# Ngô Quốc Anh

## November 16, 2011

### Conformal compactification

Filed under: Riemannian geometry — Ngô Quốc Anh @ 0:49

Start with a pseudo-Riemannian manifold $(M,g)$, let $\tilde{g}$ be another pseudo-Riemannian metric on $M$, we say that $g$ and $\tilde{g}$ are conformal if there exists a positive scalar function $\phi$ on $M$ such that $\tilde{g} = \phi g$ (sufficient smoothness of the relevant quantities are always assumed).

Observe that two conformal metrics measure angles the same way: recall that on a pseudo-Riemannian manifold $(M,g)$, given a point $p\in M$ and two non-null vectors $v,w\in T_pM$, the angle between the vectors can be defined by

$\displaystyle \frac{g(v,w)^2}{g(v,v) g(w,w) }.$

(Notice that on Euclidean space, if $v,w$ form an angle $\theta$, then $v\cdot w = |v||w| \cos\theta$.) Thus if $\tilde{g}$ is conformal to $g$, they define the same angles

$\displaystyle \frac{\tilde{g}(v,w)^2}{\tilde{g}(v,v)\tilde{g}(w,w)} = \frac{\phi^2 g(v,w)^2}{\phi g(v,v) \phi g(w,w)} = \frac{g(v,w)^2}{g(v,v)g(w,w)}$

In fact, this inference works the other way too. If $g$,$\tilde{g}$ are two pseudo-Riemannian metrics such that for any two vectors $v,w$ we have $g(v,w) = 0 \iff \tilde{g}(v,w) = 0$, then $g$,$\tilde{g}$ are conformal (up to a change of sign) by the above definition (see e.g. Exercise 14, Chapter 2 from B.O’Neill, Semi-Riemannian Geometry).

So, in plain English, two metrics are conformal if they measure angles the same way.

Now, let $(M,g)$ be a pseudo-Riemannian manifold that is non-compact. A conformal compactification of $(M,g)$ is a choice of a metric $\tilde{g}$ such that $(M,\tilde{g})$ can be isometrically embedded into a compact domain $\tilde{M}$ of a pseudo-Riemannian manifold $(M',g')$ (well, I am ignoring some regularity issues here). Let $\phi$ be the conformal factor as before. Then observe that any regular extension of $\phi$ to the conformal boundary $\partial\tilde{M} \subset M'$ must vanish on said boundary. This reflects the property of a conformal compactification that “brings infinity to a finite distance”.

The simplest example of conformal compactification is the one-point compactification of Euclidean space via the stereographic projection. In this case, the target manifold $(M',g')$ is compact itself, taken to be standard sphere. The source manifold $(M,g)$ is Euclidean space with the standard metric, and the image set $\tilde{M}$ is taken to be the sphere minus the north pole.

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