# Non-flat connection on trivial bundle?

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?

• Yes, they do exist. But you have to exclude 1-d base. – Moishe Kohan Apr 2 at 23:42
• Is it correct that on the trivial complex line bundles, no flat-connection exists? This is because then applying Weil-Chern theory we get that the first chern class cannot be trivial and thus the bundle cannot be trivial which is a contradiction? – jojo Apr 3 at 6:41

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.