Start with a pseudo-Riemannian manifold , let
be another pseudo-Riemannian metric on
, we say that
and
are conformal if there exists a positive scalar function
on
such that
(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 , given a point
and two non-null vectors
, the angle between the vectors can be defined by
(Notice that on Euclidean space, if form an angle
, then
.) Thus if
is conformal to
, they define the same angles
In fact, this inference works the other way too. If ,
are two pseudo-Riemannian metrics such that for any two vectors
we have
, then
,
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 be a pseudo-Riemannian manifold that is non-compact. A conformal compactification of
is a choice of a metric
such that
can be isometrically embedded into a compact domain
of a pseudo-Riemannian manifold
(well, I am ignoring some regularity issues here). Let
be the conformal factor as before. Then observe that any regular extension of
to the conformal boundary
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 is compact itself, taken to be standard sphere. The source manifold
is Euclidean space with the standard metric, and the image set
is taken to be the sphere minus the north pole.
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