Graphics Reference
In-Depth Information
8.5.3.
This exercise describes an alternate approach to the orientability of a smooth manifold
M
k
in
R
n
. Given any two local parameterizations F
p
:
U
p
Æ
V
p
and F
q
:
U
q
Æ
V
q
for
M
define
Ë
¯
(
)
¢
(
)
,
()
=
-
1
-
1
()
d
pq
r
det
FFF
o
r
q
p
p
for
r
Œ
V
p
«
V
q
.
Definition.
We say that
M
is
orientable
if one can choose local parameterizations F
p
for
M
in such a way that either d
p
,
q
(
r
) is positive for all
p
,
q
Œ
M
and
r
Œ
V
p
«
V
q
or
d
p
,
q
(
r
) is negative for all
p
,
q
Œ
M
and
r
Œ
V
p
«
V
q
.
Show that this definition of orientability agrees with the one in Section 8.5.
Prove that every simply connected manifold in
R
n
is orientable, where the definition of
orientable is based on the definition in Exericse 8.5.3.
8.5.4.
Hint:
Pick a point
p
0
in the manifold
M
. For any other point
q
Œ
M
let g : [0,1] Æ
M
be a path from
p
0
to
q
. Choose a partition (t
0
= 0,t
1
,...,t
k
= 1) of [0,1] so that, if
p
i
=
g(t
i
), then we have local parameterizations
F
pp
:
UV
Æ=
i
i
,
01
, ,..., ,
k
p
i
i
with the property that
V
p
i-1
«
V
p
i
π f for i = 1, 2,...k. If any d
p
i-1
,
p
i
(
r
),
r
Œ
V
p
i-1
«
V
p
i
,
is negative, then replace F
p
i
by F
p
i
a, where a :
U
p
i
Æ
U
p
i
is an orientation-reversing
diffeomorphism. Show that these steps lead to a well-defined collection of local para-
meterizations. The fact that
M
is simply connected is needed to show that the choice
for F
p
does not depend on g.
o
Consider the torus in
R
3
which is the surface obtained by rotating the circle in the x-z
plane of radius 1 and center (3,0,0) about the z-axis. Define a nonzero normal vector
field on this torus.
8.5.5.
Section 8.6
8.6.1.
Show that the definition of critical points, critical values, and nondegenerate critical
points for functions f :
M
Æ
R
in terms of local coordinates is independent of the choice
of local coordinates.
Section 8.8
Show that the coordinate neighborhoods (
U
i
,j
i
) in Example 8.8.3 induce a C
•
struc-
tures on
S
1
.
8.8.1.
(a)
Consider the coordinate neighborhoods {(
U
+
,j
+
),(
U
-
,j
-
)} for
S
n
(b)
that were defined
in Example 8.8.4. Show that
y
y
-
1
()
=
n
j
-
o
y
,
y
Œ
R
.
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