This book is based on lecture notes of a three-semester course we taught at Heidelberg University during 1985–6. It is intended for graduate students in theoretical physics. The only prerequisites on the mathematical side are linear algebra and real analysis. The physical part, with the exception of the last chapter, is logically self-contained. However, since we give no motivations nor experimental applications, the reader should already be acquainted with the basics of Yang–Mills theories and general relativity.

The mathematical part is inspired by lectures André Haefliger taught at Geneva University. In chapters 1, 2, 3, and 6 we deal with differential forms and metric structures on ℝn. It is a mere rewriting of formulas well known to physicists from tensor analysis. This formalism, standard in mathematics, serves two purposes: It suppresses indices, which is an advantage in practical calculations; and it is coordinate-free, which allows straightforward generalization to topologically nontrivial spaces. Manifolds are introduced in chapter 7, Lie groups in chapter 8, and fibre bundles in chapter 9. Since fibre bundles are rather abstract mathematical objects and their relevance in physics is not (yet) established, we have organized the subsequent material in such a way that most of it can be understood without acquaintance with bundles. In particular, in chapter 11 covering spinors, we follow again the pattern: linear algebra, open subsets of ℝn, manifolds. The mathematics presented is essentially standard. Therefore we generally do not cite original work.