I’d also like to describe some more quantitative versions of what curvature means. We saw in the previous post that curvature measures the sense in which a connection fails to be a local system, or in other words look locally like the standard connection on a trivial bundle. (This is perhaps part of the motivation of the use of curvature to construct de Rham representatives of the characteristic classes of a vector bundle, cf. Chern-Weil theory.)

1. Curvature as deviation from flatness

If you work in coordinates, it’s not immediately clear how to say to what extent a connection looks like a trivial connection, because the trivial connection can look very different if you change the frame. Curvature has the property of being tensorial and not depending on a given choice of frame.

But another way to say this is to try to express the connection in the best possible choice of coordinates, and see whether it looks like the standard one. Namely, let ${V \rightarrow M}$ be a vector bundle with connection ${\nabla}$. Choose a neighborhood of ${p \in M}$ that looks like ${\mathbb{R}^n}$ with ${p}$ at the point ${0}$. Since everything we do is local, we may just assume

$\displaystyle M = \mathbb{R}^n, p = 0.$

This gives us a particularly nice frame for ${V}$. Choose a basis ${e_1, \dots, e_m}$ for ${V_p}$ and then define sections ${E_1, \dots, E_m}$ of ${V}$ on ${\mathbb{R}^n}$ at a point ${q}$ by parallel translation along the (euclidean!) line from ${p=0}$ to ${q}$. Then ${E_1, \dots, E_m}$ is a global frame for ${V}$ and is, by construction, parallel along each straight line through the origin. One therefore has:

$\displaystyle \nabla E_i ( p) = 0, \quad 1 \leq i \leq m.$

Using this choice of frame, we get an identification of ${V}$ with the trivial bundle ${\mathbb{R}^m}$. In particular, we can write the connection ${\nabla: V \rightarrow \Omega^1(V)}$ in the form

$\displaystyle \nabla = d + \omega,$ (more…)

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…)
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