Graphics Reference
In-Depth Information
To show that a subset
A
of a manifold has measure zero, it turns out that we do
not need to show that j(
U
«
A
) has measure zero for all coordinate neighborhoods
(
U
,j), simply for any collection that covers
A
. Furthermore, since n-rectangles do not
have measure zero, a closed subset of a manifold that has measure zero is nowhere
dense in the manifold.
Definition.
Let
M
n
and
N
k
be C
r
manifolds and let f :
M
n
Æ
N
k
be a C
r
map, r ≥ 1.
A point
p
Œ
M
is called a
critical point
and f(
p
) is called a
critical value
of f if the rank
of f at
p
is less than k. If the rank of f equals k at
p
, then
p
is called a
regular point
of f. The points of
N
k
that are not critical values are called
regular values
.
The definition of
p
being a regular point is equivalent to requiring that
()
Df
pM
:
T
Æ
T
f
N
(
)
p
p
is onto. It is a good exercise for the reader to convince him/herself that the definitions
here reduce to the definitions given in Section 8.6 when we are dealing with
submanifolds of
R
n
and to the definitions in Section 4.5 when
M
=
R
n
and
N
=
R
.
8.11.1. Theorem.
Let f :
M
n
Æ
N
k
be a differentiable map between differentiable
manifolds. Let
p
Œ
M
and
q
= f(
p
). If
q
Œ
N
-∂
N
is a regular value for f and f|∂
M
,
then f
-1
(
q
) is a submanifold of
M
of dimension n - k, or equivalently, of co-
dimension k.
Proof.
See Theorem 8.3.7.
8.11.2. Example.
Consider the map
n
f
:
RR
Æ
defined by
2
2
2
...
(
)
=++ +
fx x
,
,...,
x
x
x
x
.
12
n
1
2
n
The derivative of f has rank 1 everywhere except at the origin. Therefore, f
-1
(1), which
is just the unit sphere
S
n
, is a submanifold of dimension n - 1 as guaranteed by
Theorem 8.11.1.
Definition.
Let
M
n
and
N
k
be C
r
manifolds and let f :
M
n
Æ
N
k
be a C
r
map, r ≥ 1.
We say that the map f is
transverse
to a submanifold
A
in
N
if for all points
p
Πf
-1
(
A
)
and
a
Œ
A
()
+
( ( )
=
(
.
T
Ap
f
T
T
N
a
p
a
Theorem 8.11.1 has the following generalization.
8.11.3. Theorem.
Let f :
M
n
Æ
N
k
be a differentiable map between differentiable
manifolds and
A
d
a submanifold of
N
. Assume either that
