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There have been a number of posts on related questions, such as: Geometric interpretation of Cartan's structure equations, What is the geometric significance of Cartan's structure equations? and Maurer-Cartan structure equation derivation.

While my question is related, it is I hope a bit more specific. I am trying to learn some of Cartan's methods, and getting a little confused, because they all seem to be closely related, so that differentiating between them is difficult for me.

I would like to think of Cartan's structure equations this way. You start with an adapted co-frame and apply d, then express the results in terms of old data (the co-frame), but while respecting the Lie algebra. This gives 2 new things, the connection 1-form, and torsion. We then apply d again, and express the results in terms of "old" data (maybe while respecting an underlying Lie algebra).

I am trying to make sense of this. It seems that this is the same process than the one used in Cartan's equivalence method. Am I right? So torsion is the first "invariant", which could be used to compare 2 different geometric structures locally, while curvature would be the next "invariant".

But what is the relevant EDS perhaps? It seems that we are building something recursively, so I am a little confused. Perhaps we need to go to infinity, to see the whole structure, right? As in, using infinity-structures, like Urs Schreiber seems to be suggesting, in the second link above. Can someone please comment or answer my questions?

Edit: after some thinking, and reading a good chunk of Olver's book "Equivalence, Invariants and Symmetry", here is my current understanding of Cartan's structure equations. Let's say you have a Riemannian manifold $(M,g)$, and let $(\theta^i)$, for $1 \leq i \leq m$, with $m=\dim M$, be a smooth local orthonormal coframe. Applying $d$ to the coframe gives our first set of "invariants" (or perhaps I should write $O(m)$-invariants). That the first set of "invariants" is nothing but the Levi-Civita connection is the meaning of the first structure equations. We then apply $d$ a second time, and get a second set of "invariants". That this second set of invariants can be broken in 2 parts, the first quadratic in the Levi-Civita connection and the second one nothing but the curvature of $g$, is the content of Cartan's second structure equation.

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You are right that structure equations are the result of applying the method of equivalence. You don't start with an adapted coframe. (Some people might say that you do, but that is not quite what the method does.) You start with the bundle of all adapted coframes. You then construct on it the bundle of all adapted pseudoconnections, i.e. coframes on the total space of the first bundle. And so on. As you go, at each step you reveal one more collection of structure equations. You keep going until (1) you do not have a geometric hypothesis at hand which can justify the next order frame adaptation or (2) you reach involution in the sense explained in Robert Gardner's book. Since the entire procedure is invariant under local isomorphisms of G-structures, the resulting structure equations "respect" the Lie algebra of symmetries. The relevant EDS is that of the isomorphisms of the G-structures, as explained in Gardners' Lecture 7.

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  • $\begingroup$ Thank you Ben. That makes things much clearer. I was on the right track, but I had the details wrong. Thanks $\endgroup$ – Malkoun Aug 8 '17 at 15:13
  • $\begingroup$ One question though about the EDS part of your answer. Does the $G$ in the expression $G$-structure get bigger at each step? Or does it remain the same? Or is $G$ the original $G$, the structure group of the adapted co-frames? $\endgroup$ – Malkoun Aug 8 '17 at 15:14
  • $\begingroup$ The G doesn't really get bigger, as the relevant group at each stage is actually a subgroup of the prolongation of the original Lie group G. The tower of bundles has a larger structure group at each step, but that tower is not a G structure. $\endgroup$ – Ben McKay Aug 8 '17 at 15:20

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