Example of a flat manifold with non-zero (global) holonomy group.

I'm having some trouble coming to terms with there being non-zero global holonomy but zero local holonomy. Is there an easy to visualize example of a manifold whose curvature is zero but has non-zero Riemann holonomy group?

Or maybe a flat vector bundle on $S^1 \times S^1$ with non-trivial holonomy, which is easy to visualize?

Solution 1:

Ok, I decided to make this an answer.

Take the flat Möbius band, which is the only non-trivial line bundle $E \rightarrow S^1$. This is a flat manifold having non-trivial holonomy. In fact, if you start with a non-zero vector $v \in E_x$ and go around $S^1$ once, returning to $x$, you get $-v$. Do it again, and you get back to $v$. This shows that the holonomy group is $\mathbb{Z}/2\mathbb{Z}$.