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,
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