Haelfix said:
Cartan connections is something one learns in a graduate class in differential geometry, and be sure its formally identical to the usual way of studying manifolds when restricted to say the known cases people are interested in (say GR manifolds). In fact his student Ehresman simplified this line of thought into the usual connections on principle bundles point of view so often used nowdays. This is straight out of Spivak.
I looked at Derek Wise's paper and indeed that's exactly what it is, he even makes it explicit, when he constructs MM gravity from the Palatini formulation. Lo and behold, we have the same field content between the two. They are formally isomorphic...
Spivak "Comprehensive Introduction..." is an interesting book.
https://www.amazon.com/dp/0914098713/?tag=pfamazon01-20
great cover art.
dropped out of academia and started his own press called "Publish or Perish"
I guess you would be talking about volume two of the five-volume set.
still not sure we are communicating, however.
Elie Cartan invented a lot of stuff (some I referred to earlier) including (with others like his student Ehresmann) the usual idea of connection. But what I said was I'm not talking about that other stuff.
Cartan geometry the phrase in Derek Wise title is different from just doing conventional diffy geom using apparatus invented by Cartan. Cartan called it "generalized geometry", Wise and others call it Cartan geometry.
If you want to learn about it, and see if it has "new physics" potential, then I don't think you can rely on volume 2 of Spivak. I think you may need Sharpe's textbook. Garrett Lisi, who sometimes posts here, has Sharpe. Also Derek Wise referred to Sharpe. Also by carefully reading Wise' paper you can probably find out a lot.
I haven't read Spivak volume 2, but from what I see about it, I wouldn't assume it would give an intro to Cartan ("generalized") geometry. Could be wrong of course, perhaps I'll have a look at the math department library
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PS, Haelfix here is an exerpt from a reviewer's summary of volume 2
"...Kozul's concept of the connection and this is introduced in Chapter 6. First, note that the connection here is one of the versions of the introduced by Kozul as a map of pairs of vector fields to a vector field. Another useful version, not studied in volume II, is to consider the connection as a Hessian which maps any smooth function to a bilinear form on the tangent space. Second, note that Chapter 6 is usually the starting point for most treatments of curvature in differential geometry (e.g Do Carmo's "Riemannian Geometry"). Without the motivating material from the previous chapters, it would be difficult to understand the need for(or the point of) Kozul's connection.
Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7. The author is careful to intuitively motivate Cartan's deviation from Euclidean concept as represented in the structure equations. Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established.
Building on the orthonormal frames from the previous chapter, Spivak now considers Ehresmann's theory of connections in principal bundles in Chapter 8. The main results here introduce the Ehresmann connection on the frame bundle, and gives the Kozul connection as a Lie derivative, thought of as the Cartan connection obtained from the Ehresmann connection..."
