Sunday, 28 December 2008

dg.differential geometry - Singular, holonomy-free connections on Riemannian surfaces?

Consider principal connections on the frame bundle of a compact, connected, smooth, orientable Riemannian surface embedded in mathbbR3. 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 S2 with a single point p removed. If we now compactify D to get S2 again, then we have a connection tildeomega 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?

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