Consider principal connections on the frame bundle of a compact, connected, smooth, orientable Riemannian surface embedded in $\mathbb{R}^3$. On a disk $D$, it is apparent that you can construct a connection $\omega$ with zero holonomy everywhere: for instance, map $D$ to the plane and use Euclidean translation to induce parallel transport. Further, suppose that $D$ is actually an embedding of $S^2$ with a single point $p$ removed. If we now compactify $D$ to get $S^2$ again, then we have a connection $\tilde{\omega}$ on the sphere which is well-defined for any loop that does not contain $p$, and exhibits zero holonomy around any such loop. In a similar way, we can construct a connection with a single "singular" point on a surface of any genus by removing a set of loops that generate the fundamental group rather than just a single point (though we can no longer rely on Euclidean translation to provide the connection). And more generally, we can imagine connections with zero holonomy except at a number of singularities (map a punctured disk to the plane, say).

Is there a more formal description of this type of construction, and does it have a name? Any pointers to literature?

  • $\begingroup$ If you remove a small disk from a higher-genus Riemann surface, the result can be immersed in the plane, and the trivial connection pulled back. $\endgroup$ – S. Carnahan Apr 9 '10 at 18:03

I think this concerns the moduli space of flat connections on Riemann surfaces with punctures (aka holes). If there is at least one puncture \pi_1 of the Riemann surface is a free group and the moduli space in question reduces to the moduli space of (G-valued) representations of the free group (in some letters). Hence you need to study so-called character varieties, see e.g.

You might also have a look at
For some general stuff see also


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