Graphics Reference
In-Depth Information
1
Æ
sp
o
:
SE
would have the form
(
)( )
=
[]
()
(
)
sp
o
qqqq
,
a
,
(8.27)
for some continuous map
1
Æ
-
()
=-
()
a
:
SR
with
a
q
a
q
.
(Exercise 8.9.3). Such a map a takes on both positive and negative values. Since
S
1
is
connected, the intermediate value theorem implies that a must be zero somewhere,
which contradicts the hypothesis that s was a nonzero cross-section. This finishes
Example 8.9.2.
Here are some important constructions defined for vector bundles.
Definition.
Let x
i
= (
E
i
,p
i
,
B
i
) be vector bundles. The vector bundle x
1
¥x
2
= (
E
1
¥
E
2
,p
1
¥p
2
,
B
1
¥
B
2
) is called the
vector bundle product of
x
1
and
x
2
.
It is trivial to show that x
1
¥x
2
is in fact a vector bundle.
Definition.
Let j=(
E
,p,
B
) be an n-plane bundle and let f :
B
1
Æ
B
be a map. Define
an n-plane bundle f*x=(
E
1
,p
1
,
B
1
) over
B
1
as follows:
(1)
E
1
= {(
b
1
,
e
) Œ
B
1
¥
E
| f(
b
1
) =p(
e
)}
(2) p
1
(
b
1
,
e
) =
b
1
(3) The vector space structure for each fiber p
1
-1
(
b
1
) is defined by
(
)
+
(
)
=
(
)
r
be
,
s
be
,
¢
b e e
,
r
+
s
¢
,
for r s
,
Œ
R
.
1
1
1
(4) Let
b
1
Œ
B
1
and let
n
-
1
¥Æ
()
j
:
UR
p
U
be a local coordinate chart for x over a neighborhood
U
of f(
b
1
). If
V
= f
-1
(
U
),
then the map
-
1
U ¥Æ
()
n
j
:
R
p
V
,
1
1
defined by
j
1
(
b
1
¢,
v
) = (
b
1
¢,j(f(
b
1
¢),
v
)).
is a local coordinate chart for x
1
over
V
.
