We will now study morphisms of premanifolds \(\pi\colon X\to B\) that look locally on B like a fixed morphism \(p\colon Z\to B\). We then call π a twist of p. Important special cases are fiber bundles, where \(p\colon B\times F\to B\) is a projection. Moreover, one often endows twists with an additional datum that restricts the changes between different local isomorphisms of p and π to a fixed subsheaf of the sheaf all automorphisms of p. For fiber bundles this subsheaf will usually be given by a faithful action of a Lie group G on the fiber F. An important special case of a fiber bundle is the notion of a principal bundle for a Lie group G. They are in a way the universal fiber bundles for a given structure group G (Remark 8.22).
Among the most important vector bundles is the tangent bundle whose fiber at a point p of a premanifold M is the tangent space T p (M). This will be described as a geometric vector bundle as well as a locally free module. The dual of the tangent bundle is the vector bundle of differential 1-forms and by taking exterior powers we obtain differential forms of arbitrary degree. We conclude the chapter by defining the de Rham complex of a premanifold.
In the next section we study a very important special case of fiber bundles: vector bundles. This is the main topic of this chapter. There are two ways to look at vector bundles. The first point of view is that of a fiber bundle where the typical fiber is a vector space and where the local coordinate changes are linear. This is explained in Sect. 8.2. The second point of view is to consider vector bundles as locally free modules over the structure sheaf. Hence we introduce such modules in Sect. 8.3 and prove in Sect. 8.4 that both point of views are equivalent.