This chapter contains the basic theory of analytic manifolds modelled on finite-dimensional real vector spaces. As promised, a coordinate-free approach will be used with emphasis on global definitions and properties. One of the reasons for including this chapter, instead of referring the reader to one or other of the numerous texts on manifolds, is to allow the reader to gain familiarity with this approach since it will permeate our whole treatment of Lie groups. Once one does away with coordinates it becomes obvious that large chunks of the theory of manifolds can be effortlessly generalised to manifolds modelled on infinitedimensional spaces. We will have no need here of this degree of generality for reasons explained in the Notes at the end of the chapter, but the interested reader should consult the works of Lang, [1] and [2]. Since the theory of manifolds is one of the three legs on which the study of Lie groups stands, the other two being the theory of locally compact groups and the theory of Lie algebras, it is important that the ideas in this chapter, few though they may be, are well understood.

Manifolds and differentiability

Manifolds. Let M be a nonvoid Hausdorff topological space and E a real finite-dimensional vector space.