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Question: What are some characteristic properties of constant curvature or flat spaces?

Some motivation and context:

There are a lot of things on manifolds that are called geometric structures, like pseudo-Riemannian geometry or conformal geometry. All of the examples I know (minus symplectic geometry, which is "not geometry") fit into the framework of a Cartan geometry, but there are more general formulations around that I am not familiar with.

Briefly, a Cartan geometry modelled on a homogenous space $G/P$ is a principal bundle $P\to E\to M$ together with a parallelism $\omega: TE\to\mathfrak g$ as well as some compatibility conditions.

Here one can define a curvature as a $\mathfrak g$-valued two form on $E$: $$\kappa_x[v,w]= d\omega_x(v,w)-[\omega_x(v),\omega_x(w)]_\mathfrak{g}.$$ With the parallelism $\omega$ this is recast to be a function $E\to \mathrm{Lin}(\Lambda^2 \mathfrak{g},\mathfrak g)$. It is natural to say that the space has constant curvature if the curvature is constant, or that the space is flat if the curvature is zero.

A remark about this is that $\mathrm{Lin}(\Lambda^2 \mathfrak g, \mathfrak g)$ carries a representation of $P$ from the adjoint action. The curvature as defined above is equivariant wrt this representation, ie $\kappa(x\cdot p)=\rho(p^{-1})\cdot\kappa(x)$. If the function $\kappa$ is to be constant it must take on values in trivial sub-representations in $\mathrm{Lin}(\Lambda^2\mathfrak{g},\mathfrak{g})$, which with our definition means that there might be geometries with no non-flat constant curvature realisations.

I'm interested in trying to recover "synthetic" properties of constant curvature spaces from this definition in order to see that it is a good definition. One thing that is clear is that we will have enough Killing fields (fields that flow via automorphisms) to explore a neighbourhood of any point in the manifold, but this is a property that locally homogenous manifolds also have. Beyond this we also have the "maximum possible" amount of local isotropies at any point.

But I don't really have much more intuition available with regards to what I should expect for a constant curvature space.

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    $\begingroup$ What do you mean for the curvature of a Cartan geometry to be constant? For a general Cartan geometry $(\mathcal G, \omega)$ of type $(G, P)$, the curvature is a section of $\bigwedge^2 T^* \mathcal G \otimes \mathfrak g$. In the special case of Riemannian geometry, we can extract from this section the usual curvature tensor $R_{ab}{}^c{}_d \in \Gamma(\bigwedge^2 T^*M \otimes \operatorname{End}(TM))$ and contract twice (once with the metric) to produce the scalar curvature. But for some Cartan geometries one can't mimic this process to produce from the curvature an unweighted scalar function. $\endgroup$ – Travis Willse Mar 14 at 0:18
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    $\begingroup$ I second Travis' request. For instance, suppose that we have conformal or projective structures. I understand what flatness means but constant curvature is utterly unclear to me in these cases. $\endgroup$ – Moishe Kohan Mar 14 at 0:34
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    $\begingroup$ Even in Riemannian geometry, there are different notions of "constant curvature", at least you have constant sectional curvature (which characterizes space-forms), covariantly constant curvature (which characterizes locally symmetric spaces) and constant scalar curvature (which probably is to weak to count as a condition of constant curvature). The condition of covariantly constant curvature generalizes to all geometric structures admitting an appropriate canonical connection, for the other conditions, this is less clear. $\endgroup$ – Andreas Cap Mar 14 at 9:30
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    $\begingroup$ I agree that "constant sectional curvature" is the most common notion of "constant curvature", but it is an intricate notion (depending a lot on pculiarities of Riemannian geometry). Your idea to require constant curvature function does not work out, since $P$-equivariancy contradicts being constant unless there is a trivial representation contained in $Lin(\Lambda^2\mathfrak g,\mathfrak g)$. Flatness on the other hand, is easy to characterize for Cartan geometries, it just means local isomorphism to the homogeneous model. $\endgroup$ – Andreas Cap Mar 14 at 10:52
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    $\begingroup$ That's true, but they don't exist for conformal and projective structures and, more generally, for parabolic geometries. $\endgroup$ – Andreas Cap Mar 14 at 12:55

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