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@TheoJohnson-Freyd: You may be right about that. I wouldn't have called any group that has no finite-dimensional faithful representation 'classical', but if one's only choices are 'classical' and 'exceptional', then the simply-connected covers of classical matrix groups would be classical.
@AntonPetrunin: Thanks for reminding me. I forgot about this question, even though, at the time, I expected that the answer was known and somebody else would actually give the citation. Coming back to it now, I realize that my original idea that such a metric would be conformally harmonic was based on something that I had misremembered, and turns out to be irrelevant to the question at hand. Thus, I have to say that I don't have any idea about it now. I would delete my comment, but then your comment would be mystifying, so I'm just responding to say I don't have an answer after all.
@BastamTajik: If you look at my answer to the question mathoverflow.net/questions/165304/…, you will see how I describe the approach. There is an $n$-plane field $H$ on the bundle $\pi:R\to M^\mathbb{C}$ whose integrals are the real slices. You have to compute $H$ and then determine the $n$-dimensional integral manifolds of $H$. I explain the results in the case $n=2$, but one needs to do it for $n=4$ in the Gödel case. This will give the answer; it's a finite computation, but not short.
@BastamTajik: There is a way to search for the Riemannian slices and determine whether any exist, but it requires a computation that I don't have time to do right now.
@BastamTajik: Note, however, that the whole point of Lorentzian geometry is that there is no natural separation of time and space. As Minkowski famously said, "Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
@BastamTajik: Yes, but, remember that in the paper that you cite, the authors complexify everything in the sense that they first extend the metric to a holomorphic metric on $\mathbb{C}^4$ and then consider 'real slices' of dimension 4.
@mfdmfd: I first learned about it from my PhD advisor, but then read the relevant works of Cartan and Chern on the equivalence problem of Cartan. The formulation in terms of bundle language is made explicit in papers of D. C. Spencer, but I think that the idea was 'in the air' at the time.
@MoziburUllah: It's called conformal geometry because the conformal structure is the thing that's preserved by the symmetry group (at least in the standard model that includes elliptic, Euclidean, and hyperbolic geometry). It's only in dimension $n=2$ that the appropriate undefined terms are 'point' and 'circle'. True, there is no notion of length, but there's also no notion of 'segment', 'ray', or 'angle'. 'Betweenness' axioms have to be replaced by 'separation' axioms concerning 4 distinct concircular points, etc. There is also a notion of congruence, but for two 4-tuples of points.
@MoziburUllah: Actually, the second axiom that I proposed above is redundant. I should have just written, "The first axiom might be "Any three distinct points are incident with a unique circle.", with pairs of circles classified as disjoint, tangent, or transverse depending on whether they meet in 0, 1, or 2 points. A second axiom might be, "Any circle is incident with at least 3 points." Another axiom would probably be "There exist 4 points that are not concircular." A small theorem then would be "There exist 4 circles that are pairwise transverse".
@MoziburUllah: I see. So, when $n=2$, conformal geometry would take 'point' and 'circle' as primitive, with the simplest primitive relation being 'incidence'. The first axiom might be "Any three distinct points are incident with a unique circle" and another axiom could be "Two distinct circles have at most two incident points in common", with pairs of circles classified as disjoint, tangent, or transverse depending on whether they meet in 0, 1, or 2 points. There would be a separation axiom for four distinct concircular points and a notion of congruence for quadruples of distinct points, etc.
