nLab dual vector bundle

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.

Related concepts

• direct sum of vector bundles

• tensor product of vector bundles, external tensor product of vector bundles

• tensor category

Related concepts

• Max Karoubi, §4.8 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226 Springer (1978) [pdf, doi:10.1007%2F978-3-540-79890-3]