# nLab dual vector bundle

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Idea

Given a vector bundle, its dual is the vector bundle obtained by passing fiber-wise to the dual vector space.

## Examples

• Given a smooth manifold $X$ with tangent bundle $T X$, the dual vector bundle is the cotangent bundle $T^\ast X$.
###### Proposition

Let $X$ be a topological space and let $E_i \overset{p_i}{\to} X$ be two topological vector bundles over $X$, of finite rank of a vector bundle. Then a homomorphism of vector bundles

$f \;\colon\; E_1 \rightarrow E_2$

is equivalently a section of the tensor product of vector bundles of $E_2$ with the dual vector bundle of $E_1$.

$Hom_{Vect(X)}(E_1, E_2) \;\simeq\; \Gamma_X( E_1^\ast \otimes_X E_2 ) \,.$

Moreover, this section is a trivializing section (this example) precisely if the corresponding morphism is an isomorphism.

Last revised on October 2, 2019 at 12:54:51. See the history of this page for a list of all contributions to it.