Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?

Question

Let $(M,g)$ be a (semi-)Riemannian manifold, and let $(M,[g])$ be the conformal class thereof. The following two sets are naturally torsors for the collection of positive-valued functions on $M$:

1. The collection of metrics $g'$ in the conformal class $[g]$;

2. The collection of measures on $M$ which are suitably "smooth".

Moreover, there is a canonical bijection from (1) to (2), carrying $g'$ to the associated volume form $vol_{g'}$. This bijection carries $\lambda^2 g'$ to $\lambda^d vol_{g'}$ where $d$ is the dimension of $M$. I'd like to better understand the inverse to this bijection.

Question: How can we recover $g'$ from the volume form $vol_{g'}$ and the conformal structure $[g]$ in geometric terms?

Notes:

• Ideally I'd be interested in a description which would work "synthetically" (i.e. assuming that the conformal structure is somehow given to me "directly", maybe with low regularity, and not in terms of a representative $g'$). For instance, I'd like avoid the "obvious" procedure which proceeds by choosing an arbitrary metric $g$ in the conformal class $[g]$, and observes that $g' = (vol_g/vol_{g'})^{2/d} g$.

• Ideally I'd love to see a method which doesn't require me to know the dimension $d$ of $M$. It's hard to formulate this precisely, since of course the dimension is a topological invariant which can be read off from $M$ without knowing even about $[g]$, but again maybe the measure should be that the method should work "synthetically", if the conformal information is handed to me in some "direct" manner, maybe with low regularity.

Answer 1

I suppose if $M$ is orientable, then your measure should come from a $d$-form $\omega \in \Omega^d(M)$, another torsor over positive functions on $M$. The conformal structure could be given as an angle function $\theta: (T_xM-\{0\})\times (T_xM-\{0\})\to [0,2\pi], \forall x\in M$ (I guess I'm considering here the positive definite case; this is the usual way that I think about a conformal structure, as a way of measuring angles between vectors). Then there will be a unique inner product $g$ on $T_xM$ such that the associated volume form is $\omega_x$ and $\theta(u,v)=g(u,v)/(g(u,u)g(v,v))^\frac12$. This shifts the issue from extracting $g$ from $vol_g$ to extracting $\omega$ from $vol_g$. Not sure that this is helpful, especially because your second bullet point says that you want to avoid inputting the dimension.

Answer 2

A conformal structure with Lorentzian representative produces a conformally invariant causal structure. Causal structure + volume = unique metric is claimed in Bombelli and Meyer (page 2) 1976. I'm uncertain where the result is actually proven, but it's been a long time since I've read Bombelli and Meyer.

For a synthetic approach to conformal structures that mirrors torsors I suggest looking at Kobayshi's "Transformation groups in differential geometry". Note that the connection is Cartan rather than following the form associated to an Ehresmann connection. I have no idea if this will cause issues for you.

If you preform associated vector bundles then the tractor calculus approach is better for you. Curry and Gover have an introduction which is adequate. It cites references which give more detail.

Answer 3

A conformal frame for a conformal structure $\sigma$ of signature $p,q$ on a manifold $M$ of dimension $n=p+q$ is a pair $(m,u)$ of point $m\in M$ and linear isomorphism $T_m M\xrightarrow{u}\mathbb{R}^{p,q}$ identifying conformal structures. The conformal frame bundle $F_{\sigma}$ is the set of all conformal frames, equipped with the map $(m,u)\mapsto m$ and the action $(m,u)h=(m,h^{-1}u)$, for $h\in CO_{p,q}$ a conformal linear transformation of $\mathbb{R}^{p,q}$. Clearly $CO_{p,q}\to F_{\sigma}\to M$ is a smooth principal right $CO_{p,q}$-bundle. Similarly, for each volume density $|\omega|$ on $M$ (locally represented by a volume form), a volume frame is a pair $(m,u)$ of point $m\in M$ and linear isomorphism $T_m M\xrightarrow{u}\mathbb{R}^n$ identifying the volume of open sets. The volume frame bundle $F_{\sigma}$ is the set of all volume frames, equipped with the map $(m,u)\mapsto m$ and the action $(m,u)h=(m,h^{-1}u)$, for $h\in EL_n$ (the set of volume preserving linear isomorphisms). Clearly $EL_n\to F_{|\omega|}\to M$ is a smooth principal right $EL_n$-bundle. The intersection $F_{\sigma}\cap F_{|\omega|}$ is the orthonormal frame bundle of a unique pseudo-Riemannian metric of signature $p,q$.

All of this seems obvious, but it doesn't rely on choice of a representative metric.

In local coordinates $x^i$, we can write out a local section of this intersection bundle as $u=u(x)$, i.e. $u(x)=u^i_j(x)dx^je_i$, if $e_i$ is a fixed basis of $\mathbb{R}^{p,q}$, with constant metric $\eta_{ij}$. Then the metric is $g=\eta_{ij}u^i_k(x)u^j_{\ell}(x)dx^kdx^{\ell}$.