A Characterization of Invariant Connections

A Characterization of Invariant Connections

Maximilian HANUSCH

Department of Mathematics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany E-mail: mhanusch@math.upb.de

Received December 09, 2013, in final form March 10, 2014; Published online March 15, 2014 http://dx.doi.org/10.3842/SIGMA.2014.025

Abstract. Given a principal fibre bundle with structure group S, and a fibre transitive Lie groupGof automorphisms thereon, Wang’s theorem identifies the invariant connections with certain linear maps ψ: g → s. In the present paper, we prove an extension of this theorem which applies to the general situation where G acts non-transitively on the base manifold. We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies to the situation where G admits only one orbit type. Along the way, we give applications to loop quantum gravity.

Key words: invariant connections; principal fibre bundles; loop quantum gravity; symmetry reduction

2010 Mathematics Subject Classification: 22F50; 53C05; 53C80; 83C45

1 Introduction

The set of connections on a principal fibre bundle (P, π, M, S) is closed under pullback by auto- morphisms, and it is natural to search for connections that do not change under this operation.

Especially, connections invariant under a Lie group (G,Φ) of automorphisms are of particu- lar interest as they reflect the symmetry of the whole group and, for this reason, find their applications in the symmetry reduction of (quantum) gauge field theories [1, 4, 5]. The first classification theorem for such connections was given by Wang [8], cf. Case 5.7. This applies to the case where the induced action¹ ϕ acts transitively on the base manifold and states that each point in the bundle gives rise to a bijection between the set of Φ-invariant connections and certain linear mapsψ:g→s. In [6] the authors generalize this to the situation whereϕadmits only one orbit type. More precisely, they discuss a variation² of the case where the bundle admits a submanifold P₀ with π(P₀) intersecting each ϕ-orbit in a unique point, see Case 4.5 and Example 4.6. Here, the Φ-invariant connections are in bijection with such smooth maps ψ: g×P₀ → s for which the restrictions ψ|_g×Tp

0P0 are linear for all p₀ ∈ P₀, and that fulfil additional consistency conditions.

Now, in the general case we consider Φ-coverings ofP. These are families{P_α}_α∈I of immer- sed submanifolds³ P_α of P such that each ϕ-orbit has non-empty intersection with S

α∈Iπ(P_α) and for which

TpP =TpPα+ deΦp(g) +T vpP

holds whenever p ∈P_α for some α ∈ I. Here, T v_pP ⊆ T_pP denotes the vertical tangent space at p ∈ P and e the identity in G. Observe that the intersection properties of the sets π(Pα)

1Each Lie group of automorphisms of a bundle induces a smooth action on the base manifold.

2Amongst others, they assume theϕ-stabilizer ofπ(p0) to be the same for allp0∈P0.

3For the moment, assume that Pα ⊆ P is a subset which, at the same time, is a manifold such that the inclusion mapια:Pα→P is an immersion. Here, we tacitly identifyTp_αPα with im[dp_αια]. Note that we do not requirePαto be an embedded submanifold ofP. For details, see Convention3.1.

with the ϕ- orbits in the base manifold need not to be convenient in any sense. Indeed, here one might think of situations in which ϕ admits dense orbits, or of the almost-fibre transitive case, cf. Case 5.4.

Let Θ : (G×S)×P → P be defined by ((g, s), p) 7→ Φ(g, p)·s⁻¹ for (G,Φ) a Lie group of automorphisms of (P, π, M, S). Then, the main result of the present paper can be stated as follows:

Theorem. EachΦ-covering{P_α}_α∈I ofP gives rise to a bijection between the Φ-invariant con- nections on P and the families{ψ_α}α∈I of smooth mapsψ_α:g×T P_α→s for whichψ_α|g×T_pαPα

is linear for all pα∈Pα, and that fulfil the following two (generalized Wang) conditions:

• eg(p_β) +w~p_β −s(pe _β) = dLqw~pα =⇒ ψ_β(~g, ~wp_β)−~s=ρ(q)◦ψα ~0g, ~wpα

,

• ψβ Adq(~g), ~0pβ

=ρ(q)◦ψα ~g, ~0pα

with ρ(q) := Ads and Adq(~g) := Adg(~g) for q = (g, s)∈Q.

Here, eg and sedenote the fundamental vector fields that correspond to the elements ~g ∈ g and ~s ∈ s, respectively; and of course we have ~0_pα, ~w_pα ∈ T_pαP_α, ~0_pβ, ~w_pβ ∈ T_pβP_β as well as p_β =q·p_α for p_α∈P_α, p_β ∈P_β.

Using this theorem, the calculation of invariant connections reduces to identifying a Φ- covering which makes the above conditions as easy as possible. Here, one basically has to find the balance between quantity and complexity of these conditions. Of course, the more sub- manifolds there are, the more conditions we have, so that usually it is convenient to use as few of them as possible. For instance, in the situation whereϕis transitive, it suggests itself to choose a Φ-covering that consists of one single point; which, in turn, has to be chosen appropriately.

Also if there is some m ∈ M contained in the closure of each ϕ-orbit, one single submanifold is sufficient, see Case 5.4 and Example 5.5. The same example also shows that sometimes pointwise⁴ evaluation of the above conditions proves non-existence of Φ-invariant connections.

In any case, one can use the inverse function theorem to construct a Φ-covering{P_α}_α∈I ofP such that the submanifolds P_α have minimal dimension in a certain sense, see Lemma 3.4 and Corollary 5.1. This reproduces the description of connections by means of local 1-forms on M provided that Gacts trivially or, more generally, via gauge transformations onP, see Case5.2.

Finally, since orbit structures can depend very sensitively on the action or the group, one cannot expect to have a general concept for finding the Φ-covering optimal for calculations.

Indeed, sometimes these calculations become easier if one uses coverings that seem less optimal at a first sight (as, e.g., if they have no minimal dimension, cf. calculations in Appendix B.2).

The present paper is organized as follows: In Section 2, we fix the notations. In Section 3, we introduce the notion of a Φ-covering, the central object of this paper. In Section 4, we prove the main theorem and deduce a slightly more general version of the result from [6]. In Section5, we show how to construct Φ-coverings to be used in special situations. In particular, we consider the (almost) fibre transitive case, trivial principal fibre bundles and Lie groups of gauge transformations. Along the way, we give applications to loop quantum gravity.

2 Preliminaries

We start with fixing the notations.

4Here, pointwise means to consider such elementsq ∈ G×S that are contained in the Θ-stabilizer of some fixedpα∈Pαforα∈I.

2.1 Notations

Manifolds are always assumed to be smooth. IfM,N are manifolds andf:M →N is a smooth map, then df:T M →T N denotes the differential map between their tangent manifolds. The map f is said to be an immersion iff for each x ∈ M the restriction dxf := df|_TxM:TxM → T_f(x)N is injective.

Let V be a finite dimensional vector space. A V-valued 1-form ω on the manifold N is a smooth mapω:T N →V whose restriction ωy :=ω|_TyN is linear for ally∈N. The pullback of ω by f is the V-valued 1-formf^∗ω:T M →V,~vx→ω_f(x)(dxf(~vx)).

LetGbe a Lie group with Lie algebrag. Forg∈G, we define the corresponding conjugation map by αg:G → G, h 7→ ghg⁻¹. Its differential deαg:g → g at the unit element e ∈ G is denoted by Ad_g in the following.

Let Ψ be a (left) action of the Lie groupG on the manifold M. For g ∈G and x∈ M, we define Ψg:M →M, Ψg:y 7→Ψ(g, y) and Ψx:G→M,h7→Ψ(h, x), respectively. If it is clear which action is meant, we will often writeL_ginstead of Ψ_gas well asg·yorgyinstead of Ψ_g(y).

For~g∈g and x∈M, the map eg(x) := _dt^d

t=0Ψx(exp(t~g))

is called thefundamental vector field of~g. The Lie subgroupG_x :=

g∈G

g·x=x is called thestabilizer ofx∈M (w.r.t. Ψ), and its Lie algebra g_x equals ker[dxΨ], see e.g. [3]. Theorbit of x underG is the setGx:= im[Ψx]. Ψ is said to betransitive iff Gx=M holds for one (and then each) x∈M. Analogous conventions we also use for right actions.

2.2 Invariant connections

Letπ:P →M be a smooth map between manifoldsP andM, and denote byFx:=π⁻¹(x)⊆P the fibre overx∈M inP. Moreover, letSbe a a Lie group that acts viaR:P×S→P from the right onP. If there is an open covering{U_α}_α∈IofM and a family{φ_α}_α∈I of diffeomorphisms φα:π⁻¹(Uα)→Uα×S with

φα(p·s) = π(p),[pr₂◦φα](p)·s

∀p∈π⁻¹(Uα), ∀s∈S, (2.1) then (P, π, M, S) is called principal fibre bundle with total space P, projection map π, base manifold M and structure groupS. Here, pr₂ denotes the projection onto the second factor. It follows from (2.1) thatπ is surjective, and that:

• Rs(Fx)⊆Fx for all x∈M and all s∈S,

• for each x∈M the map R_x:F_x×S→F_x, (p, s)7→p·sis transitive and free.

The subspaceT v_pP := ker[d_pπ]⊆T_pP is calledvertical tangent space atp∈P and es(p) := _dt^d

t=0p·exp(t~s)∈T v_pP ∀p∈P,

denotes the fundamental vector field of~s w.r.t. the right action ofS on P. The map s 3~s→ es(p)∈T vpP is a vector space isomorphism for allp∈P.

Complementary to that, aconnection ω is ans-valued 1-form on P with

• R^∗_sω= Ad_s⁻¹ ◦ω ∀s∈S,

• ωp(es(p)) =~s ∀~s∈s.

The subspaceT hpP := ker[ωp]⊆TpP is called thehorizontal tangent space atp (w.r.t.ω). We have dR_s(T h_pP) =T hp·sP for alls∈S, and one can show thatT_pP =T v_pP ⊕T h_pP holds for all p∈P.

A diffeomorphismκ:P →P is said to be anautomorphism iffκ(p·s) =κ(p)·sholds for all p∈P and alls∈S. It is straightforward to see that an s-valued 1-formω on P is a connection iff this is true for the pullbackκ^∗ω. ALie group of automorphisms (G,Φ) ofP is a Lie groupG together with a left action Φ of G on P such that the map Φg is an automorphism for each g ∈G. This is equivalent to say that Φ(g, p·s) = Φ(g, p)·s holds for allp ∈P, g∈G and all s∈S. In this situation, we will often writegps instead of (g·p)·s=g·(p·s). Each such a left action Φ gives rise to two further actions:

• The induced action ϕis defined by ϕ: G×M →M,

(g, m)7→(π◦Φ)(g, pm), (2.2)

wherepm∈π⁻¹(m) is arbitrary. Φ is calledfibre transitive iffϕis transitive.

• We equipQ=G×S with the canonical Lie group structure and define [8]

Θ : Q×P →P,

((g, s), p)7→Φ g, p·s⁻¹

. (2.3)

A connection ω is said to be Φ-invariant iff Φ^∗_gω =ω holds for all g∈G. This is equivalent to require that for each p ∈ P and g ∈ G the differential dpLg induces an isomorphism between the horizontal tangent spaces T h_pP andT h_gpP.⁵

We conclude this subsection with the following straightforward facts, see also [8]:

• Consider the representation ρ:Q→ Aut(s), (g, s) 7→ Ad_s. Then, it is straightforward to see that each Φ-invariant connectionω is of typeρ, i.e.,ωis an s-valued 1-form onP with L^∗_qω =ρ(q)◦ω for all q∈Q.

• An s-valued 1-form ω onP withω(s(p)) =e ~sfor all~s∈s is a Φ-invariant connection iff it is of type ρ.

• Let Q_p denote the stabilizer ofp ∈ P w.r.t. Θ, and G_π(p) the stabilizer of π(p) w.r.t. ϕ.

Then,G_π(p) =

h∈G|Lh:F_π(p)→F_π(p) , and we obtain a Lie group homomorphism φ_p:G_π(p)→S by requiring that Φ(h, p) =p·φ_p(h) for all h∈G_π(p). If q_p and g_π(p) denote the Lie algebras of Qp and G_π(p), respectively, then

Q_p ={(h, φ_p(h))|h∈G_π(p)} and q_p = ~h,d_eφ_p ~h ~h∈g_π(p) . (2.4)

3 Φ-coverings

We start this section with some facts and conventions concerning submanifolds. Then, we provide the definition of a Φ-covering and discuss some its properties.

Convention 3.1. LetM be a manifold.

1. A pair (N, τ_N) consisting of a manifoldN and an injective immersionτ:N →M is called submanifold ofM.

5In literature sometimes the latter condition is used to define Φ-invariance of connections.

2. If (N, τN) is a submanifold of M, we tacitly identify N and T N with their images τ_N(N)⊆M and dτ_N(T N)⊆T M, respectively. In particular, this means that:

• If M⁰ is a manifold andκ:M →M⁰ a smooth map, then for x∈N and~v∈T N we write κ(x) and dκ(~v) instead of κ(τN(x)) and dκ(dτ(~v)), respectively.

• If Ψ :G×M →M is a left action of the Lie groupGand (H, τ_H) a submanifold ofG, the restriction of Ψ to H×N is defined by

Ψ|_H×N(h, x) := Ψ(τH(h), τN(x)) ∀(h, x)∈H×N.

• If ω:T M →V is aV-valued 1-form on M, we let

(Ψ^∗ω)|_{T G×T N}(m, ~~ v) := (Ψ^∗ω)(m,~ dτ(~v)) ∀(m, ~~ v)∈T G×T N.

• We will not explicitly refer to the maps τ_N and τ_H in the following.

3. Open subsets U ⊆ M are equipped with the canonical manifold structure making the inclusion map an embedding.

4. IfL is a submanifold ofN, and N is a submanifold ofM, we considerL as a submanifold of M in the canonical way.

Definition 3.2. A submanifold N ⊆M is called Ψ-patch iff for each x ∈N we find an open neighbourhood N⁰ ⊆ N of x and a submanifold H of G through e, such that the restriction Ψ|_H×N⁰ is a diffeomorphism to an open subset U ⊆M.

Remark 3.3.

1. It follows from the inverse function theorem and⁶

d_(e,x)Ψ(g×TxN) = deΨx(g) + dxΨe(TxN) = deΨx(g) +TxN ∀x∈N that N is a Ψ-patch iff TxM = deΨx(g) +TxN holds for allx∈N.⁷

2. Open subsets U ⊆ M are always Ψ-patches. They are of maximal dimension, which, for instance, is necessary if there is a point inU whose stabilizer equalsG, see Lemma 3.4.1.

3. We allow zero-dimensional patches, i.e., N ={x} for some x ∈ M. Necessarily, then we have deΨx(g) =TxM as well as Ψ|_H×N = Ψx|_H for each submanifoldH ofG.

The second part of the following elementary lemma equals Lemma 2.1.1 in [3].

Lemma 3.4. Let (G,Ψ) be a Lie group that acts on the manifold M, and letx∈M.

1. If N is a Ψ-patch withx∈N, then dim[N]≥dim[M]−dim[G] + dim[Gx].

2. Let V and W be algebraic complements of deΨx(g) in TxM and of g_x in g, respectively.

Then there are submanifoldsN ofM throughx andH ofGthroughesuch thatT_xN =V, TeH =W. In particular, N is a Ψ-patch and dim[N] = dim[M]−dim[G] + dim[Gx].

6The sum is not necessarily direct.

7In fact, let V ⊆deΨx(g) be an algebraic complement ofTxN inTxM and V⁰ ⊆g a linear subspace with dim[V⁰] = dim[V] and deΨx(V⁰) =V. Then, we find a submanifoldH ofGthroughewithTeH =V⁰, so that d(e,x)Ψ :TeH×TxN→TxM is bijective.

Proof . 1. By Remark 3.3.1 and since ker[deΨx] =g_x, we have

dim[M]≤dim[d_eΨ_x(g)] + dim[T_xN] = dim[G]−dim[G_x] + dim[N]. (3.1) 2. Of course, we find submanifolds N⁰ of M through x and H⁰ of G through e such that TxN⁰ =V and TeH⁰ =W. So, if~g∈g and ~vx ∈TxN⁰, then 0 = d_(e,x)Ψ(~g, ~vx) = deΨx(~g) +~vx

implies d_eΨ_x(~g) = 0 and ~v_x = 0. Hence, ~g ∈ ker[d_eΨ_x] = g_x, so that⁸ d_(e,x)Ψ|_TeH⁰_×TeN⁰ is injective. It is immediate from the definitions that this map is surjective, so that by the inverse function theorem we find open neighbourhoodsN ⊆N⁰ ofx and H⊆Gof esuch that Ψ|_H×N is a diffeomorphism to an open subsetU ⊆M. ThenN is a Ψ-patch, and since in (3.1) equality

holds, also the last claim is clear.

Definition 3.5. Let (G,Φ) be a Lie group of automorphisms of the principal fibre bun- dle P, and recall the actions ϕ and Θ defined by (2.2) and (2.3), respectively. A family of Θ-patches {P_α}_α∈I is said to be a Φ-covering ofP iff eachϕ-orbit intersects at least one of the setsπ(P_α).

Remark 3.6.

1. If O⊆P is a Θ-patch, Lemma3.4.1 and (2.4) yield

dim[O]≥dim[P]−dim[Q] + dim[Q_p]^(2.4)= dim[M]−dim[G] + dim[G_π(p)].

2. It follows from Remark 3.3.1 and d_eΘ_p(q) = d_eΦ_p(g) +T v_pP thatO is a Θ-patch iff

TpP =TpO+ deΦp(g) +T vpP ∀p∈O. (3.2)

As a consequence,

• each Φ-patch is a Θ-patch,

• P is always a Φ-covering by itself. Moreover, ifP =M×S is trivial, then M× {e} is a Φ-covering.

3. If N is a ϕ-patch and s₀:N → P a smooth section (i.e., π◦s₀ = id_N), then s₀(N) is a Θ-patch by Lemma 3.7.2.

Conversely, if N ⊆ M is a submanifold such that s₀(N) is a Θ-patch for s₀ as above, then N is a ϕ-patch. In fact, applying dπ to (3.2), this is immediate from Remark 3.3.1 and the definition of ϕ.

Lemma 3.7. Let (G,Φ) be a Lie group of automorphisms of the principal bundle(P, π, M, S).

1. IfO⊆P is aΘ-patch, then for eachp∈Oandq ∈Qthe differentiald_(q,p)Θ : T_qQ×T_pO→ Tq·pP is surjective.

2. If N is a ϕ-patch ands₀:N →P a smooth section, then s₀(N) is a Θ-patch.

Proof . 1. Since O is a Θ-patch, the claim is clear for q = e. If q is arbitrary, then for each

~

mq∈TqQwe find some ~q∈qsuch that m~q= dLq~q. Consequently, forw~p∈TpP we have d_(q,p)Θ (m~_q, ~w_p) = d_(q,p)Θ(dL_q~q, ~w_p) = d_pL_q d_(e,p)Θ(~q, ~w_p)

. So, since left translation w.r.t. Θ is a diffeomorphism, d_pL_q is surjective.

8Recall that d(e,x)Ψ|T_eH⁰×T_eN⁰: ~h, ~vx

7→d(e,x)Ψ deτH(~h),dxτN(~vx) .

2. O := s0(N) is a submanifold of P because s0 is an injective immersion. Thus, by Re- mark 3.6.2 it suffices to show that

dim

T_s0(x)O+ d_eΦ_s0(x)(g) +T v_s0(x)P

≥dim[T_s0(x)P] ∀x∈N.

For this, let x∈N andV⁰⊆gbe a linear subspace withV⁰⊕g_x and T_xM =T_xN⊕d_eϕ_x(V⁰).

Then, we haveT_s0(x)O⊕deΦ_s0(x)(V⁰)⊕T v_s0(x)P because if dxs0(~vx) + deΦ_s0(x)(~g⁰) +~vv = 0 for

~

v_x ∈T_xN,~g⁰ ∈V⁰ and ~v_v ∈T v_s0(x)P,

0 = d_s0(x)π dxs0(~vx) + deΦ_s0(x)(~g⁰) +~vv

=~vx⊕deϕx(~g⁰)

shows~v_x= 0 and d_eφ_x(~g⁰) = 0, hence~g⁰= 0 by the choice of V⁰, i.e., ~v_v= 0 by assumption. In particular, d_eφ_x(~g⁰) = 0 if d_eΦ_s0(x)(~g⁰) = 0, hence dim[d_eΦ_s0(x)(V⁰)]≥dim[d_eϕ_x(V⁰)], from which we obtain

dim

T_s0(x)O+ deΦ_s0(x)(g) +T v_s0(x)P

≥dim

T_s0(x)O⊕deΦ_s0(x)(V⁰)⊕T v_s0(x)P

= dim[T_xN] + dim[d_eΦ_s0(x)(V⁰)] + dim[S]≥dim[T_xN] + dim[d_eϕ_x(V⁰)] + dim[S]

= dim[P].

4 Characterization of invariant connections

In this section, we will use Φ-coverings {P_α}_α∈I of the bundle P in order to characterize the set of Φ-invariant connections by families {ψ_α}α∈I of smooth maps ψα: g×T Pα → s whose restrictions ψα|_g×TpαPα are linear and that fulfil two additional compatibility conditions. Here, we will follow the lines of Wang’s original approach, which basically means that we generalize the proofs from [8] to the non-transitive case. We will proceed in two steps, the first one being performed in Subsection 4.1. There, we show that a Φ-invariant connection gives rise to a consistent family {ψ_α}_α∈I of smooth maps as described above. We also discuss the situation in [6] in order to make the two conditions more intuitive. Then, in Subsection4.2, we will verify that such families{ψ_α}_α∈I glue together to a Φ-invariant connection on P.

4.1 Reduction of invariant connections

In the following, let {P_α}_α∈I be a fixed Φ-covering ofP and ω a Φ-invariant connection on P. We define

ω_α := (Θ^∗ω)|_{T Q×T Pα} as well as ψ_α :=ω_α|_{g×T Pα}, and for q⁰ ∈Q we letα_q⁰:Q×P →Q×P, (q, p)7→ α_q⁰(q), p

. Finally, we define Ad_q(~g) := Ad_g(~g) ∀q = (g, s)∈Q, ∀~g∈g.

Lemma 4.1. Let q∈Q, pα∈Pα, pβ ∈Pβ with⁹ pβ =q·pα andw~pα ∈TpαPα. Then 1) ωβ(~η) =ρ(q)◦ωα(~0q, ~wpα) for all~η∈T Q×T Pβ with dΘ(~η) = dLqw~pα,

2) α^∗_qω_β

~ m, ~0p_β

=ρ(q)◦ωα m, ~~ 0pα

for allm~ ∈T Q.

Proof . 1. Let ~η ∈ T_q⁰Q×TpPβ for q⁰ ∈Q. Then, since¹⁰ L^∗_qω =ρ(q)◦ω for each q ∈Q and q⁰·p=q·p_α =p_β, we have

ωβ(~η) =ωq⁰·p(d_(q⁰_,p)Θ(~η)) =ωpβ(dLqw~pα) = (L^∗_qω)pα(w~pα)

=ρ(q)◦ω_pα(w~_pα) =ρ(q)◦ω_pα d_(e,pα)Θ ~0_q, ~w_pα

=ρ(q)◦ω_α ~0_q, ~w_pα .

9Recall that, by Convention3.1, this actually meansτP_β(pβ) =q·τPα(pα).

10See end of Subsection2.2.

2. Form~_q⁰ ∈T_q⁰Qlet γ: (−, )→Q be smooth with ˙γ(0) =m~_q⁰. Then α^∗_qωβ

(q⁰,pβ) m~q⁰, ~0pβ

=ωβ(αq(q⁰),p_β) Adq(m~q⁰), ~0pβ

=ω_qq⁰_q⁻¹_{q·pα dt}^d

t=0qγ(t)q⁻¹q·pα

= L^∗_qω

q⁰·pα

d dt

t=0γ(t)·pα

=ρ(q)◦ωq⁰·pα d_(q⁰_,pα)Θ m~q⁰

=ρ(q)◦ω_α(q⁰_,pα) m~_q⁰, ~0_pα

.

Corollary 4.2. Let q ∈ Q, p_α ∈ P_α, p_β ∈ P_β with p_β = q·p_α and w~_pα ∈ T_pαP_α. Then, for

~

w_pβ ∈T_pβP_β,~g∈g and~s∈s we have

i) eg(p_β) +w~_pβ −s(pe _β) = dL_qw~_pα =⇒ ψ_β(~g, ~w_pβ)−~s=ρ(q)◦ψ_α ~0_g, ~w_pα , ii) ψβ Adq(~g), ~0pβ

=ρ(q)◦ψα ~g, ~0pα

.

Proof . i) In general, forw~p∈TpP,~g∈gand ~s∈s we have

d_(e,p)Θ((~g, ~s), ~w_p) = d_(e,p)Φ(~g, ~w_p)−es(p) =eg(p) +w~_p−es(p) (4.1) and, since ω is a connection, for ((~g, ~s), ~w_pα)∈q×T P_α we obtain

ω_α((~g, ~s), ~w_pα) =ω d_(e,pα)Φ(~g, ~w_pα)−es(p_α)

=ω d_(e,pα)Φ(~g, ~w_pα)

−~s

=ωα(~g, ~wpα)−~s=ψα(~g, ~wpα)−~s. (4.2) Now, assume that deΦp_β(~g)+w~p_β−es(p) = dLqw~pα. Then d_(e,pβ)Θ((~g, ~s), ~wp_β) = dLqw~pαby (4.1) so that ωβ((~g, ~s), ~wpβ) =ρ(q)◦ωα ~0g, ~wpα

by Lemma4.1.1. Consequently, ψ_β ~g, ~w_pβ

−~s^(4.2)= ω_β((~g, ~s), ~w_pβ) =ρ(q)◦ω_α ~0_q, ~w_pα(4.2)

= ρ(q)◦ψ_α ~0_g, ~w_pα . ii) Lemma4.1.2 yields

ψ_β Ad_q(~g), ~0_pβ

= (α^∗_qω_β)_(e,pβ) ~g, ~0_pβ

=ρ(q)◦(ω_α)_(e,pα) ~g, ~0_pα

=ρ(q)◦ψ_α ~g, ~0_pα

.

Definition 4.3 (reduced connection). A family {ψ_α}_α∈I of smooth maps ψα: g×T Pα → s which are linear in the sense thatψ_α|_g×TpαPα is linear for allp_α∈P_αis called reduced connection w.r.t. {P_α}α∈I iff it fulfils the conditionsi) andii) from Corollary4.2.

Remark 4.4.

1) In particular, Corollary 4.2.i) encodes the following condition a) For allβ ∈I, (~g, ~s)∈q and w~p_β ∈Tp_βP_β we have

eg(p_β) +w~_pβ −es(p_β) = 0 =⇒ ψ_β(~g, ~w_pβ)−~s= 0.

2) Assume thata) is true and letq∈Q,pα ∈Pα,p_β ∈P_β withp_β =q·p_α. Moreover, assume that we find elements w~_pα ∈T_pαP_α and ((~g, ~s), ~w_pβ)∈q×T_pβP_β such that

d_(e,pβ)Θ((~g, ~s), ~wpβ) = dLqw~pα and ψβ(~g, ~wpβ)−~s=ρ(q)◦ψα(~0g, ~wpα) holds. Thenψ_β ~g⁰, ~w⁰_pβ

−~s⁰ =ρ(q)◦ψ_α ~0_g, ~w_pα

holds for each element¹¹ (~g⁰, ~s⁰), ~w⁰_pβ

∈ q×T_pβP_β with¹² d_(e,pβ)Θ (~g⁰, ~s⁰), ~w_p⁰

β

= dL_qw~_pα. In fact, we have d_(e,pβ)Θ (~g−~g⁰, ~s−~s⁰), ~w_pβ −w~_p⁰

β

= 0, so that by (4.1) condition a) gives

0^a)=ψβ(~g−~g⁰, ~wpβ −w~_p⁰_β)−(~s−~s⁰)) =

ψβ(~g, ~wpβ)−~s

−

ψβ(~g⁰, ~w⁰_pβ)−~s⁰

=ρ(q)◦ψα ~0g, ~wpα

−

ψβ(~g⁰, ~w⁰_pβ)−~s⁰ .

11Observe that due to surjectivity of d(e,p_β)Φ such elements always exist.

12Recall equation (4.1).

3) Assume that dLqw~pα ∈Tp_βP_β holds for all q∈Q, pα ∈Pα,p_β ∈P_β with p_β =q·pα and all w~_pα ∈T_pαP_α. Then d_(e,pβ)Θ (dL_qw~_pα) = dL_qw~_pα so that it follows from 2) that in this case we can substitutei) by a) and condition

b) Letq ∈Q,p_α∈P_α,p_β ∈P_β withp_β =q·p_α. Then ψβ ~0g,dLqw~pα

=ρ(q)◦ψα ~0g, ~wpα

∀w~pα ∈TpαPα.

Now, b) looks similar to ii) and makes it plausible that the conditions i) and ii) from Corollary 4.2 together encode the ρ-invariance of the corresponding connection ω. How- ever, usually there is no reason for dL_qw~_pα to be an element of T_pβP_β. Even for p_α =p_β and q ∈ Qpα this is usually not true. Thus, typically there is no way to split up i) into parts whose meaning is more intuitive.

Remark4.4 immediately proves

Case 4.5 (gauge fixing). Let P₀ be a Θ-patch of the bundle P such that π(P₀) intersects each ϕ-orbit in a unique point, and thatdL_q(T_pP₀)⊆T_pP₀ holds for allp∈P₀ and allq ∈Q_p. Then, a corresponding reduced connection consists of one single smooth map ψ:g×T P0 →s, and we have p =q·p⁰ for q ∈Q, p, p⁰ ∈P₀ iff p=p⁰ and q ∈Q_p holds. Thus, by Remark 4.4 the two conditions from Corollary 4.2 are equivalent to:

Let p∈P0,q = (h, φp(h))∈Qp, w~p ∈TpP0 and~g∈g,~s∈s. Then i⁰) eg(p) +w~_p−s(p) = 0e =⇒ ψ(~g, ~w_p)−~s= 0,

ii⁰) ψ ~0_g,dL_qw~_p

=ρ(q)◦ψ ~0_g, ~w_p , iii⁰) ψ Ad_h(~g), ~0_p

= Ad_φp(h)◦ψ ~g, ~0_p .

The next example is a slight generalization of Theorem 2 in [6]. There, the authors assume that ϕadmits only one orbit type so that dim[G_x] =l holds for all x∈M. Then, they restrict to the situation where one finds a triple (U₀, τ₀, s₀) consisting of an open subset U₀ ⊆ R^k for k = dim[M]−[dim[G]−l], an embedding τ0:U0 → M, and a smooth map s0:U0 → P with π◦s₀ = τ₀ and the addition property that Q_p is the same for all p ∈ im[s₀]. More precisely, they assume that G_x and the structure group of the bundle are compact. Then they show the non-trivial fact that s0 can be modified in such a way that in addition Qp is the same for all p∈im[s0].

Observe that the authors forgot to require that im[d_xτ₀] + im

d_eϕ_τ0(x)

=T_τ0(x)M holds for all x∈U0, i.e., that τ0(U0) is aϕ-patch (so that s0(U0) is a Θ-patch). Indeed, Example 4.10.2 shows that this additional condition is crucial. The next example is a slight modification of the result [6] in the sense that we do not assume G_x and the structure group to be compact but make the ad hoc requirement that Qp is the same for all p∈P0.

Example 4.6 (Harnad, Shnider, Vinet). LetP₀ be a Θ-patch of the bundleP such thatπ(P₀) intersects each ϕ-orbit in a unique point. Moreover, assume that the Θ-stabilizer L := Q_p is the same for all p∈ P0. Then, it is clear from (2.4) that H := G_π(p) and φ:=φp:H → S are independent of the choice of p∈P₀. Finally, we require that

dim[P₀] = dim[M]−[dim[G]−dim[H]]≡dim[P]−[dim[Q]−dim[H]] (4.3) holds. Now, let p∈P0 and q= (h, φ(h))∈Qp. Then, forw~p ∈TpP0 we have

dL_qw~_p = _dt^d

t=0Φ(h, γ(t))·φ⁻¹_p (h) = _dt^d

t=0[γ(t)·φ_γ(t)(h)]·φ⁻¹_p (h)

= _dt^d

t=0[γ(t)·φ_p(h)]·φ⁻¹_p (h) =w~_p

for γ: (−, ) → P0 some smooth curve with ˙γ(0) = w~p. Consequently, dLq(TpP0) ⊆ TpP0 so that we are in the situation of Case 4.5. Here,ii⁰) now readsψ ~0_g, ~w_p

= Ad_φ(h)◦ψ ~0_g, ~w_p for all h∈H and iii⁰) does not change. For i⁰), observe that the Lie algebralof L is contained in the kernel of d_(e,p0)Θ; denoting the differential of the restriction of Θ toQ×P0 for the moment.

Then, d_(e,p0)Θ is surjective by Lemma3.7.1 since P₀ is a Θ-patch, so that dim

ker

d_(e,p0)Θ

= dim[Q] + dim[P0]−dim[P]^(4.3)= dim[H],

hence ker[d_(e,p)Θ] = l holds for all p ∈ P₀. Altogether it follows that a reduced connection w.r.t. P₀ is a smooth, linear¹³ mapψ:g×T P₀ →s which fulfils the following three conditions:

i⁰⁰) ψ ~h, ~0p

(4.1)

= deφ ~h

∀~h∈h, ∀p∈P0, ii⁰⁰) ψ ~0g, ~w

= Ad_φ(h)◦ψ ~0g, ~w

∀h∈H, ∀w~ ∈T P0, iii⁰⁰) ψ Ad_h(~g), ~0_p

= Ad_φ(h)◦ψ ~g, ~0_p

∀h∈H, ∀~g∈g, ∀p∈P₀. Then, µ :=ψ|_{T P0} and Ap0(~g) := ψ ~g, ~0p0

are the maps that are used for the characterization in Theorem 2 in [6].

4.2 Reconstruction of invariant connections

Let{P_α}_α∈I be some fixed Φ-covering ofP. We are going to show that each respective reduced connection {ψ_α}_α∈I gives rise to a unique Φ-invariant connection on P. To this end, for each α∈I we define the mapsλ_α:q×T P_α →s,((~g, ~s), ~w)7→ψ_α(~g, ~w)−~sand

ω_α: T Q×T P_α→s,

~

m_q, ~w_pα

7→ρ(q)◦λ_α dL_q⁻¹m~_q, ~w_pα form~q∈TqQand w~pα ∈TpαPα.

Lemma 4.7. Let q∈Q, p_α∈P_α, p_β ∈P_β withp_β =q·p_α and w~_pα ∈T_pαP_α. Then 1) λ_β(~η) =ρ(q)◦λ_α ~0_q, ~w_pα

for all~η∈q×T_pβP withdΘ_(e,pβ)(~η) = dL_qw~_pα, 2) λβ Adq(~q), ~0pβ

=ρ(q)◦λα ~q, ~0pα

for all~q ∈q.

For each α∈I we have 3) ker

λ_α|_q×TpαPα

⊆ker

d_(e,pα)Θ

for allp_α∈P_α,

4) the map ωα is the unique s-valued 1-form on Q×Pα which extends λα and for which we have L^∗_qω_α =ρ(q)◦ω_α for all q∈Q.

Proof . 1. Write ~η= ((~g, ~s), ~wp_β) for~g∈g,~s∈s and w~p_β ∈Tp_βP_β. Then

eg(p_β) +w~_pβ −s(pe _β)^(4.1)= dΘ_(e,pβ)(~η) = dL_qw~_pα so that from conditioni) in Corollary4.2 we obtain

λ_β(~η) =ψ_β(~g, ~w_pβ)−~s=ρ(q)◦ψ_α ~0_g, ~w_pα

=ρ(q)◦λ_α ~0_q, ~w_pα . 2. Let~q= (~g, ~s) for~g∈g and~s∈s. Then, by Corollary4.2.ii) we have

λβ Adq(~q), ~0pβ

=ψβ Adq(~g), ~0pβ

−Adq(~s) =ρ(q)◦[ψα ~g, ~0pα

−~s] =ρ(q)◦λα ~q, ~0pα

.

13In the sense thatψ|g×T_pP0 is linear for allp∈P0.

3. This follows from the first part forα=β,q =eand w~pα =~0pα.

4. By definition we haveω_α|_{q×T Pα} =λ_α, and for the pullback property we calculate L^∗_q0ω_α

(q,pα) m~_q, ~w_pα

=ω_α(q⁰_q,pα) dL_q⁰m~_q, ~w_pα

=ρ q⁰q

◦λ_α dL_q⁻¹_q⁰⁻¹dL_q⁰m~_q, ~w_pα

=ρ q⁰

◦ρ(q)◦λα dL_q⁻¹m~q, ~wpα

=ρ q⁰

◦ωα(q,p_α)(m~q, ~wpα), where q, q⁰ ∈ Q and m~_q ∈ T_qQ. For uniqueness, let ω be another s-valued 1-form on Q×P_α whose restriction to q×T Pα isλα and that fulfilsL^∗_qω =ρ(q)◦ω for all q∈Q. Then

ω_(q,pα)(m~_q, ~w_pα) =ω_(q,pα) dL_q◦dL_q⁻¹m~_q, ~w_pα

= (L^∗_qω)_(e,pα) dL_q⁻¹m~_q, ~w_pα

=ρ(q)◦ω_(e,pα)(dL_q⁻¹m~q, ~wpα) =ρ(q)◦λα dL_q⁻¹m~q, ~wpα

=ω_α(dL_q⁻¹m~_q, ~w_pα).

Finally, smoothness ofω_αis an easy consequence of smoothness of the mapsρ,λ_αandµ:T Q→ q, m~q 7→ dL_q⁻¹m~q with m~q ∈ TqQ. For this, observe that µ = dτ ◦κ for τ:Q×Q → Q, (q, q⁰)7→q⁻¹q⁰ and κ:T Q→T Q×T Q,m~_q 7→ ~0_q, ~m_q

form~_q∈T_qQ.

So far, we have shown that each reduced connection{ψ_α}_α∈Igives rise to uniquely determined maps{λ_α}_α∈Iand{ω_α}_α∈I. In the final step, we will construct a unique Φ-invariant connection ωfrom the data{(P_α, λ_α)}_α∈I. Here, uniqueness and smoothness ofωwill follow from uniqueness and smoothness of the maps ω_α.

Proposition 4.8. There is one and only ones-valued1-formω onP withωα= (Θ^∗ω)|_{T Q×T Pα} for all α∈I. This 1-form is a Φ-invariant connection on P.

Proof . For uniqueness, we have to show that the values of such an ω are uniquely determined by the mapsω_α. To this end, letp∈P,α∈I andp_α ∈P_αbe such thatp=q·p_αholds for some q ∈Q. By Lemma 3.7.1 forw~_p ∈T_pP we find some~η ∈T_qQ×T_pαP_α withw~_p = d_(q,pα)Θ(~η), so that uniqueness follows from

ωp(w~p) =ωq·p_α d_(q,pα)Θ(~η)

= (Θ^∗ω)_(q,pα)(~η) =ωα(~η).

For existence, let α∈I and p_α ∈P_α. Due to surjectivity of d_(e,pα)Θ and Lemma 4.7.3, there is a (unique) map bλ_pα:T_pαP →swith

bλ_pα◦d_(e,pα)Θ =λ_{α q×T}

pαPα. (4.4)

Let bλ_α: F

pα∈P_αT_pαP → s denote the (unique) map whose restriction to T_pαP is bλ_pα for each p_α ∈P_α. Then λ_α =bλ_α◦dΘ|_{q×T Pα} and we construct the connection ω as follows. Forp ∈ P we choose someα∈I and (q, p_α)∈Q×P_α such that q·p_α=p and define

ω_p w~_p

:=ρ(q)◦bλ_α dL_q⁻¹ w~_p

∀w~_p ∈T_pP. (4.5)

We have to show that this depends neither on α ∈ I nor on the choice of (q, p_α) ∈ Q×P_α. For this, let p_α ∈ P_α, p_β ∈ P_β and q ∈ Q with p_β = q ·p_α. Then for w~ ∈ T_pαP we have

~

w= dΘ(~q, ~wpα) for some (~q, ~wpα)∈q×TpαPα, and since dLqw~pα ∈TpβP, there is~η∈q×TpβPβ

such that d_(e,pβ)Θ(~η) = dL_qw~_pα holds. It follows from the conditions 1 and 2 in Lemma4.7that bλ_β(dL_qw) =~ bλ_β((dL_q◦dΘ)(~q, ~w_pα)) =λb_β (dL_q◦dΘ) ~q, ~0_pα

+bλ_β dL_qw~_pα

(4.7)

= bλ_β◦dΘ Adq(~q), ~0p_β

+bλ_β◦dΘ(~η)

(4.4)

= λβ Adq(~q), ~0pβ

+λβ(~η) =ρ(q)◦λα ~q, ~0pα

+ρ(q)◦λα ~0q, ~wpα

=ρ(q)◦λ_α(~q, ~w_pα) =ρ(q)◦bλ_α◦dΘ(~q, ~w_pα) =ρ(q)◦bλ_α(w),~

(4.6)