Singular, holonomy-free connections on Riemannian surfaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:57:18Z http://mathoverflow.net/feeds/question/20835 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20835/singular-holonomy-free-connections-on-riemannian-surfaces Singular, holonomy-free connections on Riemannian surfaces? fuzzytron 2010-04-09T13:05:44Z 2010-04-23T21:22:18Z <p>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).</p> <p>Is there a more formal description of this type of construction, and does it have a name? Any pointers to literature?</p> http://mathoverflow.net/questions/20835/singular-holonomy-free-connections-on-riemannian-surfaces/20860#20860 Answer by Orbicular for Singular, holonomy-free connections on Riemannian surfaces? Orbicular 2010-04-09T19:53:35Z 2010-04-09T20:32:07Z <p>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.</p> <p><a href="http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/0807/0807.3317v2.pdf" rel="nofollow">http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/0807/0807.3317v2.pdf</a><br> You might also have a look at<br> <a href="http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/0907/0907.4720v2.pdf" rel="nofollow">http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/0907/0907.4720v2.pdf</a><br> For some general stuff see also<br> <a href="http://www.springerlink.com/content/r57w32lhk6346157/" rel="nofollow">http://www.springerlink.com/content/r57w32lhk6346157/</a></p>