Cartan's Structure Equations VS Cartan's Method of Equivalence

Question

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.

Answer 1

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.

Answer 2

There is a lot going on in Cartan's method of moving frames, and of Cartan's method of equivalence (or Cartan's test). In a general sense, you seek (Lie) groups G that are in a sense bigger than that of the underlying manifold M, but with the property that you can factor the (Lie) Group by a closed (left-)coset H, with the factor group G\H being isomorphic to M. The classic example is that of the Euclidean group of rigid motions, consisting of a semi-direct product of two factors, one being rigid translations of points (of M), the other being transformations restricted to a fiber over a given point p, i.e. rotations of axes at a fixed point p. If you choose H as the group of rotations at a fixed point, then G\H must be the group of translations of points p of M. It's easy to miss on a first pass through, but the Fiber bundle of all the framings is not an actual vector bundle. However, it's structure is such that the transition maps of coordinate neighbourhoods is fiber-preserving, i.e. splits into a fiber-preserving action and another action that is on the so-called Horizontal space, the coordinates themselves (in most cases). This canonical splitting is one of the reasons for such focus upon the G-Structures and more specifically upon the Principal Bundle. (P-Bundles have the basic property that the Group G, or its effective subgroup, are in some sense isomorphic to the fiber itself, so that G\H is iso M. If H is the set of rigid rotations, that means G is basically rotation plus translation, and G\H is the set of all points p that belong to a manifold called M = G\H. Lie Groups are manifolds in their own right, hence such a factoring must result in a manifold, subject to certain constraints). Such a factoring is called a Quotient Manifold, and it's true there are some pathologies that can occur in such a construct. Olver and Chen and Ivey give time to explaining the kinds of pathological behaviour you might see). If your interests are local, then these pathologies are not your concern.

If you take the case of EDS, i.e. the Pfaffian systems, then the idea as I understand it is that you can write down a bunch of contact forms in terms of the coordinates of a Jet space, typically of much greater dimension than the space of solutions to the EDS. Because the pullback operation commutes and distributes with the exterior derivative, you can use a pullback to take generally defined contact forms to the solution surface, and in so doing they have to vanish on that surface, if they are to represent the solution. A classic example is that in which you find the conditions of differentials that allow a solution to exist, the simplest such condition being the commutation of partial derivatives (if you restrict to certain coordinate systems). The essential ingredient is that if a form is pull backed to a surface upon which it vanishes, then so too its exterior derivative must vanish under the pull back. This means there is both an algebraic Ideal and a differential Ideal, and it means that by the means of exterior differentiation, you eventually reach a point of stabilisation, meaning that every condition underlying what the equivalence means has been found by a certain point of the process of pullback and exterior differentiation.

I can't say I've got the whole story in my head, so I pre-emptively apologise if this is a shortfall in explanation. I do recommend Olver's purple book, Chen, Chern, and Lam's fantastic book on Differential Geometry, and also the most recent version of Ivey et al's book on Cartan for Beginners (NB: I dispute it's for beginners, but once you have mastered both Olver and Chen et al, you'll be in a great position to understand what Ivey et al are getting at). Finally, I'd also point at the book by Sharpe, for it digs deep into Principal Bundles, etc. In short, do Olver plus Chen, then Olver plus Chen and Ivey; then, Olver plus Chen plus Ivey and Sharpe. My meaning is that you'll only get to the destination with a good reading and re-reading of each of those texts, in singular and then in conjunction. It all boils down to how do we understand differentiation, and the fact that we can use linear methods once some kind of differentiation operation is supplied. Personally, I've learned so much more than I expected, about the nature of mathematics and the sheer brilliance of mathematicians (or physicists, etc.) when at the pointy end of their craft.

Olver only says this in one or two spots in his text, but the difference between applying a tangent operation (the push-forward that a differential provides) and a pullback is that pushing forward a vector is only unique if the push forward is f-related, i.e. the function of the push forward generates a one-to-one mapping under push forward. The pullback never suffers from potential non-uniqueness that a push forward does. This means you can use the pullbacks in a much more general fashion (i.e. unrestricted), which clearly simplifies the algebra and the other matters. Pullbacks to the solution space of a system are what provide the overall solution strategy to equivalence problems with ODEs, PDEs, and more generally the Pfaffian EDS.

I don't know if any of the above is a help. Just remember this: those who bothered to write the textbooks, they were trying to find a path for us to follow. Once you can do calculus with exterior derivatives, and with connections in more generality, you'll be good to go on Cartan's Test, and Cartan's Method of Moving (Co-)Frames.