3 - Fiber Bundles

Abstract

In the previous chapter, we assigned a tangent space Tb (B) to each point b of a manifold B. This idea is effective for first order differential calculus, allowing us to define the tangent linear mapping Tbf∈ℒTbBTfbB′ of a morphism f : B → B′ at every point b ∈ B. Our next task is to define the mapping T (f): b ↦ Tb (f). To do this, we need to assemble the various tangent spaces Tb (B) of B (b ∈ B) into a single object, written as T (B).If B is a surface in everyday space, the set of its tangent planes still belongs to this space. However, it is useful to reference points in the tangent plane Tb (B) by four coordinates: two for the point b =(b1, b2) on the surface (e.g. longitude and latitude for the Earth) and two determining the tangent vector hb in Tb (B) (its components with respect to a basis of Tb (B)). The set T (B) of planes tangent to a surface B is therefore the union of {b} × Tb (B), i.e. a disjoint union of all Tb (B). After establishing these definitions, in order to perform more advanced differential calculus, we still need to equip T (B) with a manifold structure. We can then define the tangent linear mapping Tx2 (f):= Tx (T (f)) of T (f): T (B) → T (B′) at a point x ∈ T (B). The set T (B) has a fiber bundle structure ([P2], section 5.2.3(I)) with base B and projection π : T (B) → B : (b, hb) ↦ b (here, the canonical surjection). In Tb (B), each tangent vector hb is referenced by its coordinates (t1, t2) = t, and any point x ∈ T (B) is a pair (b, t), where b = π (x). The fiber bundle M is characterized by its projection π : M ↠ B. Once the manifold T (B) has been constructed, we can iterate the process and define the fiber bundle T (T (B)) = T2 (B), etc., ad lib., which allows us to perform differential calculus of arbitrary order within the framework of manifolds. However, we need to expand this framework slightly further still by making the fibers themselves manifolds rather than vector spaces, leading to the notion of fibration. The details of this construction are somewhat tedious but are given in sections 3.2, 3.3 and 3.4. Most of the proofs are omitted to lighten the presentation. The majority of these proofs are entirely trivial; they are listed in full detail in [LAN 99b], Chapter 3, and [DIE 93], Volume 3, Chapter 16, for the finite-dimensional case.