Non-flat connection on trivial bundle?

Question

From what I have read it seems like there exists non-flat connections on trivial vector/principal bundles. However I can't find any notes on it or examples.

Can anyone confirm that such connections exist and perhaps provide an example?

Answer 1

You are confusing the notion of flatness for connections with characteristic classes. A connection is flat if and only if it has (identically) zero curvature.

Thus, to construct a non-flat connection on a trivial bundle $p: E\to M$, it suffices to do so locally: Construct a non-flat connection form with compact support on $p^{-1}(U)$ for some small open subset in the base $M$ and then extend by zero to the rest of the manifold (this makes no sense for general bundles but is well-defined for the trivial bundle). I will do so for complex vector bundles (of rank $\ge 1$) since it appears that this is what you are interested in. Furthermore, it suffices to do so for complex line bundles $p: L\to M$ (since our trivial vector bundle is a direct sum of trivial line bundles). Then, a connection can be identified with a complex-valued 1-form $\omega$ on $M$ (the actual connection will equal $d+\omega$). The curvature of the connection equals $$ d\omega+ \omega\wedge \omega=d\omega$$ in our situation. Thus, all what you need is to construct a compactly supported complex-valued (actually, even real-valued is enough!) 1-form $\omega$ on the open ball $B$ in $R^n$, $n\ge 2$, such that $d\omega\ne 0$. For instance, take a smooth compactly supported function $\eta(x)$ on $B$ which is not identically zero and let $\omega= \eta(x) dx_1$. I will leave it to you to verify that $d\omega$ is not identically zero.

Answer 2

They do exist (provided that the base has dimension bigger than one) and any non-flat connection provides an example.They point here is that curvature is a local invariant of a connection, while triviality of a bundle is a global condition. So whenever you have an example of a connection and a point in which the curvature of the connection is non-trivial, then the bundle is trivial on a neighbourhood of that point, an the restriction of the connection to that trivial bundle still is non-flat.

Implications can only be obtained in the other direction, say that certain bundles do not admit (global) flat connections.