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then a tangent vector at
p
can be represented by a tuple (
U
+
,j
+
,
a
)
p
with respect to
(
U
+
,j
+
) or a tuple (
U
-
,j
-
,
b
)
p
with respect to (
U
-
,j
-
), where
a
,
b
Œ
R
n
. The two tuples
define the same tangent vector if
(
)
-
1
(
()
)( )
b
=
D jj
o
j
p
a
.
-+
+
Clearly, the tangent vector representatives (
U
+
,j
+
,
e
1
)
p
, (
U
+
,j
+
,
e
2
)
p
,..., (
U
+
,j
+
,
e
n
)
p
define a continuously varying choice of orientations s(
p
) in the tangent spaces at
points
p
in
U
+
. Similarly, we get a continuously varying choice of orientations m(
p
)
from the representatives (
U
-
,j
-
,
e
1
)
p
, (
U
-
,j
-
,
e
2
)
p
,..., (
U
-
,j
-
,
e
n
)
p
. at points
p
in
U
-
. We
will get an orientation for the tangent bundle of
S
n
from continuously varying orien-
tations s and m over
U
+
and
U
-
, respectively, if and only if s(
p
) =m(
p
) for all
p
Œ
U
+
«
U
-
or s(
p
) πm(
p
) for all
p
Œ
U
+
«
U
-
(we can then simply reverse the orientations
of m). In our case, s(
p
) =m(
p
) because by Exercise 8.8.1(b),
a
(
)
()
=
j
-
-
1
o
a
,
2
a
so that the linear transformation
(
)
-
-
1
n
n
(
()
)
D
jj j
o
pR R
:
Æ
+
which identifies tangent vector representatives at
p
is orientation preserving.
Another interesting fact is that the analog of Theorem 7.5.7 holds for differen-
tiable manifolds. We shall only restate part (1).
8.10.5. Theorem.
Every simply connected differentiable manifold is orientable.
Proof.
This follows from Theorem 8.9.9.
Differentiable maps between differentiable manifolds induce bundle maps
between their tangent bundles.
Let
M
n
and
N
k
be differentiable manifolds and let f :
M
n
Æ
N
k
Definition.
be a dif-
ferentiable map. Define a map
(
Æ
()
Df
:
E
t
E
t
M
N
by the condition that for all
p
Œ
M
,
()
=
()
,
Df
T
p
Mp
Df
where
()
( )
Æ
()
Df
pM
:T
T
f
M
(
)
p
p
is the map defined earlier by equations (8.15) or (8.19) depending on how tangent
vectors are defined. The vector bundle map
