2 Lie Groups

the formula for the inverse

A'1 = - ^ — a d j ^

detA J

(where adjA is the matrix whose entries are the signed cofactors of

each of the entries a^), we see that the coordinates of A~l are also

smooth functions of those of A. This concludes the description of

the general linear group GLnR as a manifold and as a group, with

the group operations of multiplication and inverse being smooth func-

tions. It is an important example of a Lie group. We will see more

examples of Lie groups later on, after we make a brief review of var-

ious definitions, notations and results about manifolds which will be

used later on.

2. Smooth manifolds: A review

Generally speaking, a smooth manifold is a topological space M that

locally resembles the Euclidean space Mn, with a notion of differen-

tiation that can be established in M. The formal definition is as

follows:

Definition. A smooth (or differentiable) manifold of dimension n is

a Hausdorff topological space M with a collection of pairs (Ua,(f)a)

where Ua (chart) is an open subset of M and (j)a\ Ua — Rn so that:

(a) Each (j)a is a homeomorphism of Ua onto an open subset Va

ofMn.

(b) UaUa = M.

(c) For every a, (3 the transition functions pap = (ftpoij;^1: j)a(Ua

DUp) —» /p(UanUp) are smooth, in the sense of smooth func-

tions between subsets of Rn. In this case the charts (Ua,(f)a)

and (Up, (j)p) are called compatible.

(d) The family {(Uai(pa)} is maximal relative to the conditions

(b) and (c).

Such a family of sets and maps satisfying (b), (c), and (d) constitutes

a smooth structure on M.

Remark. Condition (d) is a purely technical one. Given a family

of charts satisfying (a)-(c) it can be completed to a maximal one,