# 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.

• there's an extension to the fundamental theorem of Riemannian geometry that states there's a unique metric connection for any given torsion; this highly depends on the ambient structure of your manifold: eg there's no such result for symplectic connections (IIRC); also note that if you force curvature to be zero, you end up with teleparallel gravity, which has basically the same degrees of freedom as ordinary general relativity recast into a different geometric framework Jan 21, 2014 at 16:53
• The word "invariants" should be precised. If it is "invariant" relatively to diffeomorphisms, one thinks about scalar quantities. For instance, from the curvature tensor $R_{\mu\nu\alpha\beta}$, you have scalar quantities $R, R^2, R_{\mu\nu}R^{\mu\nu}, R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$. Here $R_{\mu\nu}$ is the Ricci tensor, and $R$ is the scalar curvature, which are obtained from contractions from the curvature tensor. Jan 21, 2014 at 18:25

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.