Over the past couple of days I have been brushing up on introductory differential geometry. I’ve blogged about this subject a fair bit in the past, but I’ve never really had a good feel for it. I’d therefore like to make this post, and the next, a “big picture” one, rather than focusing on the technical details.

**1. Curvature of a connection**

Let be a manifold, and let be a vector bundle. Suppose given a connection on . This determines, and is equivalent to, the data of parallel transport along each (smooth) curve . In other words, for each such , one gets an isomorphism of vector spaces

with certain nice properties: for example, given a concatenation of two smooth curves, the parallel transport behaves transitively. Moreover, a homotopy of curves induces a homotopy of the parallel transport operators.

In particular, if we fix a point , we get a map

that sends a loop at to the induced automorphism of given by parallel transport along it. (Here we’ll want to take to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.) (more…)