Dual Bundle

Let $\xi$ be a real $k$-plane bundle and $\eta$ be a real $l$-plane bundle over base $B$.

Definition of Hom bundle

A vector bundle $Hom(\xi ,\eta )$ is a vector bundle with fiber $Hom({\mathbb{R}}^k,{\mathbb{R}}^l)$, and total space $EHom(\xi ,\eta )={\bigcup }_{b\in B}Ho{m}_{\mathbb{R}}(F_b(\xi ),F_b(\eta ))$.

Abstract Definition

Locally, the bundle is same as $U\times Ho{m}_{\mathbb{R}}({\mathbb{R}}^k,{\mathbb{R}}^l)$. So that using abstract definition, the Hom bundle is ${\bigsqcup }_{\text{open }U\subset B}U\times Ho{m}_{\mathbb{R}}({\mathbb{R}}^k,{\mathbb{R}}^l)/\sim$.

Dual Bundle

If the bundle $\eta$ is a Trivial Bundle ${\Sigma }_B^1$.

Example

Differential Forms