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The Cartan-Kuranishi theorem guarantees that a PDE or EDS can always be completed, by prolongation, to involution. My question is, and this is quite murky to gauge from the literature, whether Cartan's equivalence method (where prolongation and normalization of essential torsion is a little bit different from the standard prolongation of PDE and EDS) terminates at involution as an easy consequence of Cartan-Kuranishi for equivalence problems of constant type.

Some authors claim that this is so while others say that termination of Cartan's method has never really been proven, notably Bryant, Griffiths and Hsu in the introduction to the paper:

Toward a geometry of differential equations, Geometry, topology, & physics, 1–76, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995. Available from https://publications.ias.edu/node/262 (MR1358612)

EDIT: Clarified that I am talking about constant type equivalence problems.

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You can easily construct examples where the torsion is not of constant type, so the next step in the method is not defined. So I imagine the question is whether you might have termination under the assumption of constant torsion type at each step. But that is somehow unnatural, because we can't really figure out how to enforce such torsion hypotheses directly in terms of a given G-structure, say in coordinates for example, without actually carrying out the prolongations. But even under such strong hypotheses, there is no proof yet, even in the analytic category. I don't see how it could be a consequence of Cartan-Kuranishi. For structures which are flat to high enough order (to acheive involution), there are results of Pierre Molino which imply flatness in the smooth category.

Edit: Sorry, I forgot that there is a paper of Bernard Malgrange, Sur le problème d’équivalence de Cartan, Annales de le Faculté des Sciences de Toulouse, Tome XXVI, no5 (2017), p. 1087-1136, which proves that Cartan's method terminates in finitely many steps, when carried out on an algebraic pseudo-group on a complex algebraic variety. The proof uses Malgrange's proof of involutivity (after sufficiently many prolongations, above the generic point) of analytic exterior differential systems: Bernard Malgrange, Systèmes différentiels involutifs, Panoramas et Synthèses, vol. 19, Société Mathématique de France, 2005, vi+106 pages. I am embarrassed to admit: I have not read either paper.

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    $\begingroup$ There is also a techical point about the definition of constant type. The method of equivalence can normalize the torsion just exactly when the stabilizer of the torsion, at each point of the bundle of the $G$-structure, lies in a fixed conjugacy class of subgroup. So one natural definition of type is the conjugacy class of the stabilizer of the torsion at each point, and then constant type occurs just when torsion normalizes. But the definition in the OP's paper is different, involving the rank of the derivatives of the torsion. Malgrange doesn't need any constant type hypothesis. $\endgroup$
    – Ben McKay
    Commented Feb 28, 2019 at 17:13

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