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.