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The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.

Question: Is there a similarly direct geometric interpretation of the Weyl conformal tensor ${C^a}_{bcd}$?

Background: My understanding is that the Weyl conformal tensor is supposed to play a role in conformal geometry analogous to the role of the Riemann curvature tensor in (pseudo)Riemannian geometry. For instance, it is conformally invariant, and (in dimension $\geq 4$) vanishes iff the manifold is conformally flat, just as the Riemann curvature tensor is a metric invariant and vanishes iff the manifold is flat. The two tensors also share many of the same symmetries. So it would be nice to have a more hands-on understanding of the Weyl tensor when studying conformal geometry.

Notes:

  • I'd be especially happy with a geometric interpretation which is manifestly conformal in nature, referring not to the metric itself but only to conformally invariant quantities like angles.

  • I'm also keen to understand any subtleties which depend on whether one is working in a Riemannian, Lorentzian, or more general pseudo-Riemannian context.

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    $\begingroup$ One aspect which is important to have in mind when dealing with the Weyl tensor is that due to its algebraic symmetries it identically vanishes if the manifold dimension is less than four. In three dimensions there is another conformal tensor built from the metric and its derivatives, called the Bach tensor, which shares many of the properties of the Weyl tensor. In Lorentzian geometry, the importance of the Weyl tensor also lies on the fact that, on space-times solving the Einstein equations, the Weyl tensor encodes all dynamically propagating degrees of freedom (e.g. gravitational waves). $\endgroup$ – Pedro Lauridsen Ribeiro Aug 6 at 18:57
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    $\begingroup$ Penrose has some way of thinking about the Weyl tensor geometrically, but only for Lorentzian signature 4-manifolds, if I remember correctly, but I think his approach splits into self-dual and anti-self-dual parts first, and then gives each one an interpretation. $\endgroup$ – Ben McKay Aug 9 at 13:40
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There is such an interpretation, with a few caveats. Essentially, there is a canonical connection on a certain vector bundle for which the "principal part" of the curvature is the Weyl tensor in dimensions $n\geq4$, and the Cotton tensor when $n=3$. I will describe this from the point of view of the tractor calculus, but avoid introducing unnecessary bundles where needed. This can also be described using the Fefferman–Graham ambient metric or using Cartan connections. This summary mostly follows Bailey–Eastwood–Gover, though Armstrong and articles written by Gover are also good references. I use abstract index notation throughout.

First, we define conformal densities. Given a conformal manifold $(M,c)$, a conformal density of weight $w\in\mathbb{R}$ is an equivalence class of pairs $(g,f)\in c\times C^\infty(M,c)$ with respect to the equivalence relation $(g,f)\sim(e^{2\Upsilon}g,e^{w\Upsilon}f)$. Let $\mathcal{E}[w]$ denote the space of conformal densities of weight $w$. We similarly define $\mathcal{E}^i[w]$ as the space of equivalence classes of pairs $(g,v^i)\in c\times\mathfrak{X}(M)$ with respect to the equivalence relation $(g,v^i)\sim(e^{2\Upsilon}g,e^{w\Upsilon}v^i)$. Here $\mathfrak{X}(M)$ is the space of vector fields on $M$.

Next, we define the space of sections of the standard tractor bundle. Fix a metric $g\in c$. Define $\mathcal{T}_g^A=\mathcal{E}[1]\oplus\mathcal{E}^i[-1]\oplus\mathcal{E}[-1]$. Given another metric $\hat g \mathrel{:=} e^{2\Upsilon}g\in c$, we identify $(\sigma,v^i,\rho)\in\mathcal{T}_g^A$ with $(\hat\sigma,\hat v^i,\hat\rho)\in\mathcal{T}_{\hat g}^A$ if $$ \begin{pmatrix} \hat\sigma \\ \hat v^i \\ \hat\rho \end{pmatrix} = \begin{pmatrix} \sigma \\ v^i + \sigma\Upsilon^i \\ \rho - \Upsilon_j v^j - \frac{1}{2}\Upsilon^2\sigma \end{pmatrix} . $$ (Recall these are densities, so exponential factors are suppressed.) The space of sections $\mathcal{T}^A$ is the result after making this identification. Note that the top-most nonvanishing component is actually conformally invariant modulo multiplication by an exponential factor. Because of this, we call the top-most nonvanishing component the projecting part.

There is a canonical connection on (the vector bundle whose space of sections is) $\mathcal{T}^A$, the standard tractor connection, which, given a choice of metric $g\in c$, is given by the formula $$ \nabla_j \begin{pmatrix} \sigma \\ v^i \\ \rho \end{pmatrix} = \begin{pmatrix} \nabla_j\sigma - v_j \\ \nabla_j v^i + \sigma P_j^i + \delta_j^i\rho \\ \nabla_j\rho - P_{ji}v^i \end{pmatrix} . $$ Here $P_{ij}=\frac{1}{n-2}\left( R_{ij} - \frac{R}{2(n-1)}g\right)$ is the Schouten tensor and $n=\dim M$. It is straightforward to check that this is well-defined, in the sense that it is independent of the choice of matrix $g\in c$.

Given a metric $g\in c$, it is straightforward to compute that $$ (\nabla_i\nabla_j - \nabla_j\nabla_i)\begin{pmatrix} \sigma \\ v^k \\ \rho \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \\ C_{ij}{}^k & W_{ij}{}^k{}_l & 0 \\ 0 & -C_{ijl} & 0 \end{pmatrix} \begin{pmatrix} \sigma \\ v^l \\ \rho \end{pmatrix} . $$ This is conformally invariant by construction. The "3-by-3" matrix is the tractor curvature, and its projecting part is $W_{ij}{}^k{}_l$ when $n\geq4$ and $C_{ij}{}^k$ when $n=3$. Standard interpretations of holonomy then give the interpretation of the Weyl tensor in terms of parallel transport around infinitesimal loops that I indicated in the first paragraph.

Finally, given your bullet points, let me emphasize that the signature of $c$ plays no role here, and everything is manifestly conformally invariant.

Added in response to a comment. There are many geometric motivations for introducing the standard tractor bundle. One is that the conformal group of the sphere is $\operatorname{SO}(n+1,1)$, so it makes sense that the right replacement of the tangent bundle of a conformal $n$-manifold should be a bundle of rank $n+2$, as is the standard tractor bundle. Note that the metric on $\mathcal{T}$ has signature $(n+1,1)$, assuming we start with a conformal manifold of Riemannian signature (if $c$ has signature $(p,q)$, the metric on the standard tractor bundle has signature $(p+1,q+1)$).

Another motivation comes from the ambient metric. First, note that the flat conformal sphere $(S^n,c)$ (i.e., the conformal class of the round $n$-sphere) can be identified with the positive null cone $\mathcal{N}$ centered at the origin in $\mathbb{R}^{n+1,1}$. This is done by noting that the projectivization of $\mathcal{N}$ is $S^n$ and identifying sections of $\pi\colon\mathcal{N}\to S^n$ with metrics in the conformal class $c$ by pullback of the Minkowski metric. (Incidentally, this leads to a proof that $\operatorname{SO}(n+1,1)$ is the conformal group of $S^n$.) In this case, a fiber $\mathcal{T}_x$ of the standard tractor bundle is identified with $T_p\mathbb{R}^{n+1,1}$ for some $p\in\pi^{-1}(x)$; this is made independent of the choice of $p\in\pi^{-1}(x)$ by identifying tangent spaces at points subject to a homogeneity condition matching that of $\mathcal{E}^i[-1]$ above. The standard tractor connection is then induced by the Levi–Civita connection in Minkowski space, after making some identifications.

For a general conformal manifold $(M^n,c)$ of Riemannian signature, Fefferman and Graham showed that there is a "unique" Lorentzian manifold $(\widetilde{\mathcal{G}},\widetilde{g})$ which is "formally Ricci flat" and in which $(M^n,c)$ isometrically embeds as a null cone. Here formally Ricci flat means that the Ricci tensor of $\widetilde{g}$ vanishes to some order, depending on the parity of $n$, along the null cone, and I write unique in quotes because the metric is only determined as a power series to some order along the cone, and this up to diffeomorphism. One recovers the standard tractor bundle and its canonical connection from that of $(\widetilde{\mathcal{G}},\widetilde{g})$ as in the previous paragraph. See Fefferman–Graham for details, or Čap–Gover for a detailed description of the relation between the tractor calculus and the ambient metric, including the identifications I didn't detail. A similar construction for other signatures works, consistent with what is described in the previous paragraph.

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  • $\begingroup$ Thanks! It's great to learn that to any conformal manifold is naturally associated this tractor bundle and a connection on it of which the Cotton and Weyl tensors are components of the curvature -- just as the Riemann curvature tensor is the curvature of the Levi-Civita connection on the tangent bundle. Right now the tractor bundle and its connection look much more mysterious to me than the tangent bundle and the Levi-Civita connection. Is there anything to be said about what they "mean geometrically" which might make them seem as natural as the tangent bundle and Levi-Civita connection? $\endgroup$ – Tim Campion Aug 8 at 21:01
  • $\begingroup$ I have added some comments which give one possible interpretation. I recommend you read the introduction of Čap--Gover, cited in my answer, for further details or interpretations. $\endgroup$ – Jeffrey Case Aug 9 at 0:04

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