Defining bundles

When introducing tangent spaces on a manifold \({M^{n}}\), we defined the tangent bundle to be the set of tangent spaces at every point within the region of a coordinate chart \({U\rightarrow\mathbb{R}^{n}}\), i.e. it was defined as the cartesian product \({U\times\mathbb{R}^{n}}\). Globally, we had to use an atlas of charts covering \({M}\), with coordinate transformations \({\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}}\) defining how to consider a vector field across charts. We now want to take the same approach to define the global version of the tangent bundle, with analogs for frames and internal spaces.

An Illustrated Handbook