Graphics Reference
In-Depth Information
Defining a C
r
map
between manifolds.
Figure 8.10.
R
m
R
m
f
M
n
p
V
N
k
F
U,V
R
n
U
u
8.3.8. Theorem.
The definition of differentiability and rank for maps between man-
ifolds in Euclidean space is well defined and equivalent to the one given in Section
4.3.
Proof.
To show that differentiability and rank are well defined we must show that
the definitions do not depend on the local C
r
parameterization F
U
,
V
. Let
U
¢ be another
open set in
R
n
,
V
¢ another open neighborhood of
p
in
M
n
, and F
U
¢,
V
¢
:
U
¢Æ
V
¢ another
C
r
parameterization. The main observation is that
(
)
-
1
f
o
F
=
f
o
F
o
F
o
F
U,V
¢
¢
U,V
U,V
U,V
¢
¢
and the map F
U
,
V
-1
F
U
¢,
V
¢
is a
C
R
diffeomorphism between open sets in
R
n
. We may
have to restrict the maps to smaller open sets so that the composites are defined, but
we shall leave the details to the reader. See also Theorem 4.3.25.
o
To show that the antipodal map f :
S
1
Æ
S
1
8.3.9. Example.
defined by f(
p
) =-
p
is
a C
•
map.
Let
p
0
Œ
S
1
and assume that
p
0
= (cos q
0
,sin q
0
). The map
Solution.
(
)
Æ
1
F:
q
-+
11
,
q
S
0
0
defined by
()
=
(
)
Fq
cos
q
,
sin
q
is a local C
•
parameterization of a neighborhood of
p
0
. Now,
(
)
Æ
2
f
o
F:
q
-+
11
,
q
R
0
0
is just
