The final result of Part I of this book, Theorem 5.10, was stated using the term differentiable manifold. Some complexities of the relation between Liegroup sand Lie algebras were encountered in Sections 7.4.3 and 7.5.2. One needs precise definitions of these mathematical objects to set these questions in their geometrical perspective. In Chapter 12 we made use of Banach and Hilbert bundles. These are special cases of fibre bundles. Fibre bundles also provide deep insights into Lie groups and Lie algebras. The purpose of this appendix is to provide the basic definitions, and brief introductions to these subjects that would suffice for our needs. We begin with the topological concept of a fibre bundle and continue with that of a differentiable manifold. The two are brought together through the notion of tangent spaces to lead us to the desired relation between Lie groups and Lie algebras.

Fibre bundles

The topological spaces called fibre bundles are generalizations of the topological product; they have the product form locally, but not globally. The simplest example is the Möbius strip. The rectangular strip of paper abcd shown in Fig. A8.1 may be formed into a cylinder by glueing the short edges together; a is joined to b, and c to d. However, if one gives the strip a twist and joins a to c and b to d, one obtains a Möbius strip.