Talk:Conformality is Equivalence Relation on Set of Riemannian Metrics

Notation

The set $\set {g}$ is supposed to denote a collection of all Riemannian metrics which are admissible to $M$. The precise condtions come from the topology and differentiable structure of $M$. Then we are supposed to pick a subset of these metrics which in addition are related by conformal transformations. Then the conformal transformation as a relation on this subset of metrics is an equivalence relation. Maybe what we need is a set of all Riemannian metrics on $M$ like $S = \set {g : g \text { is a Riemannian metric on } M}$, and then define the conformal subset $C \subseteq S$ on which the equivalence relation will be proved?--Julius (talk) 12:10, 2 June 2023 (UTC)

The notation needs to be explained properly, because it makes no mathematical sense according to what we have on $\mathsf{Pr} \infty \mathsf{fWiki}$.

I think we actually have to agree on what we want to say. In the source it is simply said in words. To write this with the set builder notation we have to choose the level of detail needed. If it is sufficient to understand the meaning of metric, then the set of admissible metrics can be written as above. But if we want something like Definition:Set of All Mappings then many more details need to be provided. What is more, if a mapping (in this case, the metric) maps onto other mappings (the inner products at each point of the manifold), we either have to introduce dummy variables to end up with a real number or define the set of all inner products. What I wrote on the page is supposed to be the minimal notation whose meaning should follow from the context. I will try to come up with a more detailed notation, but someone will have to verify its validity.--Julius (talk) 21:30, 2 June 2023 (UTC)

All good stuff, but in the end, the notation has to be explained. I'm not the man to do this because I don't understand it. --prime mover (talk) 00:15, 3 June 2023 (UTC)