Invariants of Connection Form I am somewhat going out "on a limb" here, since I am much more grounded in the physics side of things than I am in mathematics. Nonetheless, I am wondering if someone is able to comment on the following : given a connection form defined on a differential manifold, one can derive a tensorial invariant of this, the curvature form. If there is a soldering form present, one can also define torsion as an invariant of the connection form. 
My question now is - are there any invariants other than curvature and torsion that can be derived from a connection on a differentiable manifold? My interest here is again motivated by physics, specifically gravitation theory; here using the torsion-free Levi-Civita connection will give us GR, and allowing torsion into the picture will give us Einstein-Cartan gravity. To me, this raises the question if - mathematically speaking - there are other options based on other connection invariants that I am not myself aware of.
I apologise if the question turns out to be meaningless - I am not a mathematician, and was not immediately able to find conclusive answers to this through a Google search.
 A: 
Are there any other invariants other than curvature or torsion that can be derived from a connection on a differentiable manifold?

The $n$th Chern number $c_n$ of a manifold is a topological invariant. For example,
$$c_1=\frac{i}{2\pi}\int_{M} \, \mathrm{Tr}[\mathcal{R}]$$
where $\mathcal{R}$ is the curvature $2$-form. Technically, it is another quantity, even though it is derived from the curvature. Although, the fact that $c_1$ is a topological and not just geometrical invariant is not immediately obvious from the expression. If I interpret your 'invariance' as invariance under $\mathrm{Diff}(M)$, then scalars constructed from tensors through contractions such as,$^{\dagger}$
$$R^a_{bcd}R^{b}_{a}R^{cd}$$
are invariants. Another example, interpreting 'invariance' as topological invariance, is the Chern-Simons term (used in topological quantum field theory),
$$S=\frac{k}{4\pi}\int_M \, \mathrm{Tr}\left[ A \wedge dA + A\wedge A \wedge A\right]$$
which meets your requirements as $A$ is both a gauge field, and a connection on a bundle of a manifold. There are many other examples of different types of invariants; your question is quite broad and ill-defined.

$\dagger$ The reason we check curvature scalars to see whether our space truly possesses a singularity is precisely because they are invariant under coordinate transformations, and we know inadequate coordinates can lead to the metric being singular when the space is not.
