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 be a vector bundle with connection . Choose a neighborhood of that looks like with at the point . Since everything we do is local, we may just assume

This gives us a particularly nice frame for . Choose a basis for and then define sections of on at a point by parallel translation along the (euclidean!) line from to . Then is a global frame for and is, by construction, parallel along each straight line through the origin. One therefore has:

Using this choice of frame, we get an identification of with the trivial bundle . In particular, we can write the connection in the form