This implies
∂
t
ω
(
t
) =

d
M
β
(
t
) for every
t
and hence
p
(
F
∇
1
)

p
(
F
∇
0
) =
ω
(1)

ω
(0) =
Z
1
0
∂
t
om
(
t
)
dt
=

d
M
Z
1
0
β
(
t
)
dt.
Thus
p
(
F
∇
1
)

p
(
F
∇
0
) is exact and this proves (ii).
We prove (iii). In Section 8.1.5 we have seen that the curvature of the
pullback connection
f
*
∇
is in the local trivializations
f
*
ψ
α
given by the
2forms
F
f
*
∇
α
=
f
*
F
∇
α
∈
Ω
1
(
f

1
(
U
α
)
,
g
)
.
Hence it follows directly from the definitions that
p
(
F
f
*
∇
) =
f
*
p
(
F
∇
). This
proves (iii) and Theorem 8.3.2.
240
CHAPTER 8.
CONNECTIONS AND CURVATURE
8.3.3
The Euler Class of an Oriented Rank
2
Bundle
Let
π
:
E
→
M
be an oriented Riemannian real rank2 bundle over a smooth
manifold. By Example 8.1.13
E
is a vector bundle with structure group
SO(2) =
g
=
a

c
c
a
a, c
∈
R
, a
2
+
c
2
= 1
.
Its Lie algebra consists of all skewsymmetric real 2
×
2matrices:
so
(2) =
ξ
=
0

λ
λ
0
λ
∈
R
.
The linear map
e
:
so
(2)
→
R
defined by
e
(
ξ
) :=

λ
2
π
is invariant under conjugation.
(However,
e
(
g

1
ξg
) =

e
(
ξ
) whenever
g
∈
O(
n
) has determinant

1. Thus we must assume that
E
is oriented.)
Hence there is a characteristic class
e
(
E
) := [
e
(
F
∇
)]
∈
H
2
(
M
)
,
(8.3.5)
where
∇
is Riemannian connection on
E
.
If we change the Riemannian
structure on
E
then there is an orientation preserving automorphism of
E
intertwining the two inner products. (Prove this!) Thus the characteristic
class
e
(
E
) is independent of the choice of the Riemannian metric. We prove
below that (8.3.5) is the Euler class of
E
whenever
M
is a compact oriented
manifold without boundary.
Thus we have extended the definition of the
Euler class
of an oriented real rank2 bundle to arbitrary base manifolds.
Theorem 8.3.3.
If
E
is an oriented real rank
2
bundle over a compact
oriented manifold
M
without boundary then
(8.3.5)
is the Euler class of
E
.
Proof.
Choose a smooth section
s
:
M
→
E
that is transverse to the zero
section and denote
Q
:=
s

1
(0)
.
Choose a Riemannian metric on
M
and let
exp :
TQ
⊥
ε
→
U
ε
be the tubular neighborhood diffeomorphism in (7.2.11). Multiplying
s
by
a suitable positive function on
M
we may assume that
p
∈
M
\
U
ε/
3
=
⇒

s
(
p
)

= 1
.
8.3.
CHERN–WEIL THEORY
241
Next we claim that there is a Riemannian connection
∇
on
E
such that
∇
s
= 0
on
M
\
U
ε/
2
.
(8.3.6)
To see this, we choose on open cover
{
U
α
}
of
M
such that one of the sets is
U
α
0
=
M
\
U
ε/
3
and
E
admits a trivialization over each set
U
α
. In particular,
we can use
s
to trivialize
E
over
U
α
0
. Next we choose a partition of unity
where
ρ
α
0
= 1 on
M
\
U
ε/
2
. Then the formula (8.1.6) in Step 6 of the proof
of Proposition 8.1.3 defines a Riemannian connection that satisfies (8.3.6).
By (8.3.6) we have
F
∇
s
=
d
∇
∇
s
= 0 on
M
\
U
ε/
2
.
Since
F
∇
is a 2form
with values in the skewsymmetric endomorphisms of
E
we deduce that
F
∇
= 0
on
M
\
U
ε/
2
.
(8.3.7)
The key observation is that, under this assumption, the 2form
τ
ε
:= exp
*
e
(
F
∇
)
∈
Ω
2
c
(
TQ
⊥
ε
)
is a Thom form on the normal bundle of
Q
. With this understood we obtain
from Lemma 7.2.17 with
τ
Q
=
e
(
F
∇
) that
Z
M
ω
∧
e
(
F
∇
) =
Z
Q
ω
=
Z
M
ω
∧
s
*
τ
for every closed form
ω
∈
Ω
m

2
(
M
) and every Thom form
τ
∈
Ω
2
c
(
E
),
where the last equation follows from Theorem 7.3.15. By Poincar´
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