Principal bundles and connections modelled by Lie group bundles

In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article.


Introduction
A fiberwise action of a Lie group fiber bundle on a smooth bundle over the same base leads to the notion of generalized principal bundle. They share some similar properties to (standard) principal bundles [13, Ch.II], but they also show important differences. For instance, on one hand it is possible to build (generalized) associated bundles by the action of Lie group bundles on other bundles. But on the other hand, Ehresmann connections [14,21] that are equivariant with respect to the action, require a precise and correct approach providing the corresponding notion of generalized principal connections. Unlike usual principal connections, they are always associated to a certain connection on the Lie group bundle, which gives an additional term in the equivariance formula. Lie group bundles, together with their natural infinitesimal companions, the Lie algebra bundles, are classical objects in the literature (see, for example, [9]). These type of bundles naturally arise in various geometric contexts and in different applications. The reader should be aware that these bundles may involve questions concerning non-isomorphic algebraic structure between different fibers (the seminal work of Douady and Lazard [9] already discussed this situation; see [1] for a recent approach). Although this scenario is very interesting and beautiful, in this work we confine ourselves to the case where the fibers are algebraically isomorphic (that is, the bundle is locally trivial from an algebraic point of view), a decision mainly motivated by the bundles that one encounters in the applications. With respect to them, they appear in a natural way when performing reduction by local symmetries in Lagrangian field theory [11,19] and, since principal connections may be regarded from this perspective, they are also useful for classical reduction -that is, reduction by global symmetries-in mechanics [5,6,18] and field theories [3,4,10]. Lie group bundles are the unavoidable starting point in the geometric foundations of gauge theories. In particular, they provide the basic language for a theory of geometric reduction of field theories when the group of symmetries are sections of Lie group bundles. This theory (that is still in progress) will collect the main instances of gauge theories and will require a wise use of the concepts exposed in this article. Actually, our initial interest in generalized principal bundles started in that framework, from where we have taken much inspiration. We would like to mention that fibered actions can be extended to the Lie groupoid setting (for example see [7,8,16]), but we do not address this matter here.
Despite the interest and ubiquitous presence of these objects, it is remarkable to check the existence of some important gaps in the literature about the main properties, definitions, and the key geometric objects involved. In this work we aim at solving this situation with the study of fibered actions, as well as the smooth bundles arising from the quotient by these actions: generalized principal bundles. Furthermore, we define equivariant connections on these bundles (i.e., generalized principal connections) and their curvature. Before doing that, we need to define Lie group bundle connections, which are connections on a Lie group bundle that respect the multiplicative structure. Actually, the generalized principal connections will be associated to Lie group bundle connections, a situation that does not have a counterpart in the notion of (standard) principal bundle connections.
The paper is organized as follows. In Sect. 2 we investigate fibered actions and quotients by them. After that, the definition of infinitesimal generators is recalled and generalized associated bundles are defined. In Sect. 3 we introduce Lie group bundle connections, as well as the induced linear connection on the Lie algebra bundle. Then generalized principal connections are defined and characterized using parallel transport. Besides, we prove a theorem of existence and study their curvature. In Sect. 4 we present several examples to illustrate the ideas of this work. In particular, we show that usual principal bundles and connections are particular cases of the generalized objects. We have a similar situation with connections in affine bundles. Finally, the action of the gauge group on connections is modelled with Lie group bundle actions. In this case, the generalized principal bundle provides a new approach to the well-known Utiyama Theorem.
In the following, every manifold or map is smooth, meaning C ∞ . In addition, every fiber bundle π Y ,X : Y → X is assumed to be locally trivial and is denoted by π Y ,X . Given Y ,X ({x}) denotes the fiber over x. The space of (smooth) global sections of π Y ,X is denoted by (π Y ,X ). In particular, vector fields on a manifold X are denoted by X(X ) = (π T X,X ), where T X is the tangent bundle of X . Likewise, the space of local sections on an open set U ⊂ X is denoted by (U, π Y ,X ). The derivative, or tangent map, of a map f ∈ C ∞ (X , X ) between the manifolds X and X is denoted by . When working in local coordinates, we will assume the Einstein summation convention for repeated indices. A compact interval will be denoted by I = [a, b].

Actions of Lie group bundles
A Lie group fiber bundle with typical fiber a Lie group G is a fiber bundle π G,X : G → X such that for any point x ∈ X the fiber G x is equipped with a Lie group structure and there is a neighborhood U ⊂ X and a diffeomorphism x ∈ U × G → π −1 G,X (U) preserving the Lie group structure fiberwisely.
Note that the multiplication map M : G × X G → G and the inversion map · −1 : G → G are bundle morphisms covering the identity id X : X → X , where × X denotes the fibered product. Likewise, the map 1 : X → G that assigns the identity element 1 x ∈ G x to each x ∈ X is a global section (called the unit section) of π G,X . Any Lie group bundle defines a Lie algebra bundle π g,X : g → X as the vector bundle whose fiber g x at each x ∈ X is the Lie algebra of G x . That is, the Lie algebra bundle is the pull-back bundle g = 1 * (V G), where V G ⊂ T G is the vertical bundle of π G,X , i.e., the kernel of (π G,X ) * . Remark 2.1 (Jets of Lie group fiber bundles) Let π G,X : G → X be a Lie group fiber bundle and r ≥ 0 be an integer. Then the r -th jet bundle of π G,X , J r G → X , is again a Lie group fiber bundle (see, for example, [11,§3,Th. 1]). The multiplication is inherited from the Lie group bundle structure of π G,X , that is, We consider subgroups of Lie group bundles in the following sense.

Definition 2.1
A Lie group subbundle of a Lie group bundle π G,X : G → X is a Lie group bundle π H,X : H → X such that H is a submanifold of G and H x is a Lie subgroup of G x for each x ∈ X . It is said to be closed if H x is a closed Lie subgroup of G x for every x ∈ X .
Let π Y ,X be a fiber bundle and π G,X be a Lie group fiber bundle.
covering the identity id X : X → X such that (y, hg) = ( (y, h), g) and (y, For the sake of simplicity, we will denote (y, g) = y · g and we will say that π G,X acts fiberwisely on the right on π Y ,X . Note that induces a right action on each fiber, The fibered action is said to be free if y · g = y for some (y, g) ∈ Y × X G implies that g = 1 x , x = π Y ,X (y). In the same way, it is said to be proper if the bundle morphism Y × X G (y, g) → (y, y · g) ∈ Y × X Y is proper. If is free and proper, so is each action x , since the fibers of a bundle are closed.
As the fibered action is vertical (i.e., it covers the identity id X ), we may regard the quotient space Y /G as the disjoint union of the quotients of the fibers by the induced actions, that is, Obviously, the following diagram is commutative: Example 2.1 (Jet lift of fibered actions) Let : Y × X G → Y be a (right) fibered action of a Lie group bundle π G,X on a fiber bundle π Y ,X . The first jet extension of turns out to be a (right) fibered action of π J 1 G,X on π J 1 Y ,X , s, γ )).
If we regard 1-jets as differentials of sections at a point, that is, j 1 x s ≡ (ds) x and j 1 x γ ≡ (dγ ) x for certain local sections s and γ , then (1) may be seen as (1)

Smooth structure of fibered quotients
Theorem 2.1 If π G,X acts on π Y ,X freely and properly, then Y /G admits a unique smooth structure such that (i) π Y ,Y /G is a fiber bundle with typical fiber G, called generalized principal bundle.
Proof Taking trivializations of π Y ,X and π G,X on the same neighborhood U ⊂ X , the local expression of the actions is: whereŶ is the typical fiber of Y . This can be seen as a standard Lie group action acting trivially on U. The classical results on Lie group actions (for example, see [15,Theorem 7.10]) provide a unique smooth structure on (U ×Ŷ )/G that can be used as a chart on Y /G. In these trivializations, the projections π Y ,Y /G and π Y /G,X are U ×Ŷ → (U ×Ŷ )/G and (U ×Ŷ )/G → U which gives (i) and (ii).

Remark 2.2
If a Lie group G acts freely, properly and fiberwisely on a bundle π Y ,X : Y → X , then π Y /G,X : Y /G → X is a bundle with typical fiberŶ /G. However, in the case of action of Lie group bundle, since the action depends on X , the notion of typical fiber is more delicate. For example, if X is not connected, the topology of the typical fiber may differ on each component. This is the case of the action of The typical fiber is a Klein bottle for t < 0 and a torus for t > 0.
For a connected base manifold X , the quotientŶ /G is well-defined (up to diffeomorphism). Indeed, given any two points p, q ∈ X connected by a path, p = γ (0), q = γ (1), the action on Y t = π −1 Y ,X (γ (t)) can be regarded (from the local trivializations of G and Y ) as a homotopy of diffeomorphism of G onŶ . This gives a diffeomorphism of the fibers of Y /G on p and q.
We can build a trivializing atlas for π Y ,Y /G starting from a trivializing atlas and pick a local sectionŝ ∈ (V α , π Y ,Y /G ) (the existence of that local section may require the choice of a smaller U α ). We define and it is unique because the fibered action is free. It turns out that (V α , ψ α Y ) is a trivialization for π Y ,Y /G . Indeed, its inverse is given by As a result, (V α , ψ α Y ) | α ∈ is an atlas for π Y ,Y /G . Observe that, by definition of Lie group bundle, the maps ψ α are Lie group homomorphisms. It is thus straightforward that the fibered action is locally given by the right multiplication on the Lie group G.
Moreover, for each x ∈ U α the following diagram is commutative where π g,X is the Lie algebra bundle of π G,X , g is the Lie algebra of G and exp is the exponential map between the Lie algebra and its corresponding Lie group. Furthermore, note In fact, we can define a linear trivialization (U α , ψ α g ) for π g,X from (U α , ψ α G ). Namely,

Infinitesimal generators
If we fix x ∈ X , y 0 ∈ Y x and g 0 ∈ G x , we can consider the maps In the same way, we denote by L g 0 : G x → G x and R g 0 : G x → G x the left and right multiplication by g 0 ∈ G x , respectively. Infinitesimal generators (or fundamental fields) are defined in the same fashion as in classical actions of Lie groups. Namely, for each ξ ∈ g x , Of course, they are also π Y ,X -vertical. Proposition 2.1 Let π g,X be the Lie algebra bundle of π G,X . The following map is a vertical isomorphism of vector bundles over Y : In addition, for any (g, ξ) ∈ G × X g, we have: The linearity is clear from the second equality of (6). Since dim g x = dim G x = dim V y Y (the last equality is for being π Y ,Y /G a submersion), we just need to prove the injectivity to conclude. Let ξ ∈ g x be such that ξ * y = 0. Then, y · exp tξ = y for every t ∈ (− , ). Since the action is free, this says that exp tξ = 1 x for every t ∈ (− , ) and, hence, ξ = 0. The second part is a straightforward computation.

Generalized associated bundles
Let π F,X : F → X be a fiber bundle on which π G,X acts fiberwisely on the left. This yields a right fibered action of π G,X on product bundle π Y × X F,X . Namely, If the fibered action of π G,X on π Y ,X is free and proper, so is the induced action on π Y × X F,X . In such case, we have the smooth manifold We denote the equivalence classes by

Proposition 2.2
In the above conditions, ifF is the typical fiber of π F,X , then π Y × G F,Y /G is a fiber bundle with typical fiberF, called generalized associated bundle, where Proof It is clear that the projection is well-defined and surjective. For each [y 0 , f 0 ] G ∈ Y × G F, let us find a trivialization through that point. Let U ⊂ X be a trivializing set of both π G,X and π F,X , such that π Y ,X ( and suppose that it is a trivializing set of π Y ,Y /G (maybe we need to choose a smaller U in order to achieve this). Using the above trivializations, we define where the action G ×F →F is induced by the fibered action of π G,X on π F,X . Observe that this action may depend on the base point x. Nevertheless, the condition of : where again x = π Y /G,X (σ ).

Example 2.2 (Conjugacy bundle)
A Lie group bundle π G,X acts fiberwisely on the left on itself by conjugation: We can thus consider the generalized conjugacy bundle, Y × G G, which inherits the Lie group bundle structure from π G,X .

Example 2.3 (Adjoint bundle)
In the same vein, π G,X acts fiberwisely on the left on π g,X via the adjoint map: The corresponding quotient is the generalized adjoint bundle, which we will denote byg = Y × G g. It is a vector bundle equipped with a Lie algebra bundle structure.

Example 2.4 (Coadjoint bundle)
Let g * be the dual vector bundle of π g,X , where π G,X acts fiberwisely on the left via the coadjoint representation, g · η = Ad * g −1 (η), (g, η) ∈ G × X g * . Using this action we get the generalized coadjoint bundle,g * = Y × G g * , which is a vector bundle.

Lie group bundle connections
Recall that an Ehresmann connection (see for example [14]) on a fiber bundle π Z ,X : Z → X is a fiber map T Z → V Z = ker(π Z ,X ) * such that its restriction to V Z is the identity. Similarly, we can understand an Ehresmann connection as a distribution H Z ⊂ T Z complementary to V Z. Finally, an Ehresmann connection is also a section of the jet bundle If π G,X is a Lie group bundle, an Ehresmann connection can be also regarded as a vertical bundle map (denoted by the same letter for the sake of simplicity) Furthermore, it is natural to impose a compatibility of ν with the algebraic structure of G.

Definition 3.1 A Lie group bundle connection on π G,X is an Ehresmann connection
where M : G × X G → G is the fiber multiplication map.
The corresponding conditions when regarded as a map ν : Geometric interpretations of Lie group bundle connections are provided by the following results. We denote by ν the parallel transport of ν and by Hor ν g : T x X → T g G its horizontal lift at any g ∈ G, x = π G,X (g).

Then ν is a Lie group connection if and only if for any curve x
Consequently, Proof Suppose that ν is a Lie group connection. Let x : I → X be a curve and g, h ∈ G x(a) . Denote x(a) (gh), t ∈ I , we use the uniqueness of the parallel transport. It is clear that π G,X • α = x and α(a) = gh, so we only need to check that it is horizontal, since α 1 and α 2 are horizontal. Conversely, suppose that ν satisfies (9).
We conclude that ν is a Lie group connection as a straightforward consequence of (9) and the definition of covariant derivative,

Proposition 3.2 Let ν be an Ehresmann connection on π G,X and consider the corresponding jet sectionν ∈ (π J 1 G,G ). Then ν is a Lie group bundle connection if and only if
Thanks to the previous Proposition, it is enough to show that (ii) is equivalent to (9). Let x ∈ X , U x ∈ T x X and g, h ∈ G x and pick a curve α : We define the following curve: .
where Hor ν g : T x X → T g G is the horizontal lifting given by ν. Performing the same construction with h and gh we obtain curves Since the equality is valid for every U x ∈ T x X , we conclude thatν(gh) =ν(g)ν(h).
From the proof above we also deduce that the condition together with (i) of Definition 3.1 characterize Lie group connections. More generally, we have the following property.

Proposition 3.3 Let ν be a Lie group connection on π G,X . Then for each
Proof Thanks to the uniqueness of the horizontal lifting, it is enough to show that is horizontal and projects to U x . For horizontality we have that The last equality is a particular case of property (ii) of the definition of Lie group connection with U g = 0. In such case, we have (d M) (g,h) (0, U h ) = d L g (U h ) and, thus, At last, we check that it projects to U x . In order to do so, let α = (α 1 , With respect to this last Proposition, if the vector U h is horizontal, then an identity that is similar to the corresponding property of connections on standard principal bundles.

Induced connection on the Lie algebra bundle
A Lie group connection induces a linear connection on the corresponding Lie algebra bundle.
is a linear parallel transport on π g,X .
Proposition 3.5 Let x : I → X. Then x(a) Ad g (ξ ), t ∈ I , which satisfy α(a) = β(a) = Ad g (ξ ) and π g,X •α = π g,X •β = x. Thanks to the uniqueness of parallel transport, to show that α = β it is enough to check that α = β . For each t ∈ I , using the definition of g we have where ( ) is due to ν being a Lie group connection and ( ) comes from the fact that c g : G x(a) → G x(a) is a Lie group homomorphism.
We denote by ∇ g and ∇ g /dt the linear connection and the covariant derivative on π g,X corresponding to this parallel transport g , respectively. A more practical way of regarding ∇ g may be the following. Consider the Lie group bundle connection as a map ν : T G → g. Then, for any section ξ ∈ (π g,X ), we have The next result is the infinitesimal version of Proposition 3.5.

Generalized principal connections
Let π Y ,X : Y → X be a fiber bundle on which a Lie group bundle π G,X : G → X acts freely and properly on the right, and denote by : Y × X G → Y the fibered action.
Proof It is a straightforward computation using that Ehresmann connections on π Y ,Y /G are identified with forms as follows.

Proposition 3.7
There is a bijective correspondence between Ehresmann connections on π Y ,Y /G and 1-forms 1 ω ∈ 1 (Y , g) such that Furthermore, such forms satisfy Proof The equivalence between Ehresmann connections on π Y ,Y /G and the 1-forms as in the statement is a consequence of the isomorphism (7). For the second part, let (y, g, ξ) ∈ Y × X G × X g, then: where equation (8) has been taken into account.
Observe that U v y = ω y (U y ) * y for each U y ∈ T y Y , y ∈ Y , and, thus U y is horizontal if and only if ω y (U y ) = 0. We are ready to introduce generalized principal connections.

Definition 3.2 Let ν be an Ehresmann connection on the Lie group bundle π G,X . A generalized principal connection on π Y ,Y /G associated to ν is an Ehresmann connection
Thanks to Proposition 3.7, it is easy to check that the following is an equivalent way of defining a generalized principal connection.
(ii) (Ad-equivariance) For each (y, g) ∈ Y × X G and (U y , Roughly speaking, generalized principal connections extend the property given in Lemma 3.1 and Proposition 3.7 to non-necessarily vertical vectors U g with respect to ν.
The next result gives a geometric interpretation of the above Definitions in terms of the parallel transports ν and ω , in the same vein as Proposition 3.1.

Proposition 3.8
Let ν : T G → g and ω ∈ 1 (Y , g) be Ehresmann connections on π G,X and π Y ,Y /G , respectively. Then ω is a generalized principal connection associated to ν if and only if for any curve γ : I → Y /G, the corresponding parallel transports satisfy Proof Suppose that ω is a generalized principal connection associated to ν. Let γ : I → Y /G be a curve, x = π Y /G,X • γ , y ∈ Y γ (a) and g ∈ G x(a) . Denote α 1 (·) = ω γ (·) γ (a) y, α 2 (·) = ν x(·) x(a) g and α = • (α 1 , α 2 ). To show that α(·) = ω γ (·) γ (a) (y · g) we use the uniqueness of the parallel transport. It is clear that π Y ,Y /G • α = γ and α(a) = y · g, so we only need to check that α is horizontal. For each t ∈ I we have since α 1 and α 2 are horizontal for the corresponding connections. Conversely, suppose that ω and ν satisfy (11). Let (y, g) A straightforward consequence of (11) and the definition of covariant derivative is the following: Thus, we conclude that ω is a generalized principal connection associated to ν: Proposition 3.9 Let ν : T G → g and ω ∈ 1 (Y , g) be Ehresmann connections on π G,X and π Y ,Y /G , respectively, and suppose that Y /G = X. Letω ∈ (π J 1 Y ,Y ) andν ∈ (π J 1 G,G ) be the corresponding jet sections. Then ω is a generalized principal connection associated to ν if and only ifω (y · g) =ω(y) ·ν(g), where the fibered action of π J 1 G,X on π J 1 Y ,X is given by the first jet extension of (recall Example 2.1).

Conversely, if
By uniqueness of the parallel transport we conclude that β y·g = • (β y , β g ).
We denote by Hor ω y : T [y] G (Y /G) → T y Y the horizontal lift given by ω at y ∈ Y . A different geometric interpretation of generalized principal connections is the following.
Proof We just need to prove that U = (d ) (y,g) Hor ω y·g is horizontal and projects to U [y] G . With respect to the former For the latter let α = (α 1 , α 2 ) : (− , ) → Y × X G be such α (0) = Hor ω y (U [y] G ), U g . We then have Theorem 3.1 (Existence of generalized principal connections) If X is a paracompact smooth manifold, then there exist an Ehresmann connection ν on π G,X and a generalized principal connection ω on π Y ,Y /G associated to ν.
Fixed α ∈ , we define the local Ehresmann connection ν α : We show that ν α and ω α satisfy both properties of Definition 3.3: Using Corollary 2.1 and equation (4) we get Thence, η . Note that we can always choose the curves satisfying π Y /G,X • γ 1 = β 1 . Using these curves, Corollary 2.1 and Equation (13) it can be seen that This, together with property (i), lead to where we have used (5) and the fact that (dψ α Therefore, ω α is a generalized principal connection on π Y ,Y /G | V α : Y | V α → V α associated to ν α . At last, we take a smooth partition of unity {θ α | α ∈ } on X subordinated to {U α | α ∈ }. It is easy to check that { α = θ α • π Y /G,X | α ∈ } is a smooth partition of unity on Y /G subordinated to {V α | α ∈ }. Now set ω = α∈ ( α • π Y ,Y /G ) ω α and ν = α∈ (θ α • π G,X ) ν α . They are a well defined 1-form ω ∈ 1 (Y , g) and a well defined Ehresmann connection ν : T G → V G. It is straightforward that ν and ω satisfy both properties of Definition 3.3. Thus, ω is a generalized principal connection associated to ν.
A 1-form α ∈ 1 (Y , g) is said to be tensorial of the adjoint type if it is horizontal, i.e., α y (U y ) = 0 for each U y ∈ V y Y = ker(dπ Y ,Y /G ) y , y ∈ Y , and Ad-equivariant, i.e., X (y). The family of tensorial 1-forms of the adjoint type is denoted by 1 (Y , g). These forms can be reduced to Y /G in the sense that we have a bijection By an abuse of notation, we identify α ≡α. This construction can be straightforwardly generalized to p-forms.

Proposition 3.11
Let ν : T G → g be an Ehresmann connection on π G,X . The family of generalized principal connections on π Y ,Y /G associated to ν is an affine space modelled on 1 (Y , g).
On the other hand, given a generalized principal connection ω ∈ 1 (Y , g) associated to ν and a 1-form ω ∈ 1 (Y , g), a straightforward computation shows that ω + ω is a generalized principal connection, since ω vanish on vertical vectors. Now we give another interpretation of a generalized principal connection as a connection on π Y × X G,X equivariant under the fibered action.

Proposition 3.12
Let ω ∈ 1 (Y , g) be a generalized principal connection on π Y ,Y /G associated to an Ehresmann connection ν : T G → V G on π G,X . Then the map where V (Y × X G) = ker π Y × X G,X * , is an Ehresmann connection on π Y × X G,X equivariant under the map 8 : Y × X G → Y × X G defined as 8(y, g) = (y · g, g), i.e., (d8) (y,g) (y,g) U y , U g = 8(y,g) (d8) y,g (U y , U g ) , Proof To begin with, observe that is well defined, i.e., (y,g) This is a straightforward consequence of equality V (y,g) (Y × X G) = V y Y × V g G, where V y Y = ker(π Y ,X ) * , and the fact that infinitesimal generators are π Y ,X -vertical. Likewise, it is clear that is a vertical vector bundle morphism over Y × X G, since ω y , ν g and the infinitesimal generator are linear maps. To conclude, let us check the 8-equivariance. Note that (d8) (y,g) U y , U g = (d ) (y,g) So far, we have considered generalized principal connections associated to arbitrary Ehresmann connections on G → X . We now prove that this Ehresmann connections must be a Lie group bundle connection, that is, they must respect the algebraic structure of π G,X .

Proposition 3.13
Let ω ∈ 1 (Y , g) be a generalized principal connection on π Y ,Y /G associated to an Ehresmann connection ν : T G → g on π G,X . Then ν is a Lie group bundle connection.
Proof Let γ : I → Y /G projecting onto x : I → X . Thanks to (11) for each y ∈ Y γ (a) and g, h ∈ G x(a) we have Since the action is free, we conclude that property (9) holds. On the other hand, applying (11) with g = 1 x(a) and using again the fact that the action is free, we get This gives that ker ν 1 x = (d1) x (T x X ) for each x ∈ X and we conclude thanks to Proposition 3.1. Indeed, let U x ∈ ker ν 1 x = H 1 x and denote u x = (dπ G,X ) 1 x (U x ) ∈ T x X . Let x : (− , ) → X be such that x (0) = u x . Then we have that Therefore, ker The other inclusion is due to the fact that both are vector spaces of the same dimension.

Curvature
The cuvature ω as an Ehresmann connection (see, for example [14, §9.4] This is equivalent, by the identification (7), to the g-valued 2-form ∈ 2 (Y , g) which will be denoted the same. The linear connection ∇ g induced on π g,X by ν enables us to express the curvature as follows.
Proposition 3.14 Let d g be the exterior covariant derivative 2 associated to ∇ g . Then 3 As in the case of (standard) principal connections, it is possible to regard the curvature as a 2-form on the base space Y /G with values ing.

Definition 3.4
The reduced curvature of ω is the 2-form˜ ∈ 2 Y /G,g taking values in the adjoint bundle given bỹ The reduced curvature is well-defined, i.e., it does not depend on the choice of y ∈ Y . Indeed, let g ∈ G x , where x = π Y ,X (y), u i = (dπ Y /G,X ) [y] G (U i ) ∈ T x X for i = 1, 2 and γ ∈ (π G,X ) be such that γ (x) = g and ν γ (x) • (dγ ) x = 0. Proposition 3.10 gives Hence, we have y · g, y·g Hor ω y·g (U 1 ), Hor ω y·g (U 2 ) where we have used that (dγ )

Standard principal bundles and connections
Generalized principal connections reduce to usual principal connections on (standard) principal bundles (for example, [13, Ch.II], [14, Ch.III]). Let π P,X be a principal G-bundle and denote by R : P × G → P the corresponding right action. We define a fibered action of the trivial Lie group bundle G = X × G on π P,X as : P × X G −→ P (y, (x,ĝ)) −→ Rĝ(y) = y ·ĝ.
Note that P/G X , so we can regard π P,X as a generalized principal bundle with respect to this fibered action.
Let ν 0 be the trivial connection on π G,X , that is, the one given by In addition, note that the Lie algebra bundle of π G,X is g = X × g, where g is the Lie algebra of G.
Then ω is a generalized principal connection associated to ν 0 if and only ifω is a (standard) principal connection.
Subsequently, we only need to check Ad-equivariance to conclude.

Affine bundles and connections
Generalized principal connections on an affine bundle are just affine connections (see [13], [20]) associated to linear connections on the modelling vector bundle. A vector bundle π E,X is an abelian Lie group bundle with the additive structure. A Lie group connection ν : T E → E is just a linear connection, since it respects this additive structure. If we consider linear bundle coordinates (x μ , v A ) corresponding to a basis of local sections {e A : 1 ≤ A ≤ m} of π E,X , then we may write for some (local) functions ν B μ,A ∈ C ∞ (X ), 1 ≤ μ ≤ n, 1 ≤ A, B ≤ m. Now consider an affine bundle π E,X modelled on π E,X . We have the following fibered action G on g is given by the adjoint representation. The group of gauge transformations acts on connections. This action is fiberwisely expressed as . The 1-jet lift of this action is This is again a (left) fibered action, but it is not free. Nevertheless, we may consider the following Lie group subbundle of J 1 J 1 Ad(P) , J 2 0 Ad(P) = j 2 x γ ∈ J 2 Ad(P) | γ (x) = 1 Ad(P) x ⊂ J 2 Ad(P) ⊂ J 1 J 1 Ad(P) . The restriction of the action to this Lie group subbundle is free and proper, making J 1 C(P) −→ J 1 C(P)/J 2 0 Ad(P) a generalized principal bundle that is not a (standard) principal bundle. Furthermore, the quotient is isomorphic to the curvature bundle of π P,X , where we denote by F A ∈ 2 (X , ad(P)) the reduced curvature corresponding to a principal connection A ∈ 1 (P, g). This is (the main part of) the geometric statement of the wellknown Utiyama theorem (for example, see [12, §5]).

An example of generalized principal connection
Within the framework of gauge field theories analyzed above, we now present a simple example of a generalized principal connection. Even though one initially would like to have such connection on C(P) = (J 1 P)/G, the action of J 1 Ad(P) on C(P) is not free. To remedy this, we may consider the action on J 1 P instead, . that is free and and proper. Actually, this construction is conceptually close to the case of affine connections (see §4.2 above), since J 1 Ad(P) acts transitively on J 1 P.