Math 598
Feb 14, 2005
1
Geometry and Topology II
Spring 2005, PSU
Lecture Notes 7
2.7
Smooth submanifolds
Let
N
be a smooth manifold.
We say that
M
⊂
N
m
is an
n
dimensional
smooth submanifold
of
N
, provided that for every
p
∈
M
there exists a local
chart (
U,
φ
) of
N
centered at
p
such that
φ
(
U
∩
M
) =
R
n
×
{
o
}
,
where
o
denotes the origin of
R
m

k
.
Proposition 1.
A smooth submanifold
M
⊂
N
is a smooth manifold.
Proof.
Since
M
⊂
N
,
M
is Hausdorf and has a countable basis. For every
p
∈
M
, let (
U,
φ
) be a local chart of
M
with
φ
(
U
∩
M
) =
R
n
×
{
o
}
. Set
U
:=
U
∩
M
, and
φ
:=
φ

U
. Then
φ
:
U
→
R
n
×
{
o
}
R
n
is a homeomorphism,
and thus
M
is a toplogical manifold. It remains to show that
M
is smooth.
To see this note that if (
V ,
ψ
) is the restriction of another local chart of
N
to
M
. Then
ψ
◦
(
φ
)

1
=
ψ
◦
φ

1

φ
(
U
)
, which is smooth.
The above proof shows how
M
induces a di
ff
erential structure on
N
.
Whenver we talk of a submanifold
M
as a smooth manifold in its own right,
we mean that
M
is equipped with the di
ff
erential structue which it inherits
from
N
.
Theorem 2.
Let
f
:
M
n
→
N
m
be a smooth map of constant rank
k
(i.e.,
rank
(
df
p
) =
k
, for all
p
∈
M
). Then, for any
q
∈
N
,
f

1
(
q
)
is an
(
n

k
)

dimensional smooth submanifold of
M
.
Proof.
Let
p
∈
f

1
(
q
). By the rank theorem there exists local neighborhoods
(
U,
φ
) and (
V,
ψ
) of
M
and
N
centered at
p
and
q
respectively such that
˜
f
(
x
) :=
ψ
◦
f
◦
φ

1
(
x
1
, . . . , x
n
) = (
x
1
, . . . , x
k
,
0
, . . . ,
0)
.
1
Last revised: February 19, 2005
1
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Next note that
φ
(
U
∩
f

1
(
q
)) =
φ
(
U
)
∩
φ
◦
f

1
◦
ψ

1
(
o
) =
R
n
∩
˜
f

1
(
o
) =
{
o
}
×
R
n

k
.
Thus
f

1
(
q
) is a smooth submanifold of
N
(To be quite strict, we need to
show that
φ
(
U
∩
f

1
(
q
)) =
R
n

k
×
{
o
}
, but this is easily achieved if we replace
ψ
with
θ
◦
ψ
, where
θ
:
R
m
→
R
m
is the di
ff
eomorphism which switches the
first
k
and last
m

k
cordinates).
Exercise 3.
Use the previous result to show that
S
n
is smooth
n
dimensional
submanifold of
R
n
+1
.
Another application of the last theorem is as follows:
Example 4.
SL
n
is a smooth submanifold of
GL
n
.
To see this define
f
:
GL
n
→
R
by
f
(
A
) := det(
A
).
Then
SL
n
=
f

1
(1), and thus it re
mains to show that
f
has constant rank on
GL
n
.
Since this rank has to
be either 1 or 0 at each point (why?), it su
ffi
ces to show that the rank is
not zero anywhere, i.e., it is enough to show that for every
A
∈
GL
n
there
exists
X
∈
T
A
GL
n
such that
df
A
(
X
) = 0. To see this, let
X
= [
α
] where
α
: (

,
)
→
GL
n
is the curve given by
α
(
t
) := (1

t
)
A
. Note that, since
det is contiuous, det(
α
(
t
)) = 0, for all
t
∈
(

,
), once we make sure that
is small enough. Thus
α
is indeed welldefined. Now recall that
df
A
(
X
) := [
f
◦
α
]
∈
T
f
(
A
)
R
.
Further recall that there is a canonical isomorphism
θ
:
T
f
(
A
)
R
→
R
given
by
θ
([
γ
]) =
γ
(0). Thus
θ
◦
df
A
(
X
) = (
f
◦
α
) (0) = det(
A
) = 0
.
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