# nLab trivial vector bundle

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Idea

For $X$ a suitable space then a vector bundle over $X$ is called trivial if there is no twist in how the fibers vary over it.

## Definition

For $X$ a topological space, then a topological vector bundle $E \to X$ over a topological field $k$ is called trivial if its total space is the product topological space

$E = X \times k^n \overset{pr_1}{\longrightarrow} X$

with the topological vector space $k^n$, for some $n \in \mathbb{N}$. For $n = 1$, one also speaks of a trivial line bundle.

An isomorphism of vector bundles over $X$ of the form

$E \longrightarrow X \times \mathbb{R}^n$

is called a trivialization of $E$. If $E$ admits such an isomorphis, then it is called a trivializable vector bundle.

Last revised on July 22, 2017 at 09:53:58. See the history of this page for a list of all contributions to it.