I return to Friedrich's paper.

Section 2.1 continues the introduction but with some more details. The problem to be considered is an initial value problem given by a 3-dimensional smooth Riemannian manifold

Further to this we assume that

Friedrich points out, further later use, that if we assume that the Yamabe number associated tot he conformally related metric

Lastlyhe proves a lemma showing that with mild assumptions on

Section 2.2 discusses the conformal constraint equations and how to solve them. Friedrich uses standard techniques along with the assumption that on

It is important to note the construction of the functions

Section 2.3 goes over the initial data for the conformal field equation 2.1. I'm not exactly sure why these additional fields are required, but I suppose we'll find out. Four new piece of information are required. In each case it seems that they are used to restrict certain types of behavior in the unphysical metric and a new conformal connection. The new data are,

- A frame of vectors over
$S$ that are orthonormal with respect to the unphysical metric - A conformal connection that is unrelated to either metrics. Friedrich gives a reference to his paper, "Einstein's equations and conformal structure" as a justification for this addition.
- A condition on a tensor derived from the Ricci tensor of the conformal connection. I have no real idea of why this is introduced, I suppose we'll find out later.
- A condition on the rescaled Weyl tensor of the unphysical metric. Ostensibly I think this is so that we can get some local control over a power series expansion of thephysical Weyl tensor at
$i$ , but to be honest I'm once again unclear why this is needed.