Lecture 31
6.3
Examples
of Manifolds
We
begin with a review
of
the
definition
of a
manifold.
Let
X
be
a subset
of
R
n
,
let
Y
be a
subset
of
R
m
, and
let
f
:
X
Y
be a
→
continuous
map.
Definition 6.6.
The
map
f
is
C
∞
if for
every
p
∈
X
, there exists
a
neighborhood
U
p
of
p
in
R
n
and a
C
∞
map
g
p
:
U
p
→
R
m
such that
g
p
=
f
on
U
p
∩
X
.
Claim.
If
f
:
X
Y
is
continuous,
then there exists a neighborhood
U
of
X
in
R
n
→
and a
C
∞
map
g
:
U
→
R
m
such
that
g
=
f
on
U
∩
X
.
Definition 6.7.
The
map
f
:
X
→
Y
is
a
diffeomorphism
if it
is
onetoone, onto,
and
both
f
and
f
−
1
are
C
∞
maps.
We
define
the
notion of
a manifold.
Definition 6.8.
A
subset
X
of
R
N
is
an
n
dimensional
manifold
if for every
p
∈
X
,
there
exists
a neighborhood
V
of
p
in
R
N
, an
open
set
U
in
R
n
, and
a
diffeomorphism
φ
:
U
→
X
∩
V
.
Intuitively,
the
set
X
is
an
n
dimensional manifold
if locally
near
every
point
p
∈
X
, the
set
X
“looks
like
an open
subset of
R
n
.”
Manifolds
come
up in practical applications
as
follows:
Let
U
be
an open subset
of
R
N
, let
k < N
, and
let
f
:
R
N
→
R
k
be a
C
∞
map.
Suppose that
0 is
a regular
value
of
f
, that is,
f
−
1
(0)
∩
C
f
=
φ
.
Theorem 6.9.
The
set
X
=
f
−
1
(0)
is an
n
dimensional
manifold,
where
n
=
N
−
k
.
Proof.
If
p
∈
f
−
1
(0),
then
p
/
∈
C
f
. So
the map
Df
(
p
) :
R
N
R
k
is
onto. The map
→
f
is a
submersion at
p
.
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 Fall '04
 unknown
 Topology, Open set

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