A **vector bundle** \({(E,M,\pi,\mathbb{K}^{n})}\) has a vector space fiber \({\mathbb{K}^{n}}\) (assumed here to be \({\mathbb{R}^{n}}\) or \({\mathbb{C}^{n}}\)) and a structure group that is linear (\({G\subseteq GL(n,\mathbb{K})}\)) and therefore acts as a matrix across trivializing neighborhoods, i.e.

\(\displaystyle f_{i}(p)=g_{ij}f_{j}(p), \)

where the operation of \({g_{ij}}\) is now matrix multiplication on the vector components \({f_{j}(p)\in\mathbb{K}^{n}}\). If we view \({V_{x}\equiv\pi^{-1}(x)}\) as an internal space on \({M}\) with intrinsic vector elements \({v}\), the linear map \({f_{i}\colon\pi^{-1}(x)\rightarrow\mathbb{K}^{n}}\) is equivalent to choosing a basis \({e_{i\mu}}\) to get vector components, i.e. \({f_{i}(v)=v_{i}^{\mu}}\), where \({v_{i}^{\mu}e_{i\mu}=v}\) (and latin letters are labels while greek letters are the usual indices for vectors and labels for bases). The action of the structure group can then be written

\(\displaystyle v_{i}^{\mu}=(g_{ij})^{\mu}{}_{\lambda}v_{j}^{\lambda}, \)

which is equivalent to a change of basis

\(\displaystyle e_{i\mu}=(g_{ij}^{-1})^{\lambda}{}_{\mu}e_{j\lambda}, \)

or as matrix multiplication on basis row vectors

\(\displaystyle e_{j}=e_{i}g_{ij}, \)

so that the action of \({g_{ij}(x)}\) in \({U_{i}\cap U_{j}}\) is equivalent to a change of frame or gauge transformation from \({e_{i}}\) to \({e_{j}}\), which is equivalent to a transformation of internal space vector components in the opposite direction.

Δ The frame is not a part of the vector bundle, it is a way of viewing the local trivializations; therefore the view of \({g_{ij}(x)}\) as effecting a change of basis should not be confused with a group action on either \({\pi^{-1}(x)}\) or \({E}\). As the structure group of \({E}\), the action of \({G}\) is on the fiber \({\mathbb{K}^{n}}\), which is not part of \({E}\). |

If the structure group of a vector bundle is reducible to \({GL(n,\mathbb{K})^{e}}\), then it is called an **orientable bundle**; all complex vector bundles are orientable, so orientability usually refers to real vector bundles. The tangent bundle of \({M}\) (formally defined in an upcoming section) is then orientable iff \({M}\) is orientable. On a pseudo-Riemannian manifold \({M}\), the structure group of the tangent bundle is reducible to \({O(r,s)}\), and if \({M}\) is orientable then it is reducible to \({SO(r,s)}\); if the structure group can be further reduced to \({SO(r,s)^{e}}\), then \({M}\) and its tangent bundle are called **time and space orientable**. Note that this additional distinction is dependent only upon the metric, and two metrics on the same manifold can have different time and space orientabilities.

Δ The orientability of a vector bundle as a bundle is different than its orientability as a manifold itself; therefore it is important to understand which version of orientability is being referred to. In particular, the tangent bundle of \({M}\) is always orientable as a manifold, but it is orientable as a bundle only if \({M}\) is. |

A gauge transformation on a vector bundle is a smoothly defined linear transformation of the basis inferred by the components due to local trivializations at each point, i.e.

\(\displaystyle e_{i\mu}^{\prime}=(\gamma_{i}^{-1})^{\lambda}{}_{\mu}e_{i\lambda}, \)

which is equivalent to new local trivializations where

\(\displaystyle \left(v_{i}^{\mu}\right)^{\prime}=(\gamma_{i})^{\mu}{}_{\lambda}v_{i}^{\lambda}, \)

giving us new transition functions

\(\displaystyle g_{ij}^{\prime}=\gamma_{i}g_{ij}\gamma_{j}^{-1}, \)

where we have suppressed indices for pure matrix relationships. Thus the gauge group is the same as the structure group, and a gauge transformation \({\gamma_{i}^{-1}}\) is equivalent to the transition function \({g_{i^{\prime}i}}\) from \({U_{i}}\) to \({U_{i}^{\prime}}\), the same neighborhood with a different local trivialization.

The above depicts how the elements of the fiber over \({x}\) in a vector bundle can be viewed as abstract vectors in an internal space, with the local trivialization acting as a choice of basis from which the components of these vectors can be calculated. The structure group then acts as a matrix transformation between vector components, and between bases in the opposite direction. A gauge transformation is also a new choice of basis, and so can be handled similarly.

A vector bundle always has global sections (e.g. the zero vector in the fiber over each point). A vector bundle with fiber \({\mathbb{R}}\) is called a **line bundle**.