Curvature form of Cartan connections

[Ravid]

Why is the definition of the curvature form for a Cartan connection the correct definition, and what does it actually tell you? I've read that it "measures the failure of the structural equation" but I suppose I don't really understand the structural equation (of a Maurer-Cartan form) anyway, and also how exactly does it measure the failure (obviously it's zero if it holds, but apart from that?).

If it would be necessary to go via an explanation of the Riemannian curvature tensor, I would also like to understand that (bearing in mind I have studied very little Riemannian geometry).

 

[zhentil]

Do you know anything about the Levi-Civita connection? The fundamental lemma of Riemannian geometry states that there is a unique affine connection which is torsion-free and compatible with the metric (in a sense, the "natural" connection). The structural equations are defined using the Levi-Civita connection, and as such, a curved connection will fail to satisfy them.

If most of this paragraph makes sense to you, and you have questions, let me know. I would imagine Cartan connections would be very difficult to pick up with no prior exposure to Riemannian geometry.

 

[Ravid]

Do you know anything about the Levi-Civita connection? The fundamental lemma of Riemannian geometry states that there is a unique affine connection which is torsion-free and compatible with the metric (in a sense, the "natural" connection). The structural equations are defined using the Levi-Civita connection, and as such, a curved connection will fail to satisfy them.

If most of this paragraph makes sense to you, and you have questions, let me know. I would imagine Cartan connections would be very difficult to pick up with no prior exposure to Riemannian geometry.

I 'know about' the Levi-Civita connection in that I know what it is and the fundamental lemma, but I have not studied them in any particular depth. I learned about Cartan connections via Klein geometry, i.e. homogeneous spaces and Lie groups/algebras, and a principal bundle with a Cartan connection being a generalisation of the former.

How are the structural equations defined with the Levi-Civita connection? Do you mean the equations of the fundamental lemma? Just to clarify, the equation (on a Lie group) I was referring to was

$$d\omega_G+\frac{1}{2}[\omega_G\wedge\omega_G]=0$$

where $$\omega_G$$ is the Maurer-Cartan form of the Lie group. Since my first post, I have seen an alternative definition of curvature of an Ehresmann connection in Kobayashi/Nomizu as $$d\omega$$ composed with the horizontal projection, but I am not sure how this relates to Cartan connections.

 

[zhentil]

The context I have seen this is in which the connection 1-forms are defined using the Levi-Civita connection, and the curvature 2-forms are defined using the connection 1-forms. In this case, the failure of the structural equation you refer to indicates that the curvature on the manifold is non-zero.

It is my understanding that you can define all of these things analogously for any connection. However, in the absence of a metric, I think you have to define the curvature two-forms in terms of the structural equation you wrote. In that sense, it's circular (the curvature measure the failure of the structural equation by definition).

To answer your original question more succinctly, I believe they've defined it this way to be consistent with the Levi-Civita connection (in which case the curvature of the connection actually measures the curvature of the manifold).

 

[Ravid]

Yes, the definition I am talking about is in terms of the structural equation, which is why I have a hard time understanding where it comes from.

How are the connection forms defined by the Levi-Civita connection, and how does the curvature form agree with the Riemannian curvature tensor (I assume you mean this when you refer to the curvature of the manifold)? Or is this an involved question for which I'm just going to have to read a book on Riemannian geometry (any recommendations)? Thanks for your help btw.

 

[zhentil]

You probably already know this part, but the connection 1-forms $$\omega_i^j$$ for a given connection are defined as follows: let $${E_i}$$ be a local frame, and $${\phi^i}$$ the dual coframe. Then we get
$$\nabla_X E_i = \omega_i^j(X) E_j$$.

You can define this for any connection. Now the curvature two forms are defined as follows: compute the connection 1-forms using the Levi-Civita connection, and define the two forms to be
$$\Omega_i^j = \frac{1}{2} R_{kli}^j \phi^k \wedge \phi^l$$.
Here we're using the curvature tensor.

In this context, Cartan's second structural equation says that
$$\Omega_i^j = d\omega_i^j - \omega_i^k \wedge \omega_k^j$$.

This might look slightly different from how you've seen it due to sign ambiguity in the curvature tensor, and ambiguity in the way different authors define the wedge product. Authors in Riemannian geometry tend to make it as difficult as possible to read each others' books.

If you want to learn more, I'd suggest do Carmo (Riemannian Geometry) or Lee (Riemannian Manifolds).

 

[Ravid]

I am used to a co-ordinate free approach to the subject, and hence I didn't really understand much of what you just wrote. Never mind, though, I figured out the relationships myself by studying Kobayashi/Nomizu. Thanks anyway, especially for the recommandations.