I’d like to finish the series I started a while back on Chern-Weil theory (and then get back to exponential sums).

So, in the discussion of the Cartan formalism a few days back, we showed that given a vector bundle with a connection on a smooth manifold, we can associate with it a curvature form, which is an -valued 2-form; this is a generalization of the Riemann curvature tensor (as some computations that I don’t feel like posting here will show). In the case of a *line *bundle, we saw that since was canonically trivialized, we could interpret the curvature form as a plain old 2-form, and in fact it turned out to be a representative — in de Rham cohomology — of the first Chern class of the line bundle. Now we want to see what to do for a vector bundle, where there are going to be a whole bunch of Chern classes.

For a general vector bundle, the curvature (of a connection) will not in itself be a form, but rather a differential form with coefficients in , which is generally not a trivial bundle. In order to get a differential form from this, we shall have to apply an invariant polynomial. In this post, I’ll describe the proof that one indeed gets well-defined characteristic classes (that are actually independent of the connection), and that they coincide with the usually defined topological Chern classes. (more…)