# Why are conformal transformations so relevant?

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.

• I opened this post without paying attention to its author, and half way through I decided I want to leave a comment asking the OP to reach out to one of the MO experts on these conformal stuff for his perspective. Yes, I almost left a comment suggesting you to ask Romain Gicquaud... Commented Jun 27, 2023 at 0:06
• I think the answer is already contained in the 6th paragraph of the post about the "equivalent" quadratic forms. The extension of O(n) by R and the fact that CO(n) is a subgroup of GL(n) could serve to choose the best direction in GL(n) in which one has to deform a given metric. The main issue, from my viewpoint, is how to device the technique helping the choice of that direction. Yang-Mills philosophy can help a much in this regard. Commented Jun 27, 2023 at 8:54
• In indefinite pseudo-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 a smooth conformal map. Commented Jun 27, 2023 at 17:42
• @RomainGicquaud: A metric g define a basis for a direction transverse to O(g) in the frame bundle Fr. That direction determine a canonical real line bundle over O(g), which can be identified with CO(g) and considered as a R-principal bundle over O(g). One can find a metric in the conformal class of g by solving a Yang-Mills-Higgs equations for a weight p, section of CO(g) and a connection 1-form w on CO(g), w naturally extending the Levi-Civita 1-form connection of g. This account for what I understand by Yang-Mills philosophy. Commented Jun 28, 2023 at 0:04
• This can also take care about higher order jets of g, the connection 1-form being some kind of first order deformation. All of this and more can be found in Calderbank and Peterson's "spaces with complex structures, Einstein- Weyl geometry and geodesics" and references therein. Another reason for the preference of conformal for deforming metric, I think, is because SO(n) is a deformation retract of GL(n)+ and so, in pure Riemannian setting, conformal geometry is homotopically, differential geometry.. Commented Jun 28, 2023 at 0:15