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It's always nice to point out the structural similarieties between (semi-)Riemannian geometry and gauge field theories alla Classical yang Mills theories. Nevertheless, I feel the relation between the gauge group and the gauge bosons "$A_{\mu}\times T^i_{\ j}$" are much simpler than the relation betreen the diffeomorphisms/isometries and the Christoffel symbols $\Gamma_{\mu\nu}^\rho$. At the very least, at first glance, I don't see the $SO(3)$ matrix lingering in $\Gamma$. I don't really know the technicalities of the Erlangen program and I specifically don't know how that "generating a manifold from a single point using the isometry group" (as I understand it) really works.

I'd like to know if it is really directly possible to view both, that is general relativity and the classical gauge theories, just as special cases of the general Cartan connection framework, or if there is a fundamental difficulty to (I guess) general relativty which makes this not work. Especially if the answer is no, then what is it that makes the metric theory so different? (At first I'd guess it's the fact that the GR manifold looks different in each patch due to $g$, but $A$ can also be difficult and both are related to what else is in space. So I suppose it's the interplay between the isometry group and the bigger diffeomorphism group.)

Nikolaj-K
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The major problem with this approach is that the gauge theory action is quadratic in the curvature, while the general relativity action is linear in the curvature. Further, the applications of gauge theory are all in the highly quantum limit, while GR is in the classical limit.

Still, you could cover gauge theory by starting with GR and doing a Kaluza Klein compactification on a symmetrical space. This gives rise to gauge theory in the low energy limit, but you need a stabilization mechanism for the internal dimensions. This was one of the paths by which gauge theory was understood to be natural, and I think it first appears (as an exercise) in unpublished lecture notes of deWitt in the 1960s.

The technical differences between an action quadratic in curvature and an action linear in curvature is big enough to make it natural to separate the two subjects in my opinion. It is also true that quantum gravity calculations are harder by orders of magnitude than gauge theory, although the two are unified somewhat in string theory.

Ron Maimon
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I too have worried about making that relationship more concrete. I wrote a paper 10 years ago that makes the parallel very explicit. Maybe it will help. See: http://uk.arxiv.org/abs/hep-th/0205250 published in JHEP.

Mario Serna
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You could certainly cover both GR and classical (Yang-Mills) gauge theory in a one semester course. Both theories are expressed in the language of modern differential geometry: manifolds, bundles, tensors & forms, metrics, connections, and curvature. Likewise, both theories have their dynamics specified by actions.

There are some crucial physical differences though. First, as Ron mentions, the Einstein-Hilbert action for GR is linear in the curvature. Second, there is not AFAIK, anything like the metric tensor (or the frame field, if you prefer) in Yang-Mills style gauge theory. In Yang-Mills, the connection is the basic variable. In GR, there's a connection but it's derived from a metric. Thirdly, gauge theories have local observables (e.g., the curvature), whereas in GR, treating diffeomorphisms as gauge symmetries basically forbids local observables.

The mathematical machinery is quite similar -- these are both classical field theories -- but the physical content is rather different.

user1504
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