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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