I have been studying construction of initial data in general relativity for many years now and it turns out that the most efficient methods to construct such data rely at some point on conformally changing a background metric (you can look at the conformal method, the conformal thin sandwich... see e.g. https://arxiv.org/abs/1402.5585).

The idea of conformal transformations is simple: given a Riemannian manifold $(M, g)$ and a smooth positive function $u$ on $M$, the 2-tensor $u^2 g$ is again a metric on $M$.

There are simple formulas that relate how the various curvatures change under such a conformal transformation. Some of the simplest and most beautiful ones are:

- The Weyl tensor is multiplied by $u^2$ (seen as a fully covariant 4-tensor), i.e. is conformally covariant
- The conformal transformation law for the scalar curvature can be seen as an elliptic equation for $u$ (this is one of the core motivations to study the Yamabe problem).

There are other deformations one can potentially think about such as considering the "graph transformation", namely look at the metric induced on the graph of a given function $f$ in $M \times \mathbb{R}$, i.e. change the metric $g$ to $g + df \otimes df$ (see e.g. https://arxiv.org/abs/1010.4256). However, formulas appear less appealing from an analytic perspective (for example, ellipticity is not guaranteed).

I would like to understand what makes conformal transformation so relevant among all possible ways to modify a metric.

The beginning of an answer would be that the normalizer of $O(n)$ in $GL(n)$ is the conformal group $CO(n)$. This can be interpreted as saying that two quadratic forms $\Phi_1, \Phi_2$ on $\mathbb{R}^n$ have the same orthogonal group, $O(\mathbb{R}^n, \Phi_1) = O(\mathbb{R}^n, \Phi_2)$, if and only if they are proportional one to another. But I do not see how to pass from this remark to the fact that conformal transformations are the nicest ones to consider.

Could you provide further insights or explanations on the connection between the normalizer of $O(n)$ in $GL(n)$ being the conformal group $CO(n)$ and the preference for conformal transformations in modifying a metric? I would appreciate any additional clarification on this topic.

indefinitepseudo-Riemannian geometry, there's a very concrete reason as to why conformal transformations are important... two indefinite nondegenerate quadratic forms are proportional to each other iff their zero sets coincide. For Lorentzian geometry, this means that conformal transformations preserve light cones; more than that, these are precisely the transformations that preserve the causal structure. This requirement is so strong, that for $n$-dim. strongly causal space-times ($n>2$) any bijection that preserves lightlike geodesic segments must be asmoothconformal map. $\endgroup$8more comments