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
)o
f
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±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±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
f
M
and
N
centered at
p
and
q
respectively such that
˜
f
(
x
):=
ψ
◦
f
◦
φ

1
(
x
1
,...,x
n
)=(
x
1
k
,
0
,...,
0)
.
1
Last revised: February 19, 2005
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