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 ${M }$ be a manifold, and let ${V \rightarrow M}$ be a vector bundle. Suppose given a connection ${\nabla}$ on ${V}$. This determines, and is equivalent to, the data of parallel transport along each (smooth) curve ${\gamma: [0, 1] \rightarrow M}$. In other words, for each such ${\gamma}$, one gets an isomorphism of vector spaces

$\displaystyle T_{\gamma}: V_{\gamma(0)} \simeq V_{\gamma(1)}$

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 ${p \in M}$, we get a map

$\displaystyle \Omega_p M \rightarrow \mathrm{GL}( V_p)$

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