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21 21.1

Connections and connection 1-forms Connections on a principal bundle

Let (P, π, M ) be a principal G-bundle. Recall that every element of A ∈ Te G gives rise to a left invariant vector field on G which we denoted by X A . However, we will now reserve this notation for a vector field on P instead. Given A ∈ Te G, we define X A ∈ Γ(T P ) by ∼

XpA : C ∞ (P ) − →R f 7→ [f (p C exp(tA))]0 (0), where the derivative is to be taken with respect to t. We also define the maps ip : Te G → Tp P A 7→ XpA , which can be shown to be a Lie algebra homomorphism. Definition. Let (P, π, M ) be a principal bundle and let p ∈ P . The vertical subspace at p is the vector subspace of Tp P given by Vp P := ker((π∗ )p ) = {Xp ∈ Tp P | (π∗ )p (Xp ) = 0}. Lemma 21.1. For all A ∈ Te G and p ∈ P , we have XpA ∈ Vp P . Proof. Since the action of G simply permutes the elements within each fibre, we have π(p) = π(p C exp(tA)), for any t. Let f ∈ C ∞ (M ) be arbitrary. Then (π∗ )p XpA (f ) = XpA (f ◦ π) = [(f ◦ π)(p C exp(tA))]0 (0) = [f (π(p))]0 (0) = 0, since f (π(p)) is constant. Hence XpA ∈ Vp P . Alternatively, one can also argue that (π∗ )p XpA is the tangent vector to a constant curve on M . ∼

In particular, the map ip : Te G − → Vp P is now a bijection. The idea of a connection is to make a choice of how to “connect” the individual points in “neighbouring” fibres in a principal fibre bundle. Definition. Let (P, π, M ) be a principal bundle and let p ∈ P . A horizontal subspace at p a vector subspace Hp P of Tp P which is complementary to Vp P , i.e. Tp P = Hp P ⊕ Vp P.

1

The choice of horizontal space at p ∈ P is not unique. However, once a choice is made, there is a unique decomposition of each Xp ∈ Tp P as Xp = hor(Xp ) + ver(Xp ), with hor(Xp ) ∈ Hp P and ver(Xp ) ∈ Vp P . Definition. A connection on a principal G-bundle (P, π, M ) is a choice of horizontal space at each p ∈ P such that i) For all g ∈ G, p ∈ P and Xp ∈ Hp P , we have (C g)∗ Xp ∈ HpCg P, where (C g)∗ is the push-forward of the map (− C g) : P → P and it is a bijection. We can also write this condition more concisely as (C g)∗ (Hp P ) = HpCg P. ii) For every smooth X ∈ Γ(T P ), the two summands in the unique decomposition X|p = hor(X|p ) + ver(X|p ) at each p ∈ P , extend to smooth hor(X), ver(X) ∈ Γ(T M ). The definition formalises the idea that the assignment of an Hp P to each p ∈ P should be “smooth” within each fibre (i) as well as between different fibres (ii). Remark 21.2. For each Xp ∈ Tp P , both hor(Xp ) and ver(Xp ) depend on the choice of Hp P . 21.2

Connection one-forms

Technically, the choice of a horizontal subspace Hp P at each p ∈ P providing a connection is conveniently encoded in the thus induced Lie-algebra-valued one-form ∼

ωp : Tp P − → Te G Xp 7→ ωp (Xp ) := i−1 p (ver(Xp )) Definition. The map ω : p → ωp sending each p ∈ P to the Te G-valued one-form ωp is called the connection one-form with respect to the connection. Remark 21.3. We have seen how to produce a one-form from a choice of horizontal spaces (i.e. a connection). The choice of horizontal spaces can be recovered from ω by Hp P = ker(ωp ). Of course, not every (Lie-algebra-valued) one-form on P is such that ker(ωp ) gives a connection on the principal bundle. What we would now like to do is to study some crucial properties of ω. We will then elevate these properties to a definition of connection oneform absent a connection, so that we may re-define the notion of connection in terms of a connection one-form.

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Lemma 21.4. For all p ∈ P , g ∈ G and A ∈ Te G, we have (Ad

(C g)∗ XpA = XpCg g

−1 )∗ A

.

Proof. Let f ∈ C ∞ (P ) be arbitrary. We have (C g)∗ XpA (f ) = XpA (f ◦ (− C g)) = [f (p C exp(tA) C g)]0 (0) = [f (p C g C g −1 C exp(tA) C g)]0 (0) = [f (p C g C (g −1 • exp(tA) • g)]0 (0) = [f (p C g C Adg−1 (exp(tA))]0 (0) = [f (p C g C exp(t(Adg−1 )∗ A)]0 (0) (Ad

= XpCg g

−1 )∗ A

(f ),

which is what we wanted. Theorem 21.5. A connection one-form ω with respect to a connection satisfies a) For all p ∈ P , we have ωp (XpA ) = A, that is ωp ◦ ip = idTe G . ip

Te G

Vp P ω p |V p P

idTe G

Te G b) ((C g)∗ ω)|p (Xp ) = (Adg−1 )∗ (ωp (Xp )) Tp P

ωp

((Cg)∗ ω)|p

Te G (Adg−1 )∗

Te G c) ω is a smooth one-form. Proof.

a) Since XpA ∈ Vp P , by definition of ω we have A −1 A ωp (XpA ) := i−1 p (ver(Xp )) = ip (Xp ) = A.

b) First observe that the left hand side is linear in Xp . Consider the two cases b.1) Suppose that Xp ∈ Vp P . Then Xp = XpA for some A ∈ Te G. Hence ((C g)∗ ω)|p (XpA ) = ωpCg ((C g)∗ XpA ) (Ad )∗ A −1 = ωpCg XpCg g = (Adg−1 )∗ A = (Adg−1 )∗ (ωp (XpA ))

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b.2) Suppose now that Xp ∈ Hp P = ker(ωp ). Then ((C g)∗ ω)|p (Xp ) = ωpCg ((C g)∗ Xp ) = 0 since (C g)∗ Xp ∈ HpCg P = ker(ωpCg ). Let Xp ∈ Tp P . We have ((C g)∗ ω)|p (Xp ) = ((C g)∗ ω)|p (ver(Xp ) + hor(Xp )) = ((C g)∗ ω)|p (ver(Xp )) + ((C g)∗ ω)|p (hor(Xp )) = (Adg−1 )∗ (ωp (ver(Xp ))) + 0 = (Adg−1 )∗ (ωp (ver(Xp ))) + (Adg−1 )∗ (ωp (hor(Xp ))) = (Adg−1 )∗ (ωp (ver(Xp ) + hor(Xp ))) = (Adg−1 )∗ (ωp (Xp )) c) We have ω = i−1 ◦ ver and both i−1 and ver are smooth.

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Connections and connection 1-forms Connections on a principal bundle

Let (P, π, M ) be a principal G-bundle. Recall that every element of A ∈ Te G gives rise to a left invariant vector field on G which we denoted by X A . However, we will now reserve this notation for a vector field on P instead. Given A ∈ Te G, we define X A ∈ Γ(T P ) by ∼

XpA : C ∞ (P ) − →R f 7→ [f (p C exp(tA))]0 (0), where the derivative is to be taken with respect to t. We also define the maps ip : Te G → Tp P A 7→ XpA , which can be shown to be a Lie algebra homomorphism. Definition. Let (P, π, M ) be a principal bundle and let p ∈ P . The vertical subspace at p is the vector subspace of Tp P given by Vp P := ker((π∗ )p ) = {Xp ∈ Tp P | (π∗ )p (Xp ) = 0}. Lemma 21.1. For all A ∈ Te G and p ∈ P , we have XpA ∈ Vp P . Proof. Since the action of G simply permutes the elements within each fibre, we have π(p) = π(p C exp(tA)), for any t. Let f ∈ C ∞ (M ) be arbitrary. Then (π∗ )p XpA (f ) = XpA (f ◦ π) = [(f ◦ π)(p C exp(tA))]0 (0) = [f (π(p))]0 (0) = 0, since f (π(p)) is constant. Hence XpA ∈ Vp P . Alternatively, one can also argue that (π∗ )p XpA is the tangent vector to a constant curve on M . ∼

In particular, the map ip : Te G − → Vp P is now a bijection. The idea of a connection is to make a choice of how to “connect” the individual points in “neighbouring” fibres in a principal fibre bundle. Definition. Let (P, π, M ) be a principal bundle and let p ∈ P . A horizontal subspace at p a vector subspace Hp P of Tp P which is complementary to Vp P , i.e. Tp P = Hp P ⊕ Vp P.

1

The choice of horizontal space at p ∈ P is not unique. However, once a choice is made, there is a unique decomposition of each Xp ∈ Tp P as Xp = hor(Xp ) + ver(Xp ), with hor(Xp ) ∈ Hp P and ver(Xp ) ∈ Vp P . Definition. A connection on a principal G-bundle (P, π, M ) is a choice of horizontal space at each p ∈ P such that i) For all g ∈ G, p ∈ P and Xp ∈ Hp P , we have (C g)∗ Xp ∈ HpCg P, where (C g)∗ is the push-forward of the map (− C g) : P → P and it is a bijection. We can also write this condition more concisely as (C g)∗ (Hp P ) = HpCg P. ii) For every smooth X ∈ Γ(T P ), the two summands in the unique decomposition X|p = hor(X|p ) + ver(X|p ) at each p ∈ P , extend to smooth hor(X), ver(X) ∈ Γ(T M ). The definition formalises the idea that the assignment of an Hp P to each p ∈ P should be “smooth” within each fibre (i) as well as between different fibres (ii). Remark 21.2. For each Xp ∈ Tp P , both hor(Xp ) and ver(Xp ) depend on the choice of Hp P . 21.2

Connection one-forms

Technically, the choice of a horizontal subspace Hp P at each p ∈ P providing a connection is conveniently encoded in the thus induced Lie-algebra-valued one-form ∼

ωp : Tp P − → Te G Xp 7→ ωp (Xp ) := i−1 p (ver(Xp )) Definition. The map ω : p → ωp sending each p ∈ P to the Te G-valued one-form ωp is called the connection one-form with respect to the connection. Remark 21.3. We have seen how to produce a one-form from a choice of horizontal spaces (i.e. a connection). The choice of horizontal spaces can be recovered from ω by Hp P = ker(ωp ). Of course, not every (Lie-algebra-valued) one-form on P is such that ker(ωp ) gives a connection on the principal bundle. What we would now like to do is to study some crucial properties of ω. We will then elevate these properties to a definition of connection oneform absent a connection, so that we may re-define the notion of connection in terms of a connection one-form.

2

Lemma 21.4. For all p ∈ P , g ∈ G and A ∈ Te G, we have (Ad

(C g)∗ XpA = XpCg g

−1 )∗ A

.

Proof. Let f ∈ C ∞ (P ) be arbitrary. We have (C g)∗ XpA (f ) = XpA (f ◦ (− C g)) = [f (p C exp(tA) C g)]0 (0) = [f (p C g C g −1 C exp(tA) C g)]0 (0) = [f (p C g C (g −1 • exp(tA) • g)]0 (0) = [f (p C g C Adg−1 (exp(tA))]0 (0) = [f (p C g C exp(t(Adg−1 )∗ A)]0 (0) (Ad

= XpCg g

−1 )∗ A

(f ),

which is what we wanted. Theorem 21.5. A connection one-form ω with respect to a connection satisfies a) For all p ∈ P , we have ωp (XpA ) = A, that is ωp ◦ ip = idTe G . ip

Te G

Vp P ω p |V p P

idTe G

Te G b) ((C g)∗ ω)|p (Xp ) = (Adg−1 )∗ (ωp (Xp )) Tp P

ωp

((Cg)∗ ω)|p

Te G (Adg−1 )∗

Te G c) ω is a smooth one-form. Proof.

a) Since XpA ∈ Vp P , by definition of ω we have A −1 A ωp (XpA ) := i−1 p (ver(Xp )) = ip (Xp ) = A.

b) First observe that the left hand side is linear in Xp . Consider the two cases b.1) Suppose that Xp ∈ Vp P . Then Xp = XpA for some A ∈ Te G. Hence ((C g)∗ ω)|p (XpA ) = ωpCg ((C g)∗ XpA ) (Ad )∗ A −1 = ωpCg XpCg g = (Adg−1 )∗ A = (Adg−1 )∗ (ωp (XpA ))

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b.2) Suppose now that Xp ∈ Hp P = ker(ωp ). Then ((C g)∗ ω)|p (Xp ) = ωpCg ((C g)∗ Xp ) = 0 since (C g)∗ Xp ∈ HpCg P = ker(ωpCg ). Let Xp ∈ Tp P . We have ((C g)∗ ω)|p (Xp ) = ((C g)∗ ω)|p (ver(Xp ) + hor(Xp )) = ((C g)∗ ω)|p (ver(Xp )) + ((C g)∗ ω)|p (hor(Xp )) = (Adg−1 )∗ (ωp (ver(Xp ))) + 0 = (Adg−1 )∗ (ωp (ver(Xp ))) + (Adg−1 )∗ (ωp (hor(Xp ))) = (Adg−1 )∗ (ωp (ver(Xp ) + hor(Xp ))) = (Adg−1 )∗ (ωp (Xp )) c) We have ω = i−1 ◦ ver and both i−1 and ver are smooth.

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