I asked this question to Professor Alice Chang when I met her during a conference in the University of Notre Dame this June. Loosely speaking, given a Riemannian manifold , why frequently we follow the rule to change our metric in conformal geometry?
Professor Chang told me that it is because under the new metric, angles are preserved. The aim of this note is to make her answer clearer.
The best way to see this is to make use of the vector formulation of the law of cosines. Indeed, let us take two vectors and $Y$ sitting in the same tangent space, say for some . Then the angel between these two vectors can be estimated as follows
where is the norm evaluated with respect to the metric . Under the new metric given by , we first obtain
Clearly, . Moreover, if we write in local coordinates as , we then have
which immediately shows that . In other words, the angle between the two vectors and is preserved under the change of metrics.