8Connections
III Differential Geometry
8.1 Basic properties of connections
Imagine we are moving in a manifold
M
along a path
γ
:
I → M
. We already
know what “velocity” means. We simply have to take the derivative of the path
γ
(and pick the canonical tangent vector 1
∈ T
p
I
) to obtain a path
γ
:
I → T M
.
Can we make sense of our acceleration? We can certainly iterate the procedure,
treating
T M
as just any other manifold, and obtain a path
γ
:
I → T T M
. But
this is not too satisfactory, because
T T M
is a rather complicated thing. We
would want to use the extra structure we know about
T M
, namely each fiber is
a vector space, to obtain something nicer, perhaps an acceleration that again
lives in T M.
We could try the naive definition
d
dt
= lim
h→0
γ(t + h) − γ(t)
h
,
but this doesn’t make sense, because
γ
(
t
+
h
) and
γ
(
t
) live in different vector
spaces.
The problem is solved by the notion of a connection. There are (at least) two
ways we can think of a connection — on the one hand, it is a specification of
how we can take derivatives of sections, so by definition this solves our problem.
On the other hand, we can view it as telling us how to compare infinitesimally
close vectors. Here we will define it the first way.
Notation. Let E be a vector bundle on M. Then we write
Ω
p
(E) = Ω
0
(E ⊗ Λ
p
(T
∗
M)).
So an element in Ω
p
(E) takes in p tangent vectors and outputs a vector in E.
Definition
(Connection)
.
Let
E
be a vector bundle on
M
. A connection on
E
is a linear map
d
E
: Ω
0
(E) → Ω
1
(E)
such that
d
E
(fs) = df ⊗ s + fd
E
s
for all f ∈ C
∞
(M) and s ∈ Ω
0
(E).
A connection on T M is called a linear or Koszul connection.
Given a connection d
E
on a vector bundle, we can use it to take derivatives
of sections. Let
s ∈
Ω
0
(
E
) be a section of
E
, and
X ∈ Vect
(
M
). We want to
use the connection to define the derivative of
s
in the direction of
X
. This is
easy. We define ∇
X
: Ω
0
(E) → Ω
0
(E) by
∇
X
(s) = hd
E
(s), Xi ∈ Ω
0
(E),
where the brackets denote applying d
E
(
s
) :
T M → E
to
X
. Often we just call
∇
X
the connection.
Proposition.
For any
X
,
∇
X
is linear in
s
over
R
, and linear in
X
over
C
∞
(
M
).
Moreover,
∇
X
(fs) = f∇
X
(s) + X(f)s
for f ∈ C
∞
(M) and s ∈ Ω
0
(E).
This doesn’t really solve our problem, though. The above lets us differentiate
sections of the whole bundle
E
along an everywhere-defined vector field. However,
what we want is to differentiate a path in
E
along a vector field defined on that
path only.
Definition
(Vector field along curve)
.
Let
γ
:
I → M
be a curve. A vector field
along γ is a smooth V : I → T M such that
V (t) ∈ T
γ(t)
M
for all t ∈ I. We write
J(γ) = {vector fields along γ}.
The next thing we want to prove is that we can indeed differentiate these
things.
Lemma.
Given a linear connection
∇
and a path
γ
:
I → M
, there exists a
unique map D
t
: J(γ) → J(γ) such that
(i) D
t
(fV ) =
˙
fV + f D
t
V for all f ∈ C
∞
(I)
(ii)
If
U
is an open neighbourhood of
im
(
γ
) and
˜
V
is a vector field on
U
such
that
˜
V |
γ(t)
= V
t
for all t ∈ I, then
D
t
(V )|
t
= ∇
˙γ(0)
˜
V .
We call D
t
the covariant derivative along γ.
In general, when we have some notion on
R
n
that involves derivatives and
we want to transfer to general manifolds with connection, all we do is to replace
the usual derivative with the covariant derivative, and we will usually get the
right generalization, because this is the only way we can differentiate things on
a manifold.
Before we prove the lemma, we need to prove something about the locality
of connections:
Lemma.
Given a connection
∇
and vector fields
X, Y ∈ Vect
(
M
), the quantity
∇
X
Y |
p
depends only on the values of Y near p and the value of X at p.
Proof. It is clear from definition that this only depends on the value of X at p.
To show that it only depends on the values of
Y
near
p
, by linearity, we just
have to show that if
Y
= 0 in a neighbourhood
U
of
p
, then
∇
X
Y |
p
= 0. To do
so, we pick a bump function
χ
that is identically 1 near
p
, then
supp
(
X
)
⊆ U
.
Then χY = 0. So we have
0 = ∇
X
(χY ) = χ∇
X
(Y ) + X(χ)Y.
Evaluating at
p
, we find that
X
(
χ
)
Y
vanishes since
χ
is constant near
p
. So
∇
X
(Y ) = 0.
We now prove the existence and uniqueness of the covariant derivative.
Proof of previous lemma. We first prove uniqueness.
By a similar bump function argument, we know that D
t
V |
t
0
depends only
on values of V (t) near t
0
. Suppose that locally on a chart, we have
V (t) =
X
j
V
j
(t)
∂
∂x
j
γ(t)
for some V
j
: I → R. Then we must have
D
t
V |
t
0
=
X
j
˙
V
j
(t)
∂
∂x
j
γ(t
0
)
+
X
j
V
j
(t
0
)∇
˙γ(t
0
)
∂
∂x
j
by the Leibniz rule and the second property. But every term above is uniquely
determined. So it follows that D
t
V must be given by this formula.
To show existence, note that the above formula works locally, and then they
patch because of uniqueness.
Proposition. Any vector bundle admits a connection.
Proof.
Cover
M
by
U
α
such that
E|
U
α
is trivial. This is easy to do locally, and
then we can patch them up with partitions of unity.
Note that a connection is not a tensor, since it is not linear over
C
∞
(
M
).
However, if d
E
and
˜
d
E
are connections, then
(d
E
−
˜
d
E
)(fs) = df ⊗ s + fd
E
s − (df ⊗ s + f
˜
d
E
S) = f (d
E
−
˜
d
E
)(s).
So the difference is linear. Recall from sheet 2 that if
E, E
0
are vector bundles
and
α : Ω
0
(E) → Ω
0
(E
0
)
is a map such that
α(fs) = fα(s)
for all
s ∈
Ω
0
(
E
) and
f ∈ C
∞
(
M
), then there exists a unique bundle morphism
ξ : E → E
0
such that
α(s)|
p
= ξ(s(p)).
Applying this to
α
= d
E
−
˜
d
E
: Ω
0
(
E
)
→
Ω
1
(
E
) = Ω
0
(
E ⊗ T
∗
M
), we know
there is a unique bundle map
ξ : E → E ⊗ T
∗
M
such that
d
E
(s)|
p
=
˜
d
E
(s)|
p
+ ξ(s(p)).
So we can think of d
E
−
˜
d
E
as a bundle morphism
E → E ⊗ T
∗
M.
In other words, we have
d
E
−
˜
d
E
∈ Ω
0
(E ⊗ E
∗
⊗ T
∗
M) = Ω
1
(End(E)).
The conclusion is that the set of all connections on
E
is an affine space modelled
on Ω
1
(End(E)).
Induced connections
In many cases, having a connection on a vector bundle allows us to differentiate
many more things. Here we will note a few.
Proposition.
The map d
E
extends uniquely to d
E
: Ω
p
(
E
)
→
Ω
p+1
(
E
) such
that d
E
is linear and
d
E
(w ⊗ s) = dω ⊗ s + (−1)
p
ω ∧ d
E
s,
for
s ∈
Ω
0
(
E
) and
ω ∈
Ω
p
(
M
). Here
ω ∧
d
E
s
means we take the wedge on the
form part of d
E
s. More generally, we have a wedge product
Ω
p
(M) × Ω
q
(E) → Ω
p+q
(E)
(α, β ⊗ s) 7→ (α ∧ β) ⊗ s.
More generally, the extension satisfies
d
E
(ω ∧ ξ) = dω ∧ ξ + (−1)
q
ω ∧ d
E
ξ,
where ξ ∈ Ω
p
(E) and ω ∈ Ω
q
(M).
Proof.
The formula given already uniquely specifies the extension, since every
form is locally a sum of things of the form
ω ⊗ s
. To see this is well-defined, we
need to check that
d
E
((fω) ⊗ s) = d
E
(ω ⊗ (fs)),
and this follows from just writing the terms out using the Leibniz rule. The
second part follows similarly by writing things out for ξ = η ⊗ s.
Definition
(Induced connection on tensor product)
.
Let
E, F
be vector bundles
with connections d
E
,
d
F
respectively. The induced connection is the connection
d
E⊗F
on E ⊗ F given by
d
E⊗F
(s ⊗ t) = d
E
s ⊗ t + s ⊗ d
F
t
for s ∈ Ω
0
(E) and t ∈ Ω
0
(F ), and then extending linearly.
Definition
(Induced connection on dual bundle)
.
Let
E
be a vector bundle
with connection d
E
. Then there is an induced connection d
E
∗
on
E
∗
given by
requiring
dhs, ξi = hd
E
s, ξi + hs, d
E
∗
ξi,
for
s ∈
Ω
0
(
E
) and
ξ ∈
Ω
0
(
E
∗
). Here
h · , · i
denotes the natural pairing Ω
0
(
E
)
×
Ω
0
(E
∗
) → C
∞
(M, R).
So once we have a connection on
E
, we have an induced connection on all
tensor products of it.
Christoffel symbols
We also have a local description of the connection, known as the Christoffel
symbols.
Say we have a frame
e
1
, · · · , e
r
for
E
over
U ⊆ M
. Then any section
s ∈ Ω
0
(E|
U
) is uniquely of the form
s = s
i
e
i
,
where
s
i
∈ C
∞
(
U, R
) and we have implicit summation over repeated indices (as
we will in the whole section).
Given a connection d
E
, we write
d
E
e
i
= Θ
j
i
⊗ e
j
,
where Θ
j
i
∈ Ω
1
(U). Then we have
d
E
s = d
E
s
i
e
i
= ds
i
⊗ e
i
+ s
i
d
E
e
i
= (ds
j
+ Θ
j
i
s
i
) ⊗ e
j
.
We can write s = (s
1
, · · · , s
r
). Then we have
d
E
s = ds + Θs,
where the final multiplication is matrix multiplication.
It is common to write
d
E
= d + Θ,
where Θ is a matrix of 1-forms. It is a good idea to just view this just as a
formal equation, rather than something that actually makes mathematical sense.
Now in particular, if we have a linear connection
∇
on
T M
and coordinates
x
1
, · · · , x
n
on
U ⊆ M
, then we have a frame for
T M|
U
given by
∂
1
, · · · , ∂
n
. So
we again have
d
E
∂
i
= Θ
k
i
⊗ ∂
k
.
where Θ
k
i
∈
Ω
1
(
U
). But we don’t just have a frame for the tangent bundle, but
also the cotangent bundle. So in these coordinates, we can write
Θ
k
i
= Γ
k
`i
dx
`
,
where Γ
k
`i
∈ C
∞
(U). These Γ
k
`i
are known as the Christoffel symbols.
In this notation, we have
∇
∂
j
∂
i
= hd
E
∂
i
, ∂
j
i
= hΓ
k
`i
dx
`
⊗ ∂
k
, ∂
j
i
= Γ
k
ji
∂
k
.