Graphics Reference
In-Depth Information
(
)( )
=-
(
)
f
o
F
q
cos
q
,
-
sin
q
and this map is clearly C
•
. If
p
0
= (x
0
,y
0
) with y
0
> 0, then we could have chosen another
local C
•
parameterization such as
(
)
Æ
S
Y:
-
11
,
,
where
(
)
()
=
2
Y xx
,
1
-
x
.
This time
(
)
(
)( )
=- -
2
f
o
Y
x
x
,
1
-
x
,
which is also C
•
.
Definition.
Let f :
M
n
Æ
N
k
be a differentiable map between differentiable mani-
folds. If f has rank n at all points of
M
, the f is said to be an
immersion
. If f is a
homeomorphism onto f(
M
) Õ
N
and is an immersion, then f is called an
imbedding
.
If f is a homeomorphism between
M
and
N
and an immersion, then it is called a
diffeomorphism
.
Immersions may only be locally one-to-one. For example, a figure eight is an
immersion of a circle in the plane but not an imbedding.
8.3.10. Theorem.
If f :
M
n
Æ
N
k
is a diffeomorphism, then n = k and f
-1
:
N
Æ
M
is
also a diffeomorphism.
Proof.
The fact that n = k follows from the invariance of domain theorem, Theorem
7.2.3.8. Since f is a homeomorphism, it has an inverse which is also a homeomor-
phism. To prove that f
-1
is a diffeomorphism, use the inverse function theorem,
Theorem 4.4.2.
Definition.
A differentiable manifold
N
k
that is a subset of a closed manifold
M
n
is
called a (differentiable)
submanifold
of
M
n
if the inclusion map is an immersion. If
the manifold
M
n
has a nonempty boundary, then we also require that for every
p
Œ
N
k
, there is an open neighborhood
U
of
p
in
M
n
, an imbedding h :
U
Æ
R
n
, and an
open subset
V
Ã
R
k
+
Ã
R
n
, so that
N
k
«
U
= h
-1
(
V
).
The reason for the complication in the case of manifolds with boundary is that
we do not want
N
to meet the ∂
M
in a bad way. See Figure 8.11(a) for some cases of
h,
U
, and
V
. Figure 8.11(b) shows some good submanifolds and Figure 8.11(c) some
imbedded manifolds that we would not want to call submanifolds. Among other
things, unless
N
is contained in ∂
M
,
N
-∂
N
should not meet ∂
M
and ∂
N
should always
meet ∂
M
nicely (“nice” means transversally as defined in Section 8.11).
Finally, another common term is the following:
