ABSTRACT
In this last chapter, we introduce the reader to some basic results in Lie groups. On the one hand, it serves the purpose of supplying a large number of interesting examples of smooth manifolds and on the other, it gives the reader an opportunity to see some basic tools of differential topology being employed fruitfully. In the first section, we quickly recall some results from matrix theory and in the second some results from topological groups. Mastering everything in these sections is not all that necessary to go ahead. In Section 9.3, we introduce the Lie groups and in Section 9.4, the Lie algebras. In section 9.5 and 9.6, the fundamental inter-relation between the Lie group and its Lie algebra is discussed. The next two sections exploit this relation to get information on the structure of Lie groups such as subgroups, normal subgroups, conjugation action and so on. The last two sections deal with the fundamental problem of existence of Lie subgroups and we take this opportunity to introduce yet another differential topological/geometrical notion-foliation. The treatment in this chapter is somewhat different from other chapters, in the sense that we assume more maturity and indulgence on the part of the reader. As a result, exercises are spread out all over the chapter, instead of at the end of each section.
