## 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 action^{1} 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 variation^{2} of the case where the bundle
admits a submanifold P_{0} with π(P_{0}) 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×P_{0} → s for which the restrictions ψ|_{g×T}_{p}

0P0 are linear for all p_{0} ∈ P_{0}, 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 submanifolds^{3} 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_{p}P ⊆ T_{p}P 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_{α}Pα with im[dp_{α}ια]. 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_{pα}Pα

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_{p}_{α}, ~w_{p}_{α} ∈ T_{p}_{α}P_{α}, ~0_{p}_{β}, ~w_{p}_{β} ∈ T_{p}_{β}P_{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
pointwise^{4} 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|_{T}_{x}_{M}: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|_{T}_{y}_{N} 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_{g}instead of Ψ_{g}as 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∈I}ofM and a family{φ_{α}}_{α∈I} of diffeomorphisms
φα:π^{−1}(Uα)→Uα×S with

φα(p·s) = π(p),[pr_{2}◦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, pr_{2} 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_{p}P := ker[d_{p}π]⊆T_{p}P is calledvertical tangent space atp∈P and
es(p) := _{dt}^{d}

t=0p·exp(t~s)∈T v_{p}P ∀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^{∗}_{s}ω= 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_{p}P) =T hp·sP for alls∈S, and one can show thatT_{p}P =T v_{p}P ⊕T h_{p}P 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^{∗}_{g}ω =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_{p}P andT h_{gp}P.^{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^{∗}_{q}ω =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_{e}φ_{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 M^{0} is a manifold andκ:M →M^{0} 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 N^{0} ⊆ N of x and a submanifold H of G through e, such that the restriction
Ψ|_{H×N}^{0} is a diffeomorphism to an open subset U ⊆M.

Remark 3.3.

1. It follows from the inverse function theorem and^{6}

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_{x}N =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 V^{0} ⊆g a linear subspace with
dim[V^{0}] = dim[V] and deΨx(V^{0}) =V. Then, we find a submanifoldH ofGthroughewithTeH =V^{0}, 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_{e}Ψ_{x}(g)] + dim[T_{x}N] = dim[G]−dim[G_{x}] + dim[N]. (3.1)
2. Of course, we find submanifolds N^{0} of M through x and H^{0} of G through e such that
TxN^{0} =V and TeH^{0} =W. So, if~g∈g and ~vx ∈TxN^{0}, then 0 = d_{(e,x)}Ψ(~g, ~vx) = deΨx(~g) +~vx

implies d_{e}Ψ_{x}(~g) = 0 and ~v_{x} = 0. Hence, ~g ∈ ker[d_{e}Ψ_{x}] = g_{x}, so that^{8} d_{(e,x)}Ψ|_{T}_{e}_{H}^{0}_{×T}_{e}_{N}^{0} is
injective. It is immediate from the definitions that this map is surjective, so that by the inverse
function theorem we find open neighbourhoodsN ⊆N^{0} 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_{e}Θ_{p}(q) = d_{e}Φ_{p}(g) +T v_{p}P 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 s_{0}:N → P a smooth section (i.e., π◦s_{0} = id_{N}), then s_{0}(N) is
a Θ-patch by Lemma 3.7.2.

Conversely, if N ⊆ M is a submanifold such that s_{0}(N) is a Θ-patch for s_{0} 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_{q}Q×T_{p}O→
Tq·pP is surjective.

2. If N is a ϕ-patch ands_{0}:N →P a smooth section, then s_{0}(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_{p}L_{q} d_{(e,p)}Θ(~q, ~w_{p})

.
So, since left translation w.r.t. Θ is a diffeomorphism, d_{p}L_{q} is surjective.

8Recall that d(e,x)Ψ|T_{e}H^{0}×T_{e}N^{0}: ~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_{s}_{0}_{(x)}O+ d_{e}Φ_{s}_{0}_{(x)}(g) +T v_{s}_{0}_{(x)}P

≥dim[T_{s}_{0}_{(x)}P] ∀x∈N.

For this, let x∈N andV^{0}⊆gbe a linear subspace withV^{0}⊕g_{x} and T_{x}M =T_{x}N⊕d_{e}ϕ_{x}(V^{0}).

Then, we haveT_{s}_{0}_{(x)}O⊕deΦ_{s}_{0}_{(x)}(V^{0})⊕T v_{s}_{0}_{(x)}P because if dxs0(~vx) + deΦ_{s}_{0}_{(x)}(~g^{0}) +~vv = 0 for

~

v_{x} ∈T_{x}N,~g^{0} ∈V^{0} and ~v_{v} ∈T v_{s}_{0}_{(x)}P,

0 = d_{s}_{0}_{(x)}π dxs0(~vx) + deΦ_{s}_{0}_{(x)}(~g^{0}) +~vv

=~vx⊕deϕx(~g^{0})

shows~v_{x}= 0 and d_{e}φ_{x}(~g^{0}) = 0, hence~g^{0}= 0 by the choice of V^{0}, i.e., ~v_{v}= 0 by assumption. In
particular, d_{e}φ_{x}(~g^{0}) = 0 if d_{e}Φ_{s}_{0}_{(x)}(~g^{0}) = 0, hence dim[d_{e}Φ_{s}_{0}_{(x)}(V^{0})]≥dim[d_{e}ϕ_{x}(V^{0})], from which
we obtain

dim

T_{s}_{0}_{(x)}O+ deΦ_{s}_{0}_{(x)}(g) +T v_{s}_{0}_{(x)}P

≥dim

T_{s}_{0}_{(x)}O⊕deΦ_{s}_{0}_{(x)}(V^{0})⊕T v_{s}_{0}_{(x)}P

= dim[T_{x}N] + dim[d_{e}Φ_{s}_{0}_{(x)}(V^{0})] + dim[S]≥dim[T_{x}N] + dim[d_{e}ϕ_{x}(V^{0})] + 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×T}_{pα}_{P}_{α} 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 P}_{α} as well as ψ_{α} :=ω_{α}|_{g×T P}_{α},
and for q^{0} ∈Q we letα_{q}^{0}:Q×P →Q×P, (q, p)7→ α_{q}^{0}(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 with^{9} 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^{∗}_{q}ω_{β}

~
m, ~0p_{β}

=ρ(q)◦ωα m, ~~ 0pα

for allm~ ∈T Q.

Proof . 1. Let ~η ∈ T_{q}^{0}Q×TpPβ for q^{0} ∈Q. Then, since^{10} L^{∗}_{q}ω =ρ(q)◦ω for each q ∈Q and
q^{0}·p=q·p_{α} =p_{β}, we have

ωβ(~η) =ωq^{0}·p(d_{(q}^{0}_{,p)}Θ(~η)) =ωpβ(dLqw~pα) = (L^{∗}_{q}ω)pα(w~pα)

=ρ(q)◦ω_{p}_{α}(w~_{p}_{α}) =ρ(q)◦ω_{p}_{α} d_{(e,p}_{α}_{)}Θ ~0_{q}, ~w_{p}_{α}

=ρ(q)◦ω_{α} ~0_{q}, ~w_{p}_{α}
.

9Recall that, by Convention3.1, this actually meansτP_{β}(pβ) =q·τPα(pα).

10See end of Subsection2.2.

2. Form~_{q}^{0} ∈T_{q}^{0}Qlet γ: (−, )→Q be smooth with ˙γ(0) =m~_{q}^{0}. Then
α^{∗}_{q}ωβ

(q^{0},pβ) m~q^{0}, ~0pβ

=ωβ(αq(q^{0}),p_{β}) Adq(m~q^{0}), ~0pβ

=ω_{qq}^{0}_{q}^{−1}_{q·p}_{α} _{dt}^{d}

t=0qγ(t)q^{−1}q·pα

= L^{∗}_{q}ω

q^{0}·pα

d dt

t=0γ(t)·pα

=ρ(q)◦ωq^{0}·pα d_{(q}^{0}_{,p}_{α}_{)}Θ m~q^{0}

=ρ(q)◦ω_{α(q}^{0}_{,p}_{α}_{)} m~_{q}^{0}, ~0_{p}_{α}

.

Corollary 4.2. Let q ∈ Q, p_{α} ∈ P_{α}, p_{β} ∈ P_{β} with p_{β} = q·p_{α} and w~_{p}_{α} ∈ T_{p}_{α}P_{α}. Then, for

~

w_{p}_{β} ∈T_{p}_{β}P_{β},~g∈g and~s∈s we have

i) eg(p_{β}) +w~_{p}_{β} −s(pe _{β}) = dL_{q}w~_{p}_{α} =⇒ ψ_{β}(~g, ~w_{p}_{β})−~s=ρ(q)◦ψ_{α} ~0_{g}, ~w_{p}_{α}
,
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_{p}_{α})∈q×T P_{α} we obtain

ω_{α}((~g, ~s), ~w_{p}_{α}) =ω d_{(e,p}_{α}_{)}Φ(~g, ~w_{p}_{α})−es(p_{α})

=ω d_{(e,p}_{α}_{)}Φ(~g, ~w_{p}_{α})

−~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,p}_{β}_{)}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_{p}_{β}

−~s^{(4.2)}= ω_{β}((~g, ~s), ~w_{p}_{β}) =ρ(q)◦ω_{α} ~0_{q}, ~w_{p}_{α}(4.2)

= ρ(q)◦ψ_{α} ~0_{g}, ~w_{p}_{α}
.
ii) Lemma4.1.2 yields

ψ_{β} Ad_{q}(~g), ~0_{p}_{β}

= (α^{∗}_{q}ω_{β})_{(e,p}_{β}_{)} ~g, ~0_{p}_{β}

=ρ(q)◦(ω_{α})_{(e,p}_{α}_{)} ~g, ~0_{p}_{α}

=ρ(q)◦ψ_{α} ~g, ~0_{p}_{α}

.

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×T}_{pα}_{P}_{α} is linear for allp_{α}∈P_{α}is 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_{β}P_{β} we have

eg(p_{β}) +w~_{p}_{β} −es(p_{β}) = 0 =⇒ ψ_{β}(~g, ~w_{p}_{β})−~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~_{p}_{α} ∈T_{p}_{α}P_{α} and ((~g, ~s), ~w_{p}_{β})∈q×T_{p}_{β}P_{β} such that

d_{(e,p}_{β}_{)}Θ((~g, ~s), ~wpβ) = dLqw~pα and ψβ(~g, ~wpβ)−~s=ρ(q)◦ψα(~0g, ~wpα)
holds. Thenψ_{β} ~g^{0}, ~w^{0}_{p}_{β}

−~s^{0} =ρ(q)◦ψ_{α} ~0_{g}, ~w_{p}_{α}

holds for each element^{11} (~g^{0}, ~s^{0}), ~w^{0}_{p}_{β}

∈
q×T_{p}_{β}P_{β} with^{12} d_{(e,p}_{β}_{)}Θ (~g^{0}, ~s^{0}), ~w_{p}^{0}

β

= dL_{q}w~_{p}_{α}. In fact, we have
d_{(e,p}_{β}_{)}Θ (~g−~g^{0}, ~s−~s^{0}), ~w_{p}_{β} −w~_{p}^{0}

β

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

0^{a)}=ψβ(~g−~g^{0}, ~wpβ −w~_{p}^{0}_{β})−(~s−~s^{0})) =

ψβ(~g, ~wpβ)−~s

−

ψβ(~g^{0}, ~w^{0}_{p}_{β})−~s^{0}

=ρ(q)◦ψα ~0g, ~wpα

−

ψβ(~g^{0}, ~w^{0}_{p}_{β})−~s^{0}
.

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_{β}P_{β} holds for all q∈Q, pα ∈Pα,p_{β} ∈P_{β} with p_{β} =q·pα and
all w~_{p}_{α} ∈T_{p}_{α}P_{α}. Then d_{(e,p}_{β}_{)}Θ (dL_{q}w~_{p}_{α}) = dL_{q}w~_{p}_{α} 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_{q}w~_{p}_{α} to be an element of T_{p}_{β}P_{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 P_{0} be a Θ-patch of the bundle P such that π(P_{0}) intersects each
ϕ-orbit in a unique point, and thatdL_{q}(T_{p}P_{0})⊆T_{p}P_{0} holds for allp∈P_{0} 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·p^{0} for q ∈Q, p, p^{0} ∈P_{0} iff p=p^{0} 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
i^{0}) eg(p) +w~_{p}−s(p) = 0e =⇒ ψ(~g, ~w_{p})−~s= 0,

ii^{0}) ψ ~0_{g},dL_{q}w~_{p}

=ρ(q)◦ψ ~0_{g}, ~w_{p}
,
iii^{0}) ψ Ad_{h}(~g), ~0_{p}

= Ad_{φ}_{p}_{(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 (U_{0}, τ_{0}, s_{0}) consisting of an open subset U_{0} ⊆ R^{k} for
k = dim[M]−[dim[G]−l], an embedding τ0:U0 → M, and a smooth map s0:U0 → P with
π◦s_{0} = τ_{0} and the addition property that Q_{p} is the same for all p ∈ im[s_{0}]. 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_{x}τ_{0}] + im

d_{e}ϕ_{τ}_{0}_{(x)}

=T_{τ}_{0}_{(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). LetP_{0} be a Θ-patch of the bundleP such thatπ(P_{0})
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∈P_{0}. Finally, we require that

dim[P_{0}] = 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_{q}w~_{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,ii^{0}) now readsψ ~0_{g}, ~w_{p}

= Ad_{φ(h)}◦ψ ~0_{g}, ~w_{p}
for
all h∈H and iii^{0}) does not change. For i^{0}), observe that the Lie algebralof L is contained in
the kernel of d_{(e,p}_{0}_{)}Θ; denoting the differential of the restriction of Θ toQ×P0 for the moment.

Then, d_{(e,p}_{0}_{)}Θ is surjective by Lemma3.7.1 since P_{0} is a Θ-patch, so that
dim

ker

d_{(e,p}_{0}_{)}Θ

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

hence ker[d_{(e,p)}Θ] = l holds for all p ∈ P_{0}. Altogether it follows that a reduced connection
w.r.t. P_{0} is a smooth, linear^{13} mapψ:g×T P_{0} →s which fulfils the following three conditions:

i^{00}) ψ ~h, ~0p

(4.1)

= deφ ~h

∀~h∈h, ∀p∈P0,
ii^{00}) ψ ~0g, ~w

= Ad_{φ(h)}◦ψ ~0g, ~w

∀h∈H, ∀w~ ∈T P0,
iii^{00}) ψ Ad_{h}(~g), ~0_{p}

= Ad_{φ(h)}◦ψ ~g, ~0_{p}

∀h∈H, ∀~g∈g, ∀p∈P_{0}.
Then, µ :=ψ|_{T P}_{0} 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_{p}_{α}

7→ρ(q)◦λ_{α} dL_{q}^{−1}m~_{q}, ~w_{p}_{α}
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~_{p}_{α} ∈T_{p}_{α}P_{α}. Then
1) λ_{β}(~η) =ρ(q)◦λ_{α} ~0_{q}, ~w_{p}_{α}

for all~η∈q×T_{p}_{β}P withdΘ_{(e,p}_{β}_{)}(~η) = dL_{q}w~_{p}_{α},
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×T}_{pα}_{P}_{α}

⊆ker

d_{(e,p}_{α}_{)}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^{∗}_{q}ω_{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_{β}P_{β}. Then

eg(p_{β}) +w~_{p}_{β} −s(pe _{β})^{(4.1)}= dΘ_{(e,p}_{β}_{)}(~η) = dL_{q}w~_{p}_{α}
so that from conditioni) in Corollary4.2 we obtain

λ_{β}(~η) =ψ_{β}(~g, ~w_{p}_{β})−~s=ρ(q)◦ψ_{α} ~0_{g}, ~w_{p}_{α}

=ρ(q)◦λ_{α} ~0_{q}, ~w_{p}_{α}
.
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_{p}P0 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 P}_{α} =λ_{α}, and for the pullback property we calculate
L^{∗}_{q}0ω_{α}

(q,pα) m~_{q}, ~w_{p}_{α}

=ω_{α(q}^{0}_{q,p}_{α}_{)} dL_{q}^{0}m~_{q}, ~w_{p}_{α}

=ρ q^{0}q

◦λ_{α} dL_{q}^{−1}_{q}^{0−1}dL_{q}^{0}m~_{q}, ~w_{p}_{α}

=ρ q^{0}

◦ρ(q)◦λα dL_{q}^{−1}m~q, ~wpα

=ρ q^{0}

◦ωα(q,p_{α})(m~q, ~wpα),
where q, q^{0} ∈ Q and m~_{q} ∈ T_{q}Q. For uniqueness, let ω be another s-valued 1-form on Q×P_{α}
whose restriction to q×T Pα isλα and that fulfilsL^{∗}_{q}ω =ρ(q)◦ω for all q∈Q. Then

ω_{(q,p}_{α}_{)}(m~_{q}, ~w_{p}_{α}) =ω_{(q,p}_{α}_{)} dL_{q}◦dL_{q}^{−1}m~_{q}, ~w_{p}_{α}

= (L^{∗}_{q}ω)_{(e,p}_{α}_{)} dL_{q}^{−1}m~_{q}, ~w_{p}_{α}

=ρ(q)◦ω_{(e,p}_{α}_{)}(dL_{q}^{−1}m~q, ~wpα) =ρ(q)◦λα dL_{q}^{−1}m~q, ~wpα

=ω_{α}(dL_{q}^{−1}m~_{q}, ~w_{p}_{α}).

Finally, smoothness ofω_{α}is an easy consequence of smoothness of the mapsρ,λ_{α}andµ:T Q→
q, m~q 7→ dL_{q}^{−1}m~q with m~q ∈ TqQ. For this, observe that µ = dτ ◦κ for τ:Q×Q → Q,
(q, q^{0})7→q^{−1}q^{0} and κ:T Q→T Q×T Q,m~_{q} 7→ ~0_{q}, ~m_{q}

form~_{q}∈T_{q}Q.

So far, we have shown that each reduced connection{ψ_{α}}_{α∈I}gives rise to uniquely determined
maps{λ_{α}}_{α∈I}and{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 P}_{α}
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_{α}be such thatp=q·p_{α}holds for some
q ∈Q. By Lemma 3.7.1 forw~_{p} ∈T_{p}P we find some~η ∈T_{q}Q×T_{p}_{α}P_{α} withw~_{p} = d_{(q,p}_{α}_{)}Θ(~η), so
that uniqueness follows from

ωp(w~p) =ωq·p_{α} d_{(q,p}_{α}_{)}Θ(~η)

= (Θ^{∗}ω)_{(q,p}_{α}_{)}(~η) =ωα(~η).

For existence, let α∈I and p_{α} ∈P_{α}. Due to surjectivity of d_{(e,p}_{α}_{)}Θ and Lemma 4.7.3, there is
a (unique) map bλ_{p}_{α}:T_{p}_{α}P →swith

bλ_{p}_{α}◦d_{(e,p}_{α}_{)}Θ =λ_{α}
_{q×T}

pαPα. (4.4)

Let bλ_{α}: F

pα∈P_{α}T_{p}_{α}P → s denote the (unique) map whose restriction to T_{p}_{α}P is bλ_{p}_{α} for each
p_{α} ∈P_{α}. Then λ_{α} =bλ_{α}◦dΘ|_{q×T P}_{α} 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_{p}P. (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_{p}_{α}P 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,p}_{β}_{)}Θ(~η) = dL_{q}w~_{p}_{α} holds. It follows from the conditions 1 and 2 in Lemma4.7that
bλ_{β}(dL_{q}w) =~ bλ_{β}((dL_{q}◦dΘ)(~q, ~w_{p}_{α})) =λb_{β} (dL_{q}◦dΘ) ~q, ~0_{p}_{α}

+bλ_{β} dL_{q}w~_{p}_{α}

(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_{p}_{α}) =ρ(q)◦bλ_{α}◦dΘ(~q, ~w_{p}_{α}) =ρ(q)◦bλ_{α}(w),~

(4.6)