Real line bundle ($L\to X$)

A real line bundle is a rank-one real vector bundle. Over a paracompact Hausdorff space, or over a smooth manifold in the smooth category, it is orientable exactly when it is a trivial vector bundle.

To prove the nontrivial implication, choose a fiber metric. An orientation of a vector bundle selects one of the two unit vectors in each fiber. These choices agree continuously in oriented local frames, giving a global nowhere-zero section. The map $(x,t)\mapsto ts(x)$ is then a fiber-linear product isomorphism. Conversely, a trivial vector bundle has the positive standard basis as a continuous orientation of a vector bundle.

The Möbius line bundle over a circle is the basic counterexample to triviality: one circuit reverses the fiber orientation, so no global positive unit section of a vector bundle can be chosen.

The Möbius line bundle over the circle iswith projection to $t$ modulo $2\pi$. It is a smooth rank-one vector bundle. A section is represented by a function $s$ satisfying $s(t+2\pi )=-s(t)$, so the intermediate value theorem forces every continuous section to vanish somewhere. Thus it has no nowhere-zero section and is not isomorphic to the trivial vector bundle of rank one.

Real line bundles over a paracompact space are classified up to isomorphism by their first Stiefel–Whitney class in $H^1(X;{\mathbb{F}}_2)$. Tensor product corresponds to addition of classes.

For a real line bundle, the mod-two Euler class is its top Stiefel–Whitney class, namely $w_1(L)\in H^1(X;{\mathbb{F}}_2)$. Its evaluation on a closed curve records whether parallel transport reverses the fiber orientation.