Manifolds

The mathematical background required for the study of modern physical theories and, in particular, gauge theories, is rather extensive and may be divided roughly into the following parts: elements of differential geometry, fiber bundles and connections, and algebraic topology of a manifold. The first two of these parts are nowadays fairly standard background for research workers in mathematical physics. In any case, physicists are familiar with classical differential geometry, which forms the cornerstone of Einstein’s theory of gravitation (the general theory of relativity). Therefore, in this chapter we give only a summary of some results from differential geometry to establish notation and make the monograph essentially self-contained. Fiber bundles and connections are discussed in Chapter 4. Characteristic classes, which are fundamental in the algebraic topology of a manifold, are discussed in detail in Chapter 5. The study of various classical field theories is taken up in Chapter 6. The reader familiar with the mathematical material may want to start with Chapter 6 and refer back to the earlier chapters as needed.