4Vector bundles
III Differential Geometry
4.2 Vector bundles
Our aim is to consider spaces
T
p
M ⊗ T
p
M, . . . ,
Λ
r
T
p
M
etc as
p
varies, i.e. con-
struct a “tensor bundle” for these tensor products, similar to how we constructed
the tangent bundle. Thus, we need to come up with a general notion of vector
bundle.
Definition
(Vector bundle)
.
A vector bundle of rank
r
on
M
is a smooth
manifold E with a smooth π : E → M such that
(i) For each p ∈ M, the fiber π
−1
(p) = E
p
is an r-dimensional vector space,
(ii)
For all
p ∈ M
, there is an open
U ⊆ M
containing
p
and a diffeomorphism
t : E|
U
= π
−1
(U) → U × R
r
such that
E|
U
U × R
r
U
t
π
p
1
commutes, and the induced map
E
q
→ {q} × R
r
is a linear isomorphism
for all q ∈ U.
We call
t
a trivialization of
E
over
U
; call
E
the total space; call
M
the
base space; and call
π
the projection. Also, for each
q ∈ M
, the vector
space E
q
= π
−1
({q}) is called the fiber over q.
Note that the vector space structure on
E
p
is part of the data of a vector bundle.
Alternatively,
t
can be given by collections of smooth maps
s
1
, · · · , s
r
:
U → E
with the property that for each
q ∈ U
, the vectors
s
1
(
q
)
, · · · , s
r
(
q
) form a basis
for E
q
. Indeed, given such s
1
, · · · , s
r
, we can define t by
t(v
q
) = (q, α
1
, · · · , α
r
),
where v
q
∈ E
q
and the α
i
are chosen such that
v
q
=
r
X
i=1
α
i
s
i
(q).
The s
1
, · · · , s
r
are known as a frame for E over U .
Example
(Tangent bundle)
.
The bundle
T M → M
is a vector bundle. Given
any point
p
, find some coordinate charts around
p
with coordinates
x
1
, · · · , x
n
.
Then we get a frame
∂
∂x
i
, giving trivializations of
T M
over
U
. So
T M
is a vector
bundle.
Definition
(Section)
.
A (smooth) section of a vector bundle
E → M
over some
open
U ⊆ M
is a smooth
s
:
U → E
such that
s
(
p
)
∈ E
p
for all
p ∈ U
, that is
π ◦ s = id. We write C
∞
(U, E) for the set of smooth sections of E over U.
Example. Vect(M) = C
∞
(M, T M).
Definition
(Transition function)
.
Suppose that
t
α
:
E|
U
α
→ U
α
× R
r
and
t
β
: E|
U
β
→ U
β
× R
r
are trivializations of E. Then
t
α
◦ t
−1
β
: (U
α
∩ U
β
) × R
r
→ (U
α
∩ U
β
) × R
r
is fiberwise linear, i.e.
t
α
◦ t
−1
β
(q, v) = (q, ϕ
αβ
(q)v),
where ϕ
αβ
(q) is in GL
r
(R).
In fact,
ϕ
αβ
:
U
α
∩ U
β
→ GL
r
(
R
) is smooth. Then
ϕ
αβ
is known as the
transition function from β to α.
Proposition.
We have the following equalities whenever everything is defined:
(i) ϕ
αα
= id
(ii) ϕ
αβ
= ϕ
−1
βα
(iii) ϕ
αβ
ϕ
βγ
= ϕ
αγ
, where ϕ
αβ
ϕ
βγ
is pointwise matrix multiplication.
These are known as the cocycle conditions.
We now consider general constructions that allow us to construct new vector
bundles from old ones.
Proposition
(Vector bundle construction)
.
Suppose that for each
p ∈ M
, we
have a vector space E
p
. We set
E =
[
p
E
p
We let
π
:
E → M
be given by
π
(
v
p
) =
p
for
v
p
∈ E
p
. Suppose there is an open
cover {U
α
} of open sets of M such that for each α, we have maps
t
α
: E|
U
α
= π
−1
(U
α
) → U
α
× R
r
over
U
α
that induce fiberwise linear isomorphisms. Suppose the transition
functions
ϕ
αβ
are smooth. Then there exists a unique smooth structure on
E
making π : E → M a vector bundle such that the t
α
are trivializations for E.
Proof. The same as the case for the tangent bundle.
In particular, we can use this to perform the following constructions:
Definition
(Direct sum of vector bundles)
.
Let
E,
˜
E
be vector bundles on
M
. Suppose
t
α
:
E|
U
α
∼
=
U
α
× R
r
is a trivialization for
E
over
U
α
, and
˜
t
α
:
˜
E|
U
α
∼
=
U
α
× R
˜r
is a trivialization for
˜
E over U
α
.
We let
ϕ
αβ
be transition functions for
{t
α
}
and
˜ϕ
αβ
be transition functions
for {
˜
t
α
}.
Define
E ⊕
˜
E =
[
p
E
p
⊕
˜
E
p
,
and define
T
α
: (E ⊕
˜
E)|
U
α
= E|
U
α
⊕
˜
E|
U
α
→ U
α
× (R
r
⊕ R
˜r
) = U
α
× R
r+˜r
be the fiberwise direct sum of the two trivializations. Then
T
α
clearly gives a
linear isomorphism (E ⊕
˜
E)
p
∼
=
R
r+˜r
, and the transition function for T
α
is
T
α
◦ T
−1
β
= ϕ
αβ
⊕ ˜ϕ
αβ
,
which is clearly smooth. So this makes E ⊕
˜
E into a vector bundle.
In terms of frames, if
{s
1
, · · · , s
r
}
is a frame for
E
and
{˜s
1
, · · · , ˜s
˜r
}
is a frame
for
˜
E over some U ⊆ M , then
{s
i
⊕ 0, 0 ⊕ ˜s
j
: i = 1, · · · , r; j = 1, · · · , ˜r}
is a frame for E ⊕
˜
E.
Definition
(Tensor product of vector bundles)
.
Given two vector bundles
E,
˜
E
over M, we can construct E ⊗
˜
E similarly with fibers (E ⊗
˜
E)|
p
= E|
p
⊗
˜
E|
p
.
Similarly, we can construct the alternating product of vector bundles Λ
n
E
.
Finally, we have the dual vector bundle.
Definition
(Dual vector bundle)
.
Given a vector bundle
E → M
, we define the
dual vector bundle by
E
∗
=
[
p∈M
(E
p
)
∗
.
Suppose again that
t
α
:
E|
U
α
→ U
α
× R
r
is a local trivialization. Taking the
dual of this map gives
t
∗
α
: U
α
× (R
r
)
∗
→ E|
∗
U
α
.
since taking the dual reverses the direction of the map. We pick an isomorphism
(
R
r
)
∗
→ R
r
once and for all, and then reverse the above isomorphism to get a
map
E|
∗
U
α
→ U
α
× R
r
.
This gives a local trivialization.
If
{s
1
, · · · , s
r
}
is a frame for
E
over
U
, then
{s
∗
1
, · · · , s
∗
r
}
is a frame for
E
∗
over U, where {s
∗
1
(p), · · · , s
∗
r
(p)} is a dual basis to {s
1
(p), · · · , s
r
(p)}.
Definition (Cotangent bundle). The cotangent bundle of a manifold M is
T
∗
M = (T M)
∗
.
In local coordinate charts, we have a frame
∂
∂x
1
, · · · ,
∂
∂x
n
of
T M
over
U
. The
dual frame is written as dx
1
, · · · , dx
n
. In other words, we have
dx
i
|
p
∈ (T
p
M)
∗
and
dx
i
|
p
∂
∂x
j
p
!
= δ
ij
.
Recall the previously, given a function
f ∈ C
∞
(
U, R
), we defined d
f
as the
differential of f given by
df|
p
= Df|
p
: T
p
M → T
f(p)
R
∼
=
R.
Thinking of x
i
as a function on a coordinate chart U, we have
Dx
i
|
p
∂
∂x
j
p
!
=
∂
∂x
j
(x
i
) = δ
ij
for all i, j. So the two definitions of dx
i
agree.
We can now take powers of this to get more interesting things.
Definition
(
p
-form)
.
A
p
-form on a manifold
M
over
U
is a smooth section of
Λ
p
T
∗
M, i.e. an element in C
∞
(U, Λ
p
T
∗
M).
Example. A 1-form is an element of T
∗
M. It is locally of the form
α
1
dx
1
+ · · · + α
n
dx
n
for some smooth functions α
1
, · · · , α
n
.
Similarly, if ω is a p-form, then locally, it is of the form
ω =
X
I
ω
I
dx
I
,
where I = (i
1
, · · · , i
p
) with i
1
< · · · < i
p
, and dx
I
= dx
i
1
∧ · · · ∧ dx
i
p
.
It is important to note that these representations only work locally.
Definition (Tensors on manifolds). Let M be a manifold. We define
T
k
`
M = T
∗
M ⊗ · · · ⊗ T
∗
M
| {z }
k times
⊗ T M ⊗ · · · ⊗ T M
| {z }
` times
.
A tensor of type (k, `) is an element of
C
∞
(M, T
k
`
M).
The convention when k = ` = 0 is to set T
0
0
M = M × R.
In local coordinates, we can write a (k, `) tensor ω as
ω =
X
α
j
1
,...,j
k
i
1
,...,i
`
dx
j
1
⊗ · · · ⊗ dx
j
k
⊗
∂
∂x
i
1
⊗ · · · ⊗
∂
∂x
i
`
,
where the α are smooth functions.
Example. A tensor of type (0, 1) is a vector field.
A tensor of type (1, 0) is a 1-form.
A tensor of type (0, 0) is a real-valued function.
Definition
(Riemannian metric)
.
A Riemannian metric on
M
is a (2
,
0)-tensor
g
such that for all
p
, the bilinear map
g
p
:
T
p
M × T
p
M → R
is symmetric and
positive definite, i.e. an inner product.
Given such a g and v
p
∈ T
p
M, we write kv
p
k for
p
g
p
(v
p
, v
p
).
Using these, we can work with things like length:
Definition (Length of curve). Let γ : I → M be a curve. The length of γ is
`(γ) =
Z
I
k ˙γ(t)k dt.
Finally, we will talk about morphisms between vector bundles.
Definition
(Vector bundle morphisms)
.
Let
E → M
and
E
0
→ M
0
be vector
bundles. A bundle morphism from
E
to
E
0
is a pair of smooth maps (
F
:
E →
E
0
, f : M → M
0
) such that the following diagram commutes:
E E
0
M M
0
F
f
.
i.e. such that F
p
: E
p
→ E
0
f(p)
is linear for each p.
Example.
Let
E
=
T M
and
E
0
=
T M
0
. If
f
:
M → M
0
is smooth, then (D
f, f
)
is a bundle morphism.
Definition
(Bundle morphism over
M
)
.
Given two bundles
E, E
0
over the same
base
M
, a bundle morphism over
M
is a bundle morphism
E → E
0
of the form
(F, id
M
).
Example.
Given a Riemannian metric
g
, we get a bundle morphism
T M →
T
∗
M over M by
v 7→ F (v) = g(v, −).
Since each
g
(
v, −
) is an isomorphism, we have a canonical bundle isomorphism
T M
∼
=
T
∗
M.
Note that the isomorphism between
T M
and
T
∗
M
requires the existence of
a Riemannian metric.