For me, a Cartan geometry is just a "natural and reasonable" deformation of a homogeneous space of Lie type ( that's, the quotient of a Lie group by a closed subgroup, whatever the algebraic structure of the Lie supergroup), the deformation being given by a Ehresmann connection on the supergroup, with vertical vector bundle, the right translations of the Lie algebra of the Lie subgroup.

Of all these geometries, there should exist one whose characteristic connection is constructed from a "non-degenerate" symmetric covariant r-tensor as the Levi-Civita connection is canonically obtained from its Riemann metric. A Riemann-Cartan geometry is just one element of the "projective" orbit of such a higher order Riemann structure.

There are too many connections (i.e geometries) in the landscape: Levi-Civita, Weyl, Ehresmann (i.e non-linear connection) and its specializations: Cartan, Berwald, Finsler, Chern, Bismuth, Lagrangians... There should be a common source for all these geometries and it is this common source which achieves, in our sense, the unity of "mathematics" and by corollary, the unification of physical theories.

Also, for me, the problem as posed is some kind of octopus (in fact "polypus") and it's hard (very hard indeed) for us to see a field of geometry where it doesn't have a foot... This should be in straight line with the Gromov's GAFA 2000 prediction that Contact Geometry should absorbs all Geometry (if I well understand the meaning of that prediction).

By following V. I. Arnol'd 's philosophy Think "contactly" and compute "symplectically", the above comment is our contact point of view of the problem. The symplectic point of view should be highly tied with pseudo-holomorphic "calculus" which should, at its turn, open the door to a wonderful world and at same time, makes the bridges between Lorentzian Geometry, Contact Geometry and Seifert Manifolds Theory (at least).

please, what do you mean by "C-K spacetime"?

I think that the existence of a conjugate function is equivalent to the fact that the distribution generated by $L$ and $\underline{L}$ is integrable and this integrability should be (I think) equivalent to the existence of an Open Book Decomposition (perhaps a planar one) on the underlying manifold. For sure, the problem is some kind of "snake basket"...

...there should be equivalence between the last question and the fact that...

A very pedagogical answer. Thank a lot. But I think that for a flow mapping these slices, it suffices that the vector field be regular (nowhere vanishing) gradient field. The same conclusion should be true if there is a regular (differentiable) section (initial section) of the flow lines and the vector field is regular and orthogonal to that initial section, at least in the (one-sided?) neighborhood of that initial section. Also, there should be equivalence the last question and the fact that the vector field define riemannian flow.

Transverse structures on foliated manifolds.

In this same vein, the entire Heidelberg team, around A. Wiennhard, should see things very clearly, particularly for the construction of the characteristic phase 7-manifold. In this case too, I could be marveling at objects that are obsolete in their eyes...

By design itself, if such an idea of ​​the conformally Anosov flow is worth dwelling on for a few moments, then it should be clear to any worthy student of the C. Ehresmann school, headed by E. Ghys (sincerely sorry to seem to speak so lightly of such illustrious people). Furthermore, it was by reading with much more attention the article by E. Ghys (L'invariant de Godbillon-Vey, séminaire Bourbaki : volume 1988/89 exposés 700-714 155-181, 1989) on the Godbillon-Vey invariant that the consideration of the Anosov flow appeared essential to me.

Furthermore, I am convinced that T. Walpuski and, in general, any specialist in higher dimensional gauge theory, should have a lot to say about these lines of ideas, provided that these lines carry the value that I glimpse... Although these lines of ideas could have been known and developed by some for a long time... Being still an apprentice, I could therefore be marveling at junk...

There should be a close link between the characteristic invariants of the conformally Anosov flow that we are talking about and the spinorial torsion studied by A. Moreno (see Algebraic Torsion in higher-dimensional contact manifolds, arXiv:1711.01562). Unfortunately, I have not yet had the time to read this document with all the necessary seriousness. It would be very interesting, from my point of view, to have the opinion (or better, the teaching) of A. Moreno (or any other SOBD expert) on this subject.

...bring me to that problem again. In any case, it's still very speculative. But, I think that it may pay to think about the lines of thoughts...

I had some thoughts about these questions some time ago. It is just recently that I saw the link with Anosov flow theory and this brings me back to that problem...

Surely, using "double null foliation" or some generalisation should help answer all questions. The "double null foliation" structure should allow to lift the problem to a conformally Anosov flow ("bi-G_2-structure" (perhaps a tamed one in the sense of D. Joyce)) on a characteristic 7-manifold associated to the given 4-mfld (via Einstein-Yang-Mills equation or a deformation of). By using the structure given by the conformally Anosov flow (characteristic invariants), one should be able to extract necessary informations, given answer to each question and even more.

Thank you very much David Roberts for editing.

At the level of "pure" Mathematic, there are many generalized and extremely interesting Yamabe type operators called conformal operators of which the classical Yamabe operator is the simplest one. These conformal operators help in conceiving generalized relativity theory. Cf mscand.dk/article/download/12120/10136, arxiv.org/pdf/math/0309085, arxiv.org/pdf/1809.06339...