Lie groups; Subgroups: Left- and right- invariant vector fields; Lie algebras; One-parameter subgroups; Exponential mapping; Adjoint representation; Lie transformation groups; Killing vector fields
3.1 Scope of the Chapter
This chapter is devoted to a concise exposition of Lie groups that help illuminate various structural peculiarities of mappings on manifolds. These groups are so named because it was M. S. Lie who has first studied family of continuous functions forming a group and recognised their effectiveness in revealing some very important and fundamental properties of differential equations. We first define in Sec. 3.2 a Lie group as a smooth manifold endowed with a group operation in which multiplication and ...