A principal $G$-bundle has a cross section iff it is trivial (e.g. Husemoller's Fibre Bundles, 3rd ed., 8.3 in chapter 4).


A principal $G$-bundle is in particular a fiber bundle with fiber $G$.


My question: does there exist a group $G$ and a **non-trivial** principal $G$-bundle $p:E\rightarrow B$ that *does* have a cross section when considered as a mere fiber-bundle?


If so, I would be glad to see a simple example. Thanks!



Cross sections in bundles and principal G-bundles
Let $p\colon E \to B$ be a principal $G$-bundle and $s\colon B \to E$ a global section. Then the map $F\colon B\times G \to E$, $F(x,g) = s(x)\cdot g$ is a global trivialization of $E$. Here the dot denotes the right action of the group $G$. The map $F$ is surjective, because the action is transitive on fibres, and it is injective because the action is free. Continuity/smoothness is the same as the continuity/smoothness of your section $s$ and the action of the group $G$.


There's no special requierement in the definition of a section of a principal $G$-bundle, it's still the section of the underlying fibre bundle. But once you have a global section, you just use the $G$-action to trivialize the bundle.


