Curvature of a Cartan geometry

See @sharpe2000differential page 184.

Definition

Given a Cartan geometry $(P,A)$ modeled on $(G,H)$, the $\mathfrak{g}$-valued 2-form

is called the curvature of the geometry.

$\blacksquare$

The structural equation of the Maurer-Cartan form of a Lie group is telling to us that a Klein geometry is flat.

On the other hand, the measure of how much $\Omega$ is not in $\mathfrak{h}$ is called the torsion of a Cartan geometry.

Interpretation

Why is this called curvature? Only because it reminds the Cartan's second structural equation?

Idea to be developed further:

The Euclidean plane is flat, in the following sense. The Levi-Civita connection has a connection 1-form $\Theta$ (upon fixing a frame $e=\{e_1,e_2\}$) satisfying:

The Cartan's second structural equations show that the Riemann curvature tensor (and then the Gaussian curvature) is closely related to the 2-form $\Omega=d\Theta-\Theta\wedge\Theta$, so we conclude that the curvature is zero.

This flat connection induces a principal associated connection $\tilde{\Theta}$ on the orthogonal frame bundle, which, at the end of the day, is the same as the Klein geometry $(E(2),O(2))$. The associated principal connection, I guess, is also flat. I am not able to prove it, yet, but I don't mind for the moment (see here for a sketch). In any case, we again have

This principal connection is a $\mathfrak{o}(2)$-valued 1-form which is, in some sense, part of the Cartan connection (an $\mathfrak{e}(2)$-valued 1-form). In this case, the Cartan connection is the Maurer-Cartan form of $E(2)$ $\Theta_{E(2)}$, which of course satisfies

This last identity is true for any Lie group $G$. But what about (0) and (1)? I mean, in the general context of a Klein geometry $(G,H)$ we have that it is flat, in the sense that

If the Klein geometry is reductive (as it happens with $(E(2),O(2))$) we have $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$, and then, to my knowledge, $\Theta_G$ decomposes in an $\mathfrak{h}$-valued 1-form $\tilde{\Theta}$ and an $\mathfrak{m}$-valued 1-form. The form $\tilde{\Theta}$ provides a principal connection on the principal bundle $G\to G/H$. This principal bundle is a kind of frame bundle for $G/H$ so we can consider a moving frame

and the connection 1-form $\Theta:=e^*(\tilde{\Theta})$ . To try to end this, read first torsion of a Cartan geometry#Interpretation. I think the key idea is the update.

Old stuff

I suspect that the curvature of a Cartan geometry is related to the curvature of a connection, and so to the curvature of a distribution. And in this sense, it represents "the deviation of a parallel transported vector along a closed loop to belong to the horizontal distribution". The horizontal distribution would be, I think, an appropriate lifting of the tangent space of $M$... I have to think it more...

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es