\mathcal N$ by a simple rescaling of the pseudo length of the points in $\mathcal P$, in such a way that $t=1$. Hence the map from $\mathbb R$ to $S^1$ is simply given by $$u\mapsto\begin{cases}x = \frac{2u}{u^2+1}\\t=1\\y=\frac{u^2-1}{u^2+1}\end{cases},\qquad u\in\mathbb R.$$

By repeating these steps for $\mathbb R^n$ one finds the compactification of $\mathbb R^n$ to be the closed subset of $\mathbb R^{n+1,1}$ given by $$\mathbf u\mapsto\begin{cases}\mathbf x = \frac{2\mathbf u}{\Vert \mathbf u\Vert^2+1}\\t=1\\y=\frac{\Vert\mathbf u\Vert^2-1}{\Vert\mathbf u\Vert^2+1}\end{cases},\qquad \mathbf u\in\mathbb R^n.$$ For $n=2$ this is precisely the stereographic projection of $\mathbb R^2$ onto the sphere $S^2$.

The same procedure fails for the conformal compactification of any Minkowski-like spaces $\mathbb R^{n,1}$ as a closed submanifold of $\mathbb R^{n+1,2}$. In the above formula, the norm $\Vert u\Vert$ must be replaced with the pseudonorm, which can now attain the value -1, thus leading to unbounded coordinates in the parametrisation. This problem can be solved by intersecting $\mathcal N$ with the sphere $S^{n+2}$ rather than the plane $\Sigma$ (as suggested by Penrose), although it is clear that this will intersect $\mathcal N$ twice, so the resulting compactification, $S^n\times S^1$, is a double covering of the sought compactification of $\mathbb R^{n,1}$.

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