FLAT CONNECTIONS
In this note we try to prove the following statement. Let X be a
smooth manifold and π : E → X be a smooth vector bundle, equipped
with a connection whose curvature form vanishes. Then we can choose
trivialisations of E such that the transition functions are locally constant. The main ingredient is the Frobenius theorem on existence of
integral submanifolds to involutive distributions.
Definition 1. A connection D is a C linear map D : Γ(U, E) →
Γ(U, E) ⊗ Γ(U, ΩU ) defined for every open set U ⊂ X satisfying Leibniz
rule, i.e. for f ∈ Γ(U, OX ), D(f s) = s ⊗ df + f D(s).
Assume that rank(E) = n and choose a frame over an open subset U over which E P
has a trivialisation. Say the frame is given by
U
. This matrix of 1 forms is called
e1 , · · · , en . D(ei ) = nj=1 ej ⊗ ωji
the connection matrix for this trivialisation. Suppose V is another
open subset over which E is trivial, and we have chosen a frame
f1 , · · · , fn over V , thenPwe may ask what is the relation between
n
g f . Then on the one hand, D(ei ) =
ω U and ω V . Let ei =
Pn Pn j=1 ji j U Pn
Pn
U
g f ⊗ω = k=1 (gω U )ki fk . On the other
j=1
j=1 ej ⊗ωji =
P
Pn k=1 kj k Pjin
V
) =
(dgji fj + nk=1 fk ⊗ gji ωkj
hand, D(ei ) = D( j=1 gji fj ) =
j=1
Pn
V
U
k=1 (dgki + (ω g)ki )fk . Equating the coefficients of fk , we get gω =
dg + ω V g, i.e.
ω V = gω U g −1 − dgg −1
We can construct a principal GLn bundle from E. Consider the
π
bundle Hom(Cn , E) → X. The open subset Iso(Cn , E) is the required
GLn bundle. GLn acts on it naturally on the right, the action beπ
ing given by (φ, g) 7→ φ ◦ g. Let us call this bundle P → U . If
U is an open subset over which E is trivial, once we choose a frame
e1 , · · · , en , we can identify P |U ∼
= U × G, where G = GLn . This
identification is given as follows, let wi denote the standard basis for
Cn . An element of P is P
a pair (x, φ) with φ : Cn → Ex an isomorn
phism. Then φ(wi ) =
j=1 Aji ej (x). We send the element (x, φ)
to the pair (x, A). It is clear that (x, φ ◦ g) 7→ (x, Ag). If V is
another open set over which E is trivial with frame f1 , · · · , fn , then
P |V ∼
∈ U ∩ V , and letPφ : Cn →
= V × G. Let x P
PnEx be an isomorn
n
U
U
phism. Then φ(wi ) = j=1 Aji ej (x) = j=1 Aji ( k=1 gkj (x)fk (x)) =
1
Pn
P
= nk=1 AVki fk (x). This shows that gAU = AV .
Thus, the point (x, A) ∈ U × G is identified with the point (x, gA) ∈
V × G.
On the space P , we have a natural map of bundles, namely, T P →
∗
π T X → 0, which comes from the operation of pushing forward tangent
vectors. Let us describe this map locally. Let U, ei be a frame as above.
Then P |U ∼
= U × G and the above map is the projection T U ⊕ T G →
T U . Since G acts on P on the right, it acts on the tangent bundle
and this action has the following local description. Let (vx , wh ) be a
tangent vector to P at the point (x, h). Then for g ∈ G, this gets
mapped to the tangent vector (vx , (Rg )∗ (wh )) at the point (x, hg). It
is clear that the map T P → π ∗ T X is G equivariant, where G acts
trivially on π ∗ T X.
Let us describe the kernel of the above map. If G is a Lie group
acting on a manifold M on the right, then to each element ξ ∈ g
we can associate
a vector
field Xξ on M . If p ∈ M , then define
d
Xξ (p) = dt t=0 p · exp(tξ) . From this, it is clear that (Rg )∗ (Xξ (p)) =
XAdg−1 (ξ) (p · g), where Rg is the action of g on M . Since G acts on the
right of P , we can use this construction to produce vector fields, i.e.
global sections of T P , on P . This defines a map from P ×g → T P . Let
us check that this map is G equivariant for the right action on P × g
given by (p, ξ) 7→ (p · g, Adg−1 (ξ)). But this is clear from the preceding
formula.
We have shown that there is a G equivariant short exact sequence of
bundles
0 → P × g → T P → π∗T X → 0
U
k=1 (g(x)A )ki fk (x)
Definition 2. A connection D on P is a G equivariant splitting of the
above short exact sequence.
Let us analyse this splitting locally. We will freely use the natural
identification of g with Te G. Over the point (x, g) ∈ U × G, the above
short exact sequence is 0 → Te G → Tx U ⊕ Tg G → Tx U → 0. Let
sU : Tx U ⊕ Tg G → Te G be the splitting. If w ∈ Tg G, then the G
equivariance of sU forces that Adg (sU (v, w)) = sU ((Rg−1 )∗ (v, w)) =
sU (v, (Rg−1 )∗ (w)) = sU (v, 0) + (Rg−1 )∗ (w). Thus, locally giving the
splitting is same as giving a map Tx U → Te G. which is same as giving
a matrix of 1-forms, call it ω U . What is the relation between ω U and
ω V ? Let W := U ∩ V . Then g : W → G is smooth. Let us compute
the differential of the map W × G → W × G given by (x, h) 7→ (x, gh).
This map is the composite of (x, h) 7→ (x, g, h) 7→ (x, gh). If (v, w) is a
tangent vector at (x, h), it first gets mapped to (v, dg(v), w) and then to
(v, (Rh )∗ (dg(v)) + (Lg )∗ (w)). This calculation shows that (v, 0) at the
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point (x, e) is mapped to the tangent vector (v, (Re )∗ dg(v)) = (v, dg(v))
at the point (x, g). Thus,
sU (v, 0) = sV (v, dg(v)) = Adg−1 (sV (v, (Rg−1 )∗ (dg(v)))
= Adg−1 (sV (v, 0)) + Adg−1 ((Rg−1 )∗ (dg(v)))
In the case of GLn , we may rewrite this as ω V = g(ω U )g −1 − dgg −1 .
This is exactly the condition for giving a connection on the vector
bundle E. Thus, both notions are equivalent.
Definition 3. The curvature of a connection D is the composite map
E → E ⊗ ΩX → E ⊗ Ω2X → E ⊗ ∧2 ΩX . It is denoted by Θ.
n
X
Θ(ei ) = D (ei ) = D(
ej ⊗ ωji )
2
j=1
n
n
X
X
ek ⊗ ωkj ∧ ωji )
=
(ej ⊗ dωji +
=
j=1
n
X
k=1
ek ⊗ (dω + ω ∧ ω)ki
k=1
Θ(f ei ) = D2 (f ei ) = D(ei ⊗ df +
n
X
f ej ⊗ ωji )
j=1
=
n
X
(ej ⊗ ωji ∧ df + ej ⊗ df ∧ ωji + f D(ej ⊗ ωji ))
j=1
n
X
=f
ek ⊗ (dω + ω ∧ ω)ki = f Θ(ei )
k=1
The above calculations show that Θ ∈ Γ(X, End(E) ⊗ ∧2 ΩX ), and
locally after choosing frames, it is given by dω + ω ∧ ω.
Theorem 1. If Θ = 0, then we can choose the transition functions to
be locally constant.
Proof. Let us denote by s the splitting T P → B and restrict our attention to what happens over the open subset U . Recall that over
the point (x, g) ∈ U × G, the short exact sequence above looks like
0 → Te G → Tx U ⊕ Tg G → Tx U → 0. If we take the kernel of the
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splitting, then we get d := dim(X) linearly independent tangent vectors. This happens for every point (x, g), which means that we have a
d dimensional distribution on the manifold U × G. This distribution
occurs as the tangent bundle of a submanifold iff it is involutive, i.e.
closed under the Lie bracket. We now proceed to check this in steps
1 : Let (v, u) be a tangent vector at the point (x, e). Let ω be defined
by s(v, u) = ω(v) + u, where s is the splitting.
2 : We could have chosen U to be a coordinate neighbourhood with
coordinates x1 , · · · , xd . Let X = X1 = ∂x∂ 1 be a vector field on U .
Using this, we define a tangent vector at points (x, e) by Xx − ω(Xx ).
Now using the right action of G define tangent vectors on the point
(x, g) by Xx − (Rg )∗ (ω(Xx )). It is clear that when we do this for all
Xi , these give a basis for the kernel of s at every point (x, g).
3 : Suppose Xi are a bunch of independent vector fields on a manifold
M , then to check that the distribution generated by them is involutive,
it is enough to check that [Xi , Xj ] is in the distribution. This is because
[f Xi , gXj ] = f Xi (g)Xj − gXj (f )Xi + f g[Xi , Xj ].
4 : The vector fields on U × G given by X1 − ω(X1 ) are invariant
under the right action of G by construction. Thus, the Lie bracket
of any two of these will also be G invariant. Hence, it is enough to
compute the value of the Lie bracket at the point (x, e). Let η, ξ be
two left invariant vector fields on G. Let Xη , Xξ be right invariant
vector fields given by Xη (e) = η(e), Xξ (e) = ξ(e). Then the vector field
[Xη , Xξ ] is right invariant and [Xη , Xξ ](e) = [ξ, η](e). The relevance of
this remark here is that ω(Xi ) constructed are right invariant.
[X1 − ω(X1 ), X2 − ω(X2 )] = [X1 , X2 ] − [ω(X1 ), X2 ] − [X1 , ω(X2 )]
+ [ω(X1 ), ω(X2 )]
= X2 (ω(X1 )) − X1 (ω(X2 ))+
ω(X2 )ω(X1 ) − ω(X1 )ω(X2 )
= −dω(X1 , X2 ) + ω([X1 , X2 ])
− ω ∧ ω(X1 , X2 )
= −(dω + ω ∧ ω)(X1 , X2 ) = 0
In the above calculation, we have used the following formula
dω(X1 , X2 ) = X1 (ω(X2 )) − X2 (ω(X1 )) + ω([X1 , X2 ])
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Choose an integral submanifold of this distribution through the point
(x0 , e). Call this subset V . If p ∈ V , then Tp V → Tπ(p) U is an isomorphism. Let W := π(V ). If w = π(v), then sending w 7→ v defines a
section to W × G → W . Use this section to give a new trivialisation
π̃ : W × G → W . By construction, the connection matrix in this trivialisation is 0. Doing this locally over the whole manifold X, we would
have gotten trivialisations such that the connection matrix is always
0. ω V = gω U g −1 − dgg −1 shows that dgg −1 = 0, which means dg = 0,
which means g is locally constant.
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