Graphics Reference
In-Depth Information
(1) ∂
A
̶
N
and the maps f and f|∂
M
are transverse to
A
, or
(2)
A
Ã
N
-∂
N
and the maps f and f|∂
M
are transverse to both
A
and ∂
A
.
Then f
-1
(
A
) is a submanifold of
M
of dimension n - (k - d), or equivalently, of co-
dimension k - d, and ∂f
-1
(
A
) = f
-1
(∂
A
).
Proof.
See Theorem 8.3.7.
If Theorem 8.11.3 looks complicated, it is because we have to worry about bound-
aries. If there are no boundaries, then the simple conclusion is that f
-1
(
A
) is an
(n - (k - d))-dimensional submanifold whenever f is transverse to
A
.
8.11.4. Example.
The inclusion map i :
S
2
Õ
R
3
is not transverse to the plane z = 1.
Note that although i
-1
(0,0,1) = (0,0,1) is a submanifold of
S
2
, it has the wrong co-
dimension.
Definition.
We say that two submanifolds
A
and
B
of a manifold
M
intersect trans-
versally
if
T
()
+
AB M
T
()
=
T
()
p
p
p
for all points
p
Œ
A
«
B
.
Note that
A
and
B
intersect transversally if and only if the inclusion map
A
Õ
M
is transverse to
B
, so that the definition of submanifolds intersecting transversally
can be thought of as a special case of the definition of maps being transverse.
Note also that the concept of manifolds intersecting transversally basically general-
izes what it means for two planes in
R
n
to transverse. See Exercise 1.5.18.
8.11.5. Example.
The lines x + y = 0 and x - y = 0 intersect transversally at the
origin. The unit circle
S
1
does not intersect the line x = 1 transversally at (1,0).
If two submanifolds
A
r
and
B
s
of a manifold
M
n
intersect transversally, then
their intersection is a (r + s - n)-dimensional submanifold. In particular, if a
compact r-dimensional submanifold and a compact (n - r)-dimensional submanifold
intersect transversally in
M
, then their intersection consists of a finite collection of
points.
(The Morse-Sard Theorem) Let
M
n
,
N
k
8.11.6. Theorem.
be manifolds of dimen-
sion n, k, respectively, and let
f:
MN
Æ
be a C
r
map. Let
C
be the set of critical points of f. If
(
)
r
>
max 0,
n
-
k
,
then f(
C
) has measure zero in
N
. The set of regular values of f is dense in
M
.
