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Parabolic Geometries I by Cap and Slovak outlines an equivalence between regular normal Cartan geometries and regular infinitesimal flag structures associated to a $|k|$-graded parabolic geometry $(G,P)$. In the case of $|1|$-graded parabolic geometries like projective and conformal geometry, the interpretation of the Cartan bundle is fairly clear: the points in the fiber are $2$-jets of approximations of the manifold by the homogeneous model space. Alternately, we can think of the Cartan bundle as the extension of a certain frame bundle with structure group $G_0=P/P_+$ by the collection of torsion-free principal connections at each point. But what is the interpretation of the Cartan bundle for larger $k$? For example, how can we interpret the bundle for CR or contact structures?

I know that if the prolongation procedure of extending by torsion-free connections does not stabilize after the first step, then it will never stabilize. Is there some other way to make sense of the series of quotients $$\mathcal{G}\to \mathcal{G}/P_k\to \mathcal{G}/P_{k-1}\to \cdots \to \mathcal{G}/P=M?$$

Given an infinitesimal flag on a $G_0$ frame bundle $E\to M$, Cap/Slovak construct a Cartan bundle as $E\times_{G_0} P$, but I'm not sure how to interpret that geometrically.

subrosar
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I am not sure what you are after. Even in the $|1|$-graded case, the interpretation via 2-jets is not always available. There are $|1|$-graded parabolic geometries that have non-trivial torsion (and even ones for which the torsion is the only obstruction to local isomorphism to the homogeneous model). Even for those you have to start with compatible connections with torsion and an interpretation in jet language needs more exotic concepts like semi-holonomic jets.

I am also not sure, whether a "geometric understanding" (whatever this means) of the Cartan bundle is particularly important. Classical literature often focusses on smart constructions of the Cartan bundle, but you have to keep in mind that locally all bundles in question are trivial, so locally these are just elaborate constructions of a trivial bundle. In the classical approachest the role of these descriptions ususually mainly is to later construct the Cartan connection using "tautological" forms. The construction of the Cartan bundle as $E\times_{G_0}P$ that you refer to follows an opposite approach: It is clear that any extension of $E$ to structure group $P$ has to be isomorphic to this, since the group $P_+$ for which $G_0=P/P_+$ is contractible. Starting from this observation, the remaining information needed for the construction of a Cartan connection can be deduced from purely algebraic properties of the homogeneous model.

Anyway, any interpretation that is available in the $|1|$-graded case admits a generalization to the $|k|$-graded cases, but there are substantial technical complications to work this out. My preferred "geometric" interpretation of the Cartan bundle in the $|1|$-graded case is as the pullback of the bundle of compatible connections with normalized torsion to the principal $G_0$-bundle that defines the underlying $G_0$-structure. (This also makes the relation to the prolongation of the Lie algebra $\mathfrak{g_0}\subset\mathfrak{gl}(\mathfrak g_{-1})$ very clear.) The analog of this for $|k|$-graded geometries has to use Weyl structures, which consist of a compatible connection, a splitting of the filtration of the tangent bundle and an additional tensor. For jet interpretations, you have to go to the concept of weighted jets (where the "order" of a single derivative depends on the direction in which you differentiate) and for non-trivial torsion (which becomes inreasingly important in higher graded cases) you need some concept of semi-holonomic weighted jets.

Definitely only parts of these ideas have been worked out explicitly, but the necessary ingredients are there.

Andreas Cap
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