Appendix D
Manifolds and Lie Groups
The purpose of this appendix is to collect the essential parts of manifold and Lie
group theory in a convenient form for the body of the book. The philosophy of this
appendix is to give the main definitions and to prove many of the basic theorems.
Some of the more difficult results are stated with appropriate references that the
careful reader who is unfamiliar with differential geometry can study.
D.1
C
∞
Manifolds
D.1.1
Basic Definitions
Let
X
be a Hausdorff topological space with a countable basis for its topology. Then
an
n

chart
for
X
is a pair
(
U,
Φ)
of an open subset
U
of
X
and a continuous map
Φ
of
U
into
R
n
such that
Φ(
U
)
is open in
R
n
and
Φ
is a homeomorphism of
U
onto
Φ(
U
)
. A
C
∞
n

atlas
for
X
is a collection
{
(
U
α
,
Φ
α
)
}
α
∈
I
of
n
charts for
X
such
that
(1) the collection of sets
{
U
α
}
α
∈
I
is an open covering of
X
,
(2) the maps
Φ
β
◦
Φ

1
α
: Φ
α
(
U
α
∩
U
β
)
d47
Φ
β
(
U
α
∩
U
β
)
are of class
C
∞
for all
α,β
∈
I
.
Examples
1.
Let
X
=
R
n
and take
U
=
R
n
and
Φ
the identity map. Then
(
U,
Φ)
is an
n
chart
for
X
and
{
(
U,
Φ)
}
is a
C
∞
atlas.
2.
Let
X
=
S
n
=
{
(
x
1
,...,x
n
+1
)
∈
R
n
+1
:
x
2
1
+
· · ·
+
x
2
n
+1
= 1
}
with the
topology as a closed subset of
R
n
+1
. Let
S
n
i,
+
=
{
(
x
1
,...,x
n
+1
)
∈
S
n
:
x
i
>
0
}
and
S
i,

=
{
(
x
1
,...,x
n
+1
)
∈
S
n
:
x
i
<
0
}
for
i
= 1
,...,n
+ 1
. Define
Φ
i,
±
(
x
) = (
x
1
,...,x
i

1
,x
i
+1
,...,x
n
+1
)
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670
APPENDIX D. MANIFOLDS AND LIE GROUPS
for
x
∈
S
n
i,
±
(the projection of
S
i,
±
onto the hyperplane
{
x
i
= 0
}
). Then
Φ
i,
±
(
S
n
i,
±
) =
B
n
=
{
x
∈
R
n
:
x
2
1
+
· · ·
+
x
2
n
<
1
}
and
Φ

1
i,
±
(
x
1
,...,x
n
) = (
x
1
,...,x
i

1
,
±
radicalBig
1

x
2
1
 · · · 
x
2
n
,x
i
,...,x
n
)
.
(D.1)
Thus each
(
U,
Φ
i,
±
)
is a chart. The sets
S
n
i,
±
cover
S
n
. From (D.1) it is clear that
{
(
S
n
i,epsilon1
,
Φ
n,epsilon1
) : 1
≤
i
≤
n
+ 1
,epsilon1
=
±}
is a
C
∞
n
atlas for
X
.
3.
Let
f
1
,...,f
k
be
C
∞
realvalued functions on
R
n
with
k
≤
n
. Let
X
be the set
of points
x
∈
R
n
so that
f
i
(
x
) = 0
for all
i
= 1
,...,k
and some
k
×
k
minor of
the
k
×
n
matrix
D
(
x
) =
bracketleftbigg
∂f
i
∂x
j
(
x
)
bracketrightbigg
is nonzero (note that
X
might be empty). Give
X
the subspace topology in
R
n
. For
1
≤
i
1
<
· · ·
< i
k
≤
n
let
D
i
1
,i
2
,...,i
k
(
x
)
be the
k
×
k
matrix formed by rows
i
1
,...,i
k
of
D
(
x
)
. Define
U
i
1
,i
2
,...,i
k
=
{
x
∈
X
: det
D
i
1
,i
2
,...,i
k
(
x
)
negationslash
= 0
}
.
Then these sets constitute an open covering of
X
.
We now construct an
(
n

k
)
atlas for
X
as follows: Given
x
∈
X
, choose
1
≤
i
1
<
· · ·
<i
k
≤
n
so that
x
∈
U
i
1
,i
2
,...,i
k
. Let
1
≤
p
1
<
· · ·
<p
n

k
≤
n
be
the complementary set of indices:
{
i
1
,...,i
k
} ∪ {
p
1
,...,p
n

k
}
=
{
1
,...,n
}
.
For
y
∈
R
n
we define
u
q
(
y
) =
f
i
q
(
y
)
for
1
≤
q
≤
k
and
u
k
+
q
(
y
) =
y
p
q
for
1
≤
q
≤
n

k
. Then
det
bracketleftbigg
∂u
i
∂x
j
(
y
)
bracketrightbigg
negationslash
= 0
for
y
∈
U
i
1
,i
2
,...,i
k
.
Set
Ψ(
y
) = (
u
1
(
y
)
,...,u
n
(
y
))
. The inverse function theorem (see Lang [1993])
implies that then there exists an open subset
V
x
⊂
R
n
containing
x
such that
W
x
=
Ψ(
V
x
)
is open in
R
n
and
Ψ
is a bijection from
V
x
onto
W
x
with
C
∞
inverse map.
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 Spring '10
 N.R.Wallach
 Topology, Group Theory, Topological space, Lie group, lie groups, Differentiable manifold

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