nLab trivial fiber bundle

Idea

The trivial fiber bundle over $B$ with fibre $F$ is simply the Cartesian product $B \times F$, equipped with the projection map to $B$ to make it a fiber bundle over $B$

$$\array{ B \times F \\ {}^{\mathllap{pr_1}}\downarrow \\ B }$$

A fiber bundle which not the trivial bundle but is isomorphic to the trivial bundle is called trivializable and the chosen isomorphism is called the corresponding trivialization.

Examples

• If $F$ is a vector space, then the trivial $F$-fiber bundle is a trivial vector bundle, etc.

• If a field bundle is trivial with fiber $F$, then the corresponding field theory is called a sigma model with target space $F$.