A Characterization of Invariant Connections

Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms thereon, Wang's theorem identifies the invariant connections with certain linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$. 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.


Introduction
The set of connections on a principal fibre bundle (P, π, M, S) is closed under pullback by automorphisms 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 particular 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 maps ψ : g → s. In [6] the authors generalize this to the situation where ϕ admits 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×Tp 0 P 0 are linear for all p 0 ∈ P 0 and that fulfil additional consistency conditions. Now, in the general case we consider Φ-coverings of P . These are families {P α } α∈I of immersed submanifolds 3 P α of P such that each ϕ-orbit has non-empty intersection with α∈I π(P α ) and for which T p P = T p P α + d e Φ p (g) + T v p P 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 of G. Observe that the intersection properties of the sets π(P α ) with the ϕ-orbits in the base manifold need not to be convenient in any sense. Here one might think of situations in which ϕ admits dense orbits, or of the almost-fibre transitive case, cf. Case 5.4.
Here q ∈ G × S, p α ∈ P α , p β ∈ P β with p β = q · p α and w pα ∈ T pα P α , w p β ∈ T p β P β . Moreover, g, s denote the fundamental vector fields assigned to the elements g ∈ g and s ∈ s, respectively.
Using this theorem the calculation of invariant connections reduces to identifying a Φ-covering that makes the above conditions as easy as possible. Here one has to find the balance between quantity and complexity of these conditions. Of course, the more submanifolds 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 where ϕ is 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 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 of P such that the submanifolds P α have minimal dimension in a certain sense, see see Lemma 3.4 and Corollary 5.1. This reproduces the description of connections by means of local 1-forms on M provided that G acts trivially or, more generally, via gauge transformations on P , see Case 5.2.
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 5 .
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 Section 5 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.

Preliminaries
We start with fixing the notations.

Notations
Manifolds are always assumed to be smooth. If M , N are manifolds and f : 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 d x f := df | TxM : Let V be a finite dimensional vector space. A V -valued 1-form ω on the manifold N is a smooth map ω : T N → V whose restriction ω y := ω| TyN is linear for all y ∈ N . The pullback of ω by f is the V -valued 1-form f * ω : Let G be a Lie group and g its Lie algebra. For g ∈ G we define the corresponding conjugation map by α g : G → G, h → ghg −1 . Its differential d e α g : 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 group G on the manifold M . If g ∈ G, then Ψ g : M → M denotes the map Ψ g : x → Ψ(g, x). We often write L g instead of Ψ g as well as g · x or gx instead of Ψ g (x) if it is clear, which action is meant. If x ∈ M , let Ψ x : G → M , g → Ψ(g, x). Then for g ∈ g and x ∈ M the map g(x) := d dt t=0 Ψ x (exp(t g )) is called the fundamental vector field w.r.t. g. The Lie subgroup G x := g ∈ G g · x = x is called the stabilizer of x ∈ M (w.r.t. Ψ) and its Lie algebra g x equals ker[d x Ψ], see e.g. [3]. The orbit of x under G is the set Gx := im[Ψ x ], and Ψ is said to be transitive iff Gx = M for one and then each x ∈ M . Analogous conventions also hold for right actions.

Invariant connections
Let π : P → M be a smooth (surjective) map between manifolds P and M , and denote by F x := π −1 (x) ⊆ P the fibre over x ∈ M in P . Assume that (S, R) is a Lie group that acts from the right on P . If there is an open covering {U α } α∈I of M and a family {φ α } α∈I of diffeomorphisms φ α : then (P, π, M, S) is called principal fibre bundle with total space P , projection map π, base manifold M and structure group S. It follows from (2.1) that • if x ∈ M and p, p ∈ F x , then p = p · s for a unique element s ∈ S.
The subspace T v p P := ker[d p π] ⊆ T p P is called vertical tangent space at p ∈ P and denotes the fundamental vector field of s w.r.t. the right action of S on P . The map s s → s(p) ∈ T v p P is a vector space isomorphism for all p ∈ P . Complementary to that, a connection ω is an s-valued 1-form on P with The subspace T h p P := ker[ω p ] ⊆ T p P is called the horizontal tangent space at p (w.r.t. ω). We have dR s (T h p P ) = T h p·s P for all s ∈ S and one can show that T p P = T v p P ⊕ T h p P for all p ∈ P .
A diffeomorphism κ : P → P is said to be an automorphism iff κ(p · s) = κ(p) · s for all p ∈ P and all s ∈ S. It is straightforward to see that an s-valued 1-form ω on P is a connection iff this is true for the pullback κ * ω. A Lie group of automorphisms (G, Φ) of P is a Lie group G together with a left action Φ of G on P such that the map Φ g is an automorphism for each g ∈ G. This is equivalent to say that Φ(g, p · s) = Φ(g, p) · s for all p ∈ P , g ∈ G and all s ∈ S. In this situation we often write gps instead of (g · p) · s = g · (p · s). Each such a left action Φ gives rise to two further actions: • The induced action ϕ is defined by where p m ∈ π −1 (m) is arbitrary. Φ is called fibre transitive iff ϕ is transitive.
• We equip Q = G × S with the canonical Lie group structure and define [8] Θ : Q × P → P, This is equivalent to require that for each p ∈ P and g ∈ G the differential d p L g induces an isomorphism between the horizontal tangent spaces T h p P and T h gp P . 7 We conclude this subsection with the following straightforward facts, see also [8]: • Consider the representation ρ : Q → Aut(s), (g, s) → Ad s . Then it is immediate that each Φ-invariant connection ω is of type ρ, i.e., ω is an s-valued 1-form on P with L * q ω = ρ(q)•ω for all q ∈ Q.
• An s-valued 1-form ω on P with ω( s(p)) = s for all s ∈ s is a Φ-invariant connection iff it is of type ρ.
• Let Q p denote the stabilizer of p ∈ P w.r.t. Θ and G π(p) the stabilizer of π(p) w.r.t. ϕ. Then G π(p) = {g ∈ G | L g : F p → F p } and we obtain a smooth homomorphism (Lie group homomorphism) φ p : G π(p) → S by requiring that Φ(j, p) = p · φ p (j) for all j ∈ G π(p) . If q p and g π(p) denote the Lie algebras of Q p and G π(p) , respectively, then

Φ-coverings
We start this section with some facts and conventions concerning submanifolds. Then we give the definition of a Φ-covering and discuss some its properties. 2. If (N, τ N ) 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 is a manifold and κ : M → M is a smooth map, then for x ∈ N and v ∈ T N we write κ(x) and dκ( v) instead of κ(τ N (x)) and dκ(dτ ( v)), respectively.
• We will not explicitly refer to the maps τ N and τ H in the following. 7 In literature sometimes the latter condition is used to define Φ-invariance of connections.
3. Open subsets U ⊆ M are equipped with the canonical manifold structure making the inclusion map an embedding.
4. If L is a submanifold of N and N is a submanifold of M , we consider L as a submanifold of M in the canonical way.  1. It follows from the inverse function theorem and 8 2. Open subsets U ⊆ M are always Ψ-patches. They are of maximal dimension which, for instance, is necessary if there is a point in U whose stabilizer equals G, see Lemma 3.4.1.
3. We allow zero-dimensional patches, i.e., N = {x} for x ∈ M . Necessarily, then d e Ψ x (g) = T x M and Ψ| H×N = Ψ x | H for each submanifold H of G.
The second part of the next elementary lemma equals Lemma 2.1.1 in [3].
and v x = 0. Hence, g ∈ ker[d e Ψ x ] = g x so that 10 d (e,x) Ψ| TeH ×TeN is injective. It is immediate from the definitions that this map is surjective so that by the inverse function theorem we find open neighbourhoods N ⊆ N of x and H ⊆ G of e such that Ψ| H×N is a diffeomorphism to an open subset U ⊆ M . Then N 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 bundle P and recall the actions ϕ and Θ defined by (2.2) and (2.3), respectively. A family of Θpatches {P α } α∈I is said to be a Φ-covering of P iff each ϕ-orbit intersects at least one of the sets π(P α ).
As a consequence • each Φ-patch is a Θ-patch, • P is always a Φ-covering by itself and if P = 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 O := s 0 (N ) is a Θ-patch as Lemma 3.7.2 shows.
Lemma 3.7. Let (G, Φ) be a Lie group of automorphisms of the principal bundle (P, π, M, S).

1.
If O ⊆ P is a Θ-patch, then for each p ∈ O and q ∈ Q the differential d (q,p) Θ : T q Q×T p O → T q·p P is surjective.

2.
If N is a ϕ-patch and s 0 : N → P a smooth section, then O := 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 m q ∈ T q Q we find some q ∈ q such that m q = dL q q. Consequently, for w p ∈ T p P 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 ) .
But, left translation w.r.t. Θ is a diffeomorphism so that d p L q is surjective. 2. First observe that O is a submanifold of P because s 0 is an injective immersion. By Remark 3.6.2 it suffices to show that For this, let x ∈ N and V ⊆ g be a linear subspace such that so that v x = 0 and g = 0 by assumption,

Characterization of invariant connections
In this section we use Φ-coverings {P α } α∈I of the bundle P to characterize the set of Φ-invariant connections by families {ψ α } α∈I of smooth maps ψ α : g×T P α → s whose restrictions ψ α | g×Tp α Pα are linear and that fulfil two additional compatibility conditions. Here we follow the lines of Wang's original approach, which means that we generalize the proofs from [8] to the nontransitive case. We will proceed in two steps where the first one is done in the next subsection.
Here 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 Subsection 4.2, we verify that such families {ψ α } α∈I glue together to a Φ-invariant connection on P .

Reduction of invariant connections
In the following let {P α } α∈I be a fixed Φ-covering of P and ω a Φ-invariant connection on P . We define ω α := (Θ * ω)| T Q×T Pα and ψ α := ω α | g×T Pα . For q ∈ Q we let α q : Q × P → Q × P denote the map α q (q, p) := α q (q), p for α q : Q → Q the conjugation map w.r.t. q as defined in Section 2.1.
Proof . i) In general, for w p ∈ T p P , g ∈ g and s ∈ s we have and, since ω is a connection, for (( g, s), w pα ) ∈ q × T P α we obtain Remark 4.4.

1) In particular, Corollary 4.2.i) encodes the following condition
a) For all β ∈ I, ( g, s) ∈ q and w p β ∈ T p β P β we have 2) Assume that a) is true and let q ∈ Q, p α ∈ P α , p β ∈ P β with p β = q · p α . Moreover, assume that we find elements w pα ∈ T pα P α and (( g, s), w p β ) ∈ q × T p β P β such that 3) Assume that dL q w pα ∈ T p β P β holds for all q ∈ Q, p α ∈ P α , p β ∈ P β with p β = q · p α and all w pα ∈ T pα P α . Then d (e,p β ) Θ (dL q w pα ) = dL q w pα so that it follows from 2) that in this case we can substitute i) by a) and condition b) Let q ∈ Q, p α ∈ P α , p β ∈ P β with p β = q · p α . Then 13 Observe that due to surjectivity of d (e,p β ) Φ such elements always exist. 14 Recall equation (4.1). Now, b) looks similar to ii) and makes it plausible that the conditions i) and ii) from Corollary 4.2 encode the ρ-invariance of the corresponding connection ω. However, usually there is no reason for dL q w pα to be an element of T p β P β . Even for p α = p β and q ∈ Q pα this is not true in general. So, typically there is no way to split up i) into parts whose meaning is more intuitive.

Remark 4.4 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 dL q (T p P 0 ) ⊆ T p P 0 for all p ∈ P 0 and all q ∈ Q p . Then a corresponding reduced connection consists of one single smooth map ψ : g × T P 0 → s and we have p = q · p for q ∈ Q, p, p ∈ P 0 iff p = p and q ∈ Q p . Then, by Remark 4.4 the two conditions from Corollary 4.2 are equivalent to: The next example is a slight generalization of Theorem 2 in [6]. Here the authors assume that ϕ admits only one orbit type so that dim[G x ] = l for all x ∈ M . Then they restrict to the situation where we find a triple , an embedding τ 0 : U 0 → M and a smooth map s 0 : U 0 → 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 s 0 can be modified in such a way that in addition Q p is the same for all Observe that the authors omitted to require that im[d x τ 0 ] + im d e ϕ τ 0 (x) = T τ 0 (x) M holds for all x ∈ U 0 , i.e., that τ 0 (U 0 ) is a ϕ-patch (so that s 0 (U 0 ) 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 Q p is the same for all p ∈ P 0 . Example 4.6 (Harnad, Shnider, Vinet). 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 assume that the Θ-stabilizer L := Q p is the same for all p ∈ P 0 . Then it is clear from (2.4) that H := G π(p) and φ := φ p : H → S are independent of the choice of p ∈ P 0 . Finally, we require that Now, let p ∈ P 0 and q = (j, φ(j)) ∈ Q p . Then for w p ∈ T p P 0 we have Consequently, dL q (T p P 0 ) ⊆ T p P 0 so that we are in the situation of Case 4.5. Here ii ) now reads ψ 0 g , w p = Ad φ(j) • ψ 0 g , w p for all j ∈ H and iii ) does not change. For i ) observe that the Lie algebra l of L is contained in the kernel of d (e,p 0 ) Θ. But d (e,p 0 ) Θ is surjective since P 0 is a Θ-patch 15 so that This shows ker[d (e,p) Θ] = l for all p ∈ P 0 . Altogether, it follows that a reduced connection w.r.t. P 0 is a smooth, linear 16 map ψ : g × T P 0 → s which fulfils the following three conditions: Then µ := ψ| T P 0 and A p 0 ( g ) := ψ g, 0 p 0 are the maps that are used for the characterization in Theorem 2 in [6].

Reconstruction of invariant connections
Let {P α } α∈I be a Φ-covering of P . We now show that each corresponding 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 maps λ α : q × T P α → s, (( g, s), w) → ψ α ( g, w) − s and where m q ∈ T q Q and w pα ∈ T pα P α .
Finally, smoothness of ω α is an easy consequence of smoothness of the maps ρ, λ α and µ : T Q → q, m q → dL q −1 m q with m q ∈ T q Q. For this observe that µ = dτ • κ for τ : 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 ω out of the data {(P α , λ α )} α∈I . Here, uniqueness and smoothness of ω will follow from uniqueness and smoothness of the maps ω α . Proposition 4.8. There is one and only one s-valued 1-form ω on P such that ω α = (Θ * ω)| T Q×T Pα holds for all α ∈ I. Moreover, ω 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 maps ω α . To this end, let p ∈ P , α ∈ I and p α ∈ P α such that p = q · p α for some q ∈ Q. By Lemma 3.7.1 for w p ∈ T p P we find some η ∈ T q Q × T pα P α with w p = d (q,pα) Θ( η), so that uniqueness follows from 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 λ pα : T pα P → s with Let λ α : pα∈Pα T pα P → s denote the (unique) map whose restriction to T pα P is λ pα for each p α ∈ P α . Then λ α = λ α • dΘ| q×T Pα and we construct the connection ω as follows. For p ∈ P we choose some α ∈ I and (q, p α ) ∈ Q × P α such that q · p α = p and define 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 pα P we have w = dΘ( q, w pα ) for some ( q, w pα ) ∈ q × T pα P α , and since dL q w pα ∈ T p β P , there is η ∈ q × T p β P β such that d (e,p β ) Θ( η ) = dL q w pα . It follows from the conditions 1 and 2 in Lemma 4.7 that where for the third equality we have used that Consequently, if q · p β = p with ( q, p β ) ∈ Q × P β for some β ∈ I, then p β = (q −1 q) −1 · p α and well-definedness follows from where the last step is due to (4.6) with w = dL q −1 w p ∈ T pα P . Next, we show that ω fulfils the pullback property. For this, let ( m, w pα ) ∈ T q Q × T pα P α . Then In the third step we have used that L q −1 • Θ = Θ(L q −1 (·), ·). Finally, we have to verify that ω is a Φ-invariant, smooth connection. For this let p ∈ P and ( q, p α ) ∈ Q × P α with p = q · p α . Then for q ∈ Q and w p ∈ T p P we have So, it remains to show smoothness of ω and that ω p ( s(p)) = s holds for all p ∈ P and all s ∈ s. For the second property let p = q · p α for (q, p α ) ∈ Q × P α . Then q = (g, s) for some g ∈ G and s ∈ S and we obtain For smoothness let p α ∈ P α and choose a submanifold Q of Q through e, an open neighbourhood P α ⊆ P α of p α and an open subset U ⊆ P such that the restriction Θ := Θ| Q ×P α is a diffeomorphism to U . Then p α ∈ U because e ∈ Q , hence Since ω α is smooth and Θ is a diffeomorphism, ω| U is smooth as well. Finally, if p = q · p α for q ∈ Q, then L q (U ) is an open neighbourhood of p and is smooth because ω| U and L q −1 are smooth.

Corollary 4.2 and Proposition 4.8 now prove
Theorem 4.9. Let G be a Lie group of automorphisms of the principal fibre bundle P . Then for each Φ-covering {P α } α∈I of P there is a bijection between the corresponding set of reduced connections and the Φ-invariant connections on P .
As already mentioned in the preliminary remarks to Example 4.6, the second part of the next example shows the importance of the transversality condition for the formulation in [6]. 1. Let P = X × S for an n-dimensional R-vector space X and an arbitrary structure group S. Moreover, let G ⊆ X be a linear subspace of dimension 1 ≤ k ≤ n acting via Φ : G×P → P , (g, (x, σ)) → (g+x, σ). If W is an algebraic complement of G in X and P 0 := W ×{e S } ⊆ P , then P 0 is a Φ-covering since Θ : (G × S) × P 0 → P is a diffeomorphism and each ϕ-orbit intersects W in a unique point. Consequently, the Φ-invariant connections on P are in bijection with the smooth maps ψ : G × T W → s such that ψ w := ψ| G×TwW is linear for all w ∈ W . This is because the conditions i) and ii) from Corollary 4.2 give no further restrictions in this case. The Φ-invariant connection that corresponds to ψ is given by 2. In the situation of the previous part let X = R 2 , G = span R ( e 1 ), W = span R ( e 2 ) and P 0 = W × {e}. We fix 0 = s ∈ s and define ψ : g × T P 0 → s by . Then (U 0 , τ 0 , s 0 ) fulfils the conditions in [6] but we have im[d 0 τ 0 ]+im d e ϕ τ 0 (0) = span R ( e 1 ) = T 0 X = T 0 R 2 = R 2 . 17 As a consequence, ψ : g × T U 0 → s defined by ψ t := (Φ * ω ψ )| g×TtU 0 is smooth because for t = 0 and r ∈ T t U 0 = R we have ψ t (λ e 1 , r) = Φ * ω ψ λ e 1 , r · e 1 + 3t 2 r · e 2 = ω ψ ((t,t 3 ),e) (λ + r) · e 1 + 3t 2 r · e 2 = ψ t 3 (λ + r) · e 1 , 3t 2 r · e 2 = 3tr · s, as well as ψ 0 (λ e 1 , r) = 0 if t = 0. For the first step keep in mind that (Φ * ω ψ )| g×TtU 0 ( g, r) = (Φ * ω ψ )( g, d t s 0 (r)) by Convention 3.1.2. Then the maps µ := ψ | T U 0 and A t 0 ( g) := ψ g, 0 t 0 fulfil the conditions from Theorem 2 in [6] because ψ fulfils the three algebraic conditions in Example 4.6 18 . This, however, contradicts that ω ψ is not a smooth connection.

Particular cases and applications
In the first part of this section we consider Φ-coverings of P that arise from the induced action ϕ on the base manifold M of P . Then we discuss the case where Φ acts via gauge transformations on P . This leads to a straightforward generalization of the description of connections by consistent families of local 1-forms on M . In the second part we discuss the (almost-)fibre transitive case and deduce Wang's original theorem [8] from Theorem 4.9. Finally, we consider the situation where P is trivial and give examples in loop quantum gravity. Proof . This is immediate from Lemma 3.7.2.

Φ-coverings and the induced action
We now consider the case where (G, Φ) is a Lie group of gauge transformations of P , i.e., ϕ g = id M for all g ∈ G. Here we show that Theorem 4.9 can be seen as a generalization of the description of smooth connections by consistent families of local 1-forms on the base manifold M . For this let {U α } α∈I be an open covering of M and {s α } α∈I a family of smooth sections s α : U α → P . We define U αβ := U α ∩ U β and consider the smooth maps δ αβ : for v x ∈ T x U αβ and g ∈ G. Then we have Proof . By Corollary 5.1 {s α (U α )} α∈I is a Φ-covering of P . So, let {ψ α } α∈I be a reduced connection w.r.t. this covering. We first show that condition i) from Corollary 4.2 implies For this observe that condition a) from Remark 4.4 means that for all β ∈ I, p ∈ s β (U β ), w p ∈ T p s β (U β ) and g ∈ g, s ∈ s we have s ∈ s and all p ∈ P β .
But, since G x = G for all x ∈ M , this just means 22 ψ β g, 0 p = d e φ p ( g) for all g ∈ g and is in line with condition ii) from Corollary 4.2 as φ p is a Lie group homomorphism. Consequently, we can ignore this condition in the following. Now, we have p β = q · p α for q ∈ Q, p α ∈ P α , p β ∈ P β iff π(p α ) = π(p β ) = x ∈ U αβ and q = g, δ −1 αβ (g, x) . Consequently, the left hand side of condition i) from Corollary 4.2 reads 19 It is always possible to choose I = M . 20 This is that π • sα = idM α . 21 Observe that δ αβ (g, where v α , v β ∈ T x M and g ∈ G. This is true for v α = v β = v x , g = 0 and s = µ αβ (g, v x ), which follows from Consequently, by Corollary 4.2.i) we have 23 for all g ∈ G and all v x ∈ T x U αβ . Due to part 2) in Remark 4.4 the condition i) from Corollary 4.2 now gives no further restrictions so that for χ β := ψ β • ds β we have Then ψ β is uniquely determined by χ β for each β ∈ I so that (5.2) yields the consistency condition (5.1) for the maps {χ α } α∈I .
Example 5.3 (trivial action). If G acts trivially, then for each x ∈ U αβ we have so that δ αβ is independent of g ∈ G. Here Case 5.2 just reproduces the description of smooth connections by means of consistent families of local 1-forms on the base manifold M .

(Almost-)fibre transitivity
In this subsection we discuss the situation where M admits an element that is contained in the closure of each ϕ-orbit. For instance, this holds for all x ∈ M if each ϕ-orbit is dense in M and, in particular, is true for fibre transitive actions.
Case 5.4 (almost-fibre transitivity). Let x ∈ M be contained in the closure of each ϕ-orbit and let p ∈ π −1 (x). Then each Θ-patch P 0 ⊆ P with p ∈ P 0 is a Φ-covering of P . Hence, the Φ-invariant connections on P are in bijection with the smooth maps ψ : g × T P 0 → s for which ψ| g×TpP 0 is linear for all p ∈ P 0 and that fulfil the two conditions from Corollary 4.2.
The next example to Case 5.4 shows that evaluating the conditions i) and ii) from Corollary 4.2 at one single point can be sufficient to verify non-existence of invariant connections. Then M = w∈Sn G · π(w), G · π(e) = π(e) and π(e) ∈ G · π(w) for all w ∈ S n . Now, im[d e Θ e ] = g, since d e Θ e ( g ) = g for all g ∈ g. Moreover, So, condition i) from Corollary 4.2 gives ψ g, 0 e − g = 0, hence ψ g, 0 e = g for all g ∈ g.
Corollary 5.6. If Φ is fibre transitive, then {p} is a Φ-covering for all p ∈ P .
Proof . It suffices to show that {π(p)} is a ϕ-patch, since then {p} is a Θ-patch by Corollary 5.
showing that T x M = d e ϕ x (g).
Let φ be transitive and p ∈ P . Then {p} is a Φ-covering by Corollary 5.6 and T p {p} is the zero vector space. Moreover, we have p α = q · p β iff p α = p β = p and q ∈ Q p . It follows that a reduced connection w.r.t. {p} can be seen as a linear map ψ : g → s that fulfils the following two conditions: • d e Θ p ( g, s) = 0 =⇒ ψ( g ) = s for g ∈ g, s ∈ s, Since ker[d e Θ p ] = q p , we have shown Case 5.7 (Hsien-Chung Wang, [8]). Let (G, Φ) be a fibre transitive Lie group of automorphisms of P . Then for each p ∈ P there is a bijection between the Φ-invariant connections on P and the linear maps ψ : g → s that fulfil This bijection is explicitly given by ω → Φ * p ω.
1. Homogeneous connections. In the situation of Example 4.10 let k = n, X = R n . Then Φ is fibre transitive and for p = (0, e) we have G π(p) = e and g π(p) = {0}. Consequently, the reduced connections w.r.t. {p} are just the linear maps ψ : R n → s and the corresponding homogeneous connections are given by 2. Homogeneous isotropic connections. Let P = R 3 × SU(2) and : SU(2) → SO(3) be the universal covering map. We consider the semi direct product E := R 3 SU(2) whose multiplication is given by Since E equals P as a set, we can define the action Φ of E on P just by · . Then E is a Lie group that resembles the euclidean one, and it follows from Wang's theorem that the Φ-invariant connections are of the form (see e.g. Appendix A.3 in [5]) Here c runs over R and µ : and { e 1 , e 2 , e 3 } the standard basis in R 3 .
We close this section with a remark concerning the relations between sets of invariant connections that correspond to different lifts of the same Lie group action on the base manifold of a principal fibre bundle.
Remark 5.9. Let P be a principal fibre bundle and Φ, Φ : G × P → P be two Lie groups of automorphisms with ϕ = ϕ . Then the respective sets of invariant connections can differ significantly. In fact, in the situation of the second part of Example 5.8 let Φ ((v, σ), (x, s)) := (v + (σ)(x), s). Then ϕ = ϕ and Appendix B.1 shows that ω 0 ( v x , σ s ) := s −1 σ s for ( v x , σ s ) ∈ T (x,s) P is the only Φ -invariant connection on P .

Trivial bundles -applications to LQG
In this section we determine the set of isotropic connections on R 3 × SU(2) to be used for the description of isotropic gravitational systems in the framework of loop quantum gravity. To this end, we reformulate Theorem 4.9 for trivial bundles.
1. Let S be a Lie group and P = R n × S. We consider the action Φ : R >0 × P → P , (λ, (x, s)) → (λx, s) and claim that the only Φ-invariant connection is given by In fact, P ∞ := R n × {e} is a Φ-covering of P by Corollary 5.1 and it is straightforward to see 25 that condition i) from Corollary 4.2 is equivalent to the conditions a) and b) from Remark 4.4. Let ψ : g×T P ∞ be a reduced connection w.r.t. P ∞ and define ψ x := ψ| g×T (x,e) . Since the exponential map exp : g → R >0 is just given by λ → e λ for λ ∈ R = g, we have g((x, e)) = g · x ∈ T (x,e) P ∞ for g ∈ g. Then for w := − g · x ∈ T (x,e) P ∞ from a) we obtain In particular, ψ 0 g, 0 = 0 and since Q (0,e) = R >0 × {e}, for q = (λ, e) condition b) yields hence ψ 0 = 0. Analogously, for x = 0, w ∈ T (λx,e) P ∞ , λ > 0 and q = (λ, e), we obtain i.e., ψ λx 0 g , w = 1 λ ψ x 0 g , w . Here, in the second step, we have used the canonical identification of the linear spaces T (x,e) P ∞ and T (λx,e) P ∞ . Using the same identification, from continuity (smoothness) of ψ and ψ 0 = 0 we obtain so that ψ x 0 g , · = 0 for all x ∈ R n , hence ψ = 0 by (5.3). Finally, it is straightforward to see that (Φ * ω 0 )| g×T P∞ = ψ = 0.
2. Let P = R n \{0} × S and Φ be defined as above. Then K × {e}, for the unit-sphere K := {x ∈ R n | x = 1}, is a Φ-covering of P with the properties from Example 4.6.
Evaluating the corresponding conditions i ), ii ), iii ) immediately shows that the set of Φ-invariant connections on P is in bijection with the smooth maps ψ : R × T K → s for which ψ| R×T k K is linear for all k ∈ K. The corresponding invariant connections are given by Here pr denotes the projection onto the axis defined by x ∈ R n and pr ⊥ the projection onto the corresponding orthogonal complement in R n .
Also in the spherically symmetric case the ϕ-stabilizer of the origin has full dimension and it turns out to be convenient 26 to use the Φ-covering R 3 × {e} in this situation, too. Since the choice P ∞ := M × {e} is always reasonable 27 if there is a point in the base manifold M (of the trivial bundle M × S) whose stabilizer is the whole group, we now adapt Theorem 4.9 to this situation. For this, we identify T x M with T (x,e) P ∞ for each x ∈ M in the sequel.
Case 5.11 (trivial principal fibre bundles). Let (G, Φ) be a Lie group of automorphisms of the trivial principal fibre bundle P = M × S. Then the Φ-invariant connections are in bijection with the smooth maps ψ : g × T M → s for which ψ| g×TxM is linear for all x ∈ M and that fulfil the following properties. Let ψ ± ( g, v y , s) := ψ ( g, v y ) ± s for (( g, s), v y ) ∈ q × T y M . Then for q ∈ Q, x ∈ M with q · (x, e) = (y, e) ∈ M × {e} and all (( g, s), Proof . The elementary proof can be found in Appendix A.
for ( v x , σ s ) ∈ T (x,s) P and with rotation invariant maps a, b, c : R 3 → R for which the whole expression is a smooth connection. We claim that the functions a, b, c can be assumed to be smooth as well. More precisely, we show that we can assume that for smooth functions f, g, h : (− , ∞) → R with > 0. Then each pullback of such a spherically symmetric connection by the global section x → (x, e) can be written in the form for smooth functions f , g , h : (− , ∞) → R with > 0.
In particular, there are spherically symmetric connections on R 3 \{0}×SU(2) which cannot be extended to those on P . For instance, if b = c = 0 and a(x) := 1/ x for x ∈ R 3 \{0}, then ω abc cannot be extended smoothly to an invariant connection on R 3 × SU(2) since elsewise a n could be extended to a continuous (smooth) function on R.

Conclusions
We conclude with a short review of the particular cases that follow from Theorem 4.9. For this let (G, Φ) be a Lie group of automorphisms of the principal fibre bundle (P, π, M, S) and ϕ the induced action on M .
• If P = M × S is trivial, then M × {e} is a Φ-covering of P . As we have demonstrated in the spherically symmetric and scale invariant case (cf. Examples 5.10 and 5.12), this choice can be useful for calculations if there is a point in M whose ϕ-stabilizer is the whole group G.
• There is an element x ∈ M which is contained in the closure of each ϕ-orbit. Then every Θ-patch that contains some p ∈ π −1 (x) is a Φ-covering of P , see Example 5.5. If ϕ acts transitively on M , then for each p ∈ P the zero-dimensional submanifold {p} is a Φ-covering of P leading to Wang's original theorem, see Case 5.7 and Example 5.8.
• Let Φ act via gauge transformations on P . In this case each open covering {U α } α∈I of M together with smooth sections s α : U α → P provides the Φ-covering {s α (U α )} α∈I of P . If G acts trivially, this specializes to the usual description of smooth connections by means of consistent families of local 1-forms on the base manifold M .
• If we find a Θ-patch P 0 such that π(P 0 ) intersects each ϕ-orbit in a unique point, then P 0 is a Φ-covering. If, in addition, the stabilizer Q p does not depend on p ∈ P 0 , then we are in the situation of [6], see Example 4.6.
• Assume there is a collection of ϕ-orbits forming an open subset U ⊆ M .  [7], giving rise to a canonical Φ-covering (of P and O) consisting of convenient patches. So, using the present characterization theorem there is a realistic chance to get some general classification results in the compact case 28 .
In the general situation one can always construct Φ-coverings of P from families of ϕ-patches in M as Corollary 5.1 shows. In particular, the first three cases arise in this way.