Routh reduction of Palatini gravity in vacuum

An interpretation of Einstein-Hilbert gravity equations as Lagrangian reduction of Palatini gravity is made. The main techniques involved in this task are those developed in a previous work for Routh reduction in classical field theory. As a byproduct of this approach, a novel set of conditions for the existence of a vielbein for a given metric is found.


INTRODUCTION
The relationship between Einstein-Hilbert and Palatini formulation of gravity has been studied in several places. Nevertheless, the main theoretical tool used in the discussion of the connection between these formulations of gravity appears to be some flavor of Hamiltonian reduction. For instance, [12] and [28] use ADM formalism [1] in order to establish the connection; it has been explored also in [10,21], where the correspondence is set by using a Hamiltonian structure on the set of fields at the boundary.
From this viewpoint, it becomes interesting to find a reduction scheme relating the Lagrangian formulation of Palatini and Einstein-Hilbert gravity directly, without the detour through Hamiltonian formalism. So far there exists two ways to implement reduction at the Lagrangian level, namely Lagrange-Poincaré reduction [9,8,14] and Routh reduction [25,11,24,15,4,6]. Moreover, there are physical considerations that can be said in support of this kind of reduction: They deal not only with the reduction problem, but also with the reconstruction problem, and it is argued in [27] that reconstruction can be relevant from the physical point of view.
The problem with these approaches to Lagrangian reduction is that they work in the setting of classical variational problems, that is, variational problems where the velocity space is a jet space of the field bundle, and where the restrictions imposed on the fields are prescribed by the contact structure of the jet space. In the present work we want to study the reduction of a variational problem of a more general nature, namely, by using the formulation of Palatini gravity given in [3], where it was interpreted as an example of the so called Griffiths variational problems [19,20]. In this approach, the field bundle is the bundle of frames (whose sections are the vielbein) but the jet space is replaced by a submanifold of the jet space of the frame bundle, namely, by the submanifold corresponding to the torsion zero constraint; also, the contact structure is changed by a set of differential constraints implementing the metricity conditions. Therefore, it is necessary to find a formulation of a Lagrangian reduction scheme taking into consideration these characteristics; because of its versatility, we will choose to work with the Routh reductions as formulated in [6].
These considerations set the purposes of the following article: In one hand, to carry out a proof of concept for the generalization of Routh reduction to variational problems more general than those corresponding to first order field theory, generalizing the techniques employed in [6]; on the other hand, to apply Routh reduction of field theory in the context of a meaningful example, namely, a formulation of gravity with basis.
In order to be more specific about the nature of this generalization, let us briefly describe how Routh reduction in field theory is performed: • First, a unified formulation along the lines of [17] is constructed, and its ability for representing the extremals of the original variational problem is proved. This procedure must be done both for the original variational problem and the reduced one. The set of differential forms encoding the restrictions to be imposed on the fields are used at this stage. • The momentum map is defined in the unified setting, and its momentum level sets are determined. It should be proved that the equations of motion can be naturally restricted to these sets. • A connection on the bundle obtained by quotienting out the symmetry must be provided. Because of the characteristics of the contact structure on a jet space, this connection allows us to split the fields of the unified formalism. The splitting induced by the chosen connection allows us to define the Routhian and the force term for the reduced system. • It is necessary to set a common ground for the comparison of the extremals of the unreduced and reduced variational problems. This is done by considering an affine subbundle of the bundle of forms on a fibred product of bundles; the factors in this product are the bundles of the unreduced and the reduced system. • The equivalence between the extremals of the original variational problem and the reduced variational problem is checked by a map involving the translation along the force term (in the space of forms associated to the unified formalism).
In the reconstruction of the extremals of the unreduced system from the reduced dynamics, it is necessary to impose some integrability conditions.
The two first items can also be done for Griffiths variational problems; when trying to reproduce the third item in this generalized context, we have to face the problem that the splitting induced by the chosen connection strongly depends on features of the contact structure. Nevertheless, the metricity constraints can be formulated using forms belonging to the contact structure, and so the hopes of reproducing the third item in this context increase. A solution to this problem is provided by Lemma 8.1. No difficulties must arise from the last two items, as they are based on geometrical operations of general nature; the main results of the article are Theorem 10.3 and Theorem 10.10, which describe reduction and reconstruction respectively. The paper is organized as follows. In Section 2 we will review some geometrical tools necessary for the construction of the variational problem for Palatini gravity we will use in this article; the actual construction of this variational problem, as well as the associated unified problem, is done in Section 3. The symmetry considerations necessary to carry out the reduction are discussed in Section 4; Section 5 is technical, and contains some calculations used in the reduction and reconstruction theorems. In Section 6 the results achieved in the previous section are employed in the search of identifications between geometrical structures present in both the reduced and unreduced spaces: A remarkable fact in this vein is that the metricity constraints correspond after projection onto the quotient, with the contact structure of a jet bundle. Construction of the first order formalism for Einstein-Hilbert gravity (and its correspondence with the usual second order formalism) is delayed until Section 7; also, an unified formalism for this variational problem is discussed in this section. The choice of a connection induces a splitting in the contact structure on the jet space of the frame bundle; in Section 8 the effects of this splitting in the variational formulation of Palatini gravity are analyzed. The Routhian is constructed in Section 9: It is shown that the Routhian for Palatini gravity is the (first order) Einstein-Hilbert Lagrangian. Finally, in Section 10 the reduction theorem and the reconstruction theorem are proved. The main result of this section is the notion of flat condition for a metric, which is a helpful hypothesis in the proof of the reconstruction theorem.
Notations. We are adopting here the notational conventions from [29] when dealing with bundles and its associated jet spaces. Also, if Q is a manifold, Λ p (Q) = ∧ p (T * Q) denotes the p-th exterior power of the cotangent bundle of Q. Moreover, for k ≤ l the set of khorizontal l-forms on the bundle π : P → N is For the same bundle, the set of vectors tangent to P in the kernel of T π will be represented with the symbol V π ⊂ T P. In this regard, the set of vector fields which are vertical for a bundle map π : P → N will be indicated by X V π (P). The space of differential p-forms, sections of Λ p (Q) → Q, will be denoted by Ω p (Q). We also write Λ • (Q) = dim Q j=1 Λ j (Q). If f : P → Q is a smooth map and α x is a p-covector on Q, we will sometimes use the notation α f (x) • T x f to denote its pullback f * α x . If P 1 → Q and P 2 → Q are fiber bundles over the same base Q we will write P 1 × Q P 2 for their fibred product, or simply P 1 × P 2 if there is no risk of confusion. Unless explicitly stated, the canonical projections onto its factor will be indicated by Given a manifold N and a Lie group G acting on N, the symbol [n] G for n ∈ N will indicate the G-orbit in N containing n; the canonical projection onto its quotient will be denoted by Also, if g is the Lie algebra for the group G, the symbol ξ N will represent the infinitesimal generator for the G-action asssociated to ξ ∈ g. Finally, Einstein summation convention will be used everywhere.

GEOMETRICAL TOOLS FOR PALATINI GRAVITY
We will give a brief account of the construction carried out in [5]. The basic bundle is the frame bundle τ : LM → M on the spacetime manifold M (dimM = m); because it is a principal bundle with structure group GL (m), we can lift this action to the jet bundle J 1 τ, so that we obtain a commutative diagram is the so called connection bundle of LM, whose sections can be naturally identified with the principal connections of the bundle τ (for details, see [7] and references therein). It is interesting to note that there exists an affine isomorphism GL(m) and under this correspondence, the GL (m)-action is isolated to the first factor in the product, namely It means that a section of the bundle τ 1 is equivalent to a connection on LM plus a moving frame (X 1 , · · · , X m ) on M; although this moving frame has no direct physical interpretation, we can associate a metric to it, namely, in contravariant terms, for some nondegenerate symmetric matrix η (see Equation (2.1) below). It is the same to declare that the metric g is the unique metric on M making the moving frame (X 1 , · · · , X m ) (pseudo)orthonormal, with the signature given by η.

By means of the canonical basis {e
we can define the collection of 1-forms θ i , ω i j on J 1 τ such that We also have the formula GL(m) J 1 τ is the infinitesimal generator associated to A ∈ gl (m) for the lifted action. It can be proved that ω is a connection form for a principal connection on the bundle because every u ∈ LM is a collection u = (X 1 , · · · , X m ) of vectors on τ (u) ∈ M, and θ i is a τ 1 -horizontal 1-form on J 1 τ, we can define the set of forms and let η i j its (i, j)-entry; we will represent with the symbol η i j the (i, j)-entry of its inverse. With these ingredients we can construct the Palatini Lagrangian where Ω := Ω i j E j i is the curvature of the canonical connection ω. This m-form will determine the dynamics of the vacuum gravity in this formulation.
Finally, let us describe a decomposition on gl (m) induced by η. In fact, this matrix yields to a compact real form u in gl (m, C), given by u = ξ ∈ gl (m, C) : ξ † η + ηξ = 0 and thus we have a Cartan decomposition gl (m, C) = u ⊕ s.
we obtain the decomposition gl (m) = k ⊕ p.

Restrictions in Palatini gravity: Zero torsion submanifold and metricity forms.
It is time to discuss the restrictions we must impose on the sections of τ 1 in order to have a characterization of a gravity field in this description. Our aim is to describe a metric and a connection on the spacetime, and the restrictions to be considered will establish the relationship between them. There are two types of conditions to be imposed to a section of J 1 τ, each of them motivated on physical grounds (which we will not discuss here): (1) The connection which is a solution for the field equations of Palatini gravity must be torsionless, and (2) this connection must be metric for the solution metric.
The canonical forms defined in the previous section allow us to set the torsion form Now, every connection Γ : M → C (LM) gives rise to a section σ Γ : LM → J 1 τ of the bundle τ 10 : J 1 τ → LM, as the equivariance of the following diagram shows The interesting fact is that the pullback form σ * Γ T coincides with the torsion of the connection Γ. Additionally, it can be proved that T is a 1-horizontal form on τ 1 : J 1 τ → M, so that there exists a maximal (respect to the inclusion) submanifold i 0 : T 0 ֒→ J 1 τ such that (1) T 0 is transversal to the fibers of τ 1 : x s J 1 τ ), and (2) it annihilates the torsion, i.e.
The transformation properties of the form T allow us to conclude that T 0 is GL (m)invariant. The connections associated to sections of J 1 τ taking values in T 0 are torsionless, so that the zero torsion restriction can be achieved through the requirement that these sections would take values in this submanifold. Accordingly, we can use the affine isomorphism F : J 1 τ → LM × C (LM), to define the bundle of torsionless connections as the bundle τ ′ : C 0 (LM) → M obtained by restricting F to T 0 C 0 (LM) := pr 2 (F (T 0 )) .
Moreover, the following lemma can be proved using standard facts about principal bundles [22].
These considerations give rise to the commutative diagram . Let x µ , e ν k be a set of adapted coordinates for LM induced on τ −1 (U) by a set of coordinates (x µ ) on U ⊂ M; as usual, it induces coordinates On this open set we have On the other hand, the metricity condition has differential nature: As we mentioned before, matrix η determines a factorization of gl (m) in a subalgebra k (the subalgebra of η-Lorentz transformations) and an invariant subspace p. The explicit formulas for this decomposition are given by the projectors The metricity condition is imposed on a section Σ : M → J 1 τ by requiring that where ω p is the p-component of the canonical connection ω respect to this decomposition. Taking into account the affine isomorphism F :

2.2.
Contact-like structure and admissible sections. The scheme we will use for Routh reduction relies on the notion of unified formulation of a variational problem; it is a necessary step in order to avoid issues regarding the non regularity of the Lagrangian to be reduced [15]. In this vein, we should mention that our approach to the unified formalism is strongly based on the groundbreaking work of Gotay [17]. Although the aim in the previously cited article is to extend the definition of the Poincaré-Cartan form to a generalized family of variational problems (the so called Griffiths variational problems, see [19,20] and references), it can be readily seen that it can serve as a generalization of the unified formalism (as defined in [2,13,26] and references therein) to these kind of variational problems, namely, when restricted to the particular case of the classical variational problem (the variational problem underlying the first order classical field theory, see [18]), this construction reduces to the construction associated to the usual formulation of the unified formalism.
There are two crucial differences between a classical variational problem and a more general Griffiths variational problem: • First of all, a classical variational problem (of first order) is formulated in a first order jet bundle, whereas a Griffiths variational problem can use in principle any bundle. • More important (at least from the viewpoint of the present work) is the fact that the sections are integral for some characteristic set of forms. In the classical case, these set of forms are the contact forms, allowing us to restrict the set of forms to be varied to the holonomic sections of the jet bundle; the contact structure is replaced by another set of forms in the more general case.

Remark 2.3 (On the description of Griffiths variational problems).
As it is said in the previous paragraph, a Griffiths variational problem consists into three kind of data: A bundle p : W → M, whose sections will be the fields of the theory, a Lagrangian form λ ∈ Ω m (W ) setting the dynamics, and a set of forms I ⊂ Ω • (W ) (more precisely, an exterior differential system) describing the set of differential restrictions on the fields. Accordingly, we will often specify a variational problem of this kind with the symbol The variational problem underlying such triple consist into finding the extremals of the action where the sections s : M → W of the bundle p must be integral for the set of forms in I , namely, The variational problem we will consider here for the Palatini gravity is not a classical one; it will differ from a variational problem of this kind in both of the aspects mentioned above: • The relevant bundle is not the first order jet J 1 τ; instead, it is the subset T 0 consisting into the jets associated to torsionless connections. Due to this fact, we will consider the pullback of the canonical forms and the restriction of maps from J 1 τ to T 0 ; unless explicitly stated, the new forms and maps will be indicated with the same symbols. An exception to this rule will be the restriction of the bundle maps τ 10 and τ 1 , which will be indicated as τ ′ 10 : T 0 → LM and τ ′ 1 : T 0 → M. • The forms we will use for the restriction of the sections of τ ′ 1 : T 0 → M are not the whole set of contact forms ω i j , but a geometrically relevant subset, namely, the components of the metricity forms ω p .
In order to establish the unified version of the equations of motion for Palatini gravity, it will be necessary to define the metricity subbundle I m PG on T 0 , With the metricity subbundle in mind, we can define the affine subbundle Because this is a subbundle in the set of m-forms on T 0 , it has a canonical m-form λ PG on it given by

THE VARIATIONAL PROBLEM FOR PALATINI GRAVITY
The variational problem we will work with in the present article is the following.
Definition 3.1 (Griffiths variational problem for Palatini gravity). The variational problem for Palatini gravity is given by the action where · indicates the exterior differential system generated by the set of forms enclosed in the brackets.
The relevance of the unified formalism in dealing with variational problems is guaranteed by the following result [5].

Proposition 3.2. A section s : U ⊂ M → T 0 is critical for the variational problem established in Definition 3.1 if and only if there exists a section
Γ is called a solution of the Palatini gravity equations of motion.
Remark 3.3. Although the proof in [5] refers to sections of τ 1 : J 1 τ → M, it can be also readily adapted to cover this case; in this regard, see Appendix B.
The situation described by Proposition 3.2is summarized in the following diagram: We will see below (Section 7) that the same can be done for (first order) Einstein-Hilbert variational problem; the reduction and reconstruction theorems (see Section 10) will be proved using these lifted systems.

SYMMETRY AND MOMENTUM
We now discuss the presence of natural symmetries and their momentum maps for the unified formulation of Palatini gravity.
As we said above (see Lemma 2.1), there exists a GL (m)-action on T 0 ; nevertheless, the Lagrangian L PG is preserved by the action of the subgroup K ⊂ GL (m) composed of the linear transformations keeping invariant the matrix η, We can lift the GL (m)-action to ∧ m (T 0 ); it results that the subbundle I m PG is also preserved by the action of K, and so It is our aim to find a momentum map for this action, in the sense of the following definition.
Thus, we obtain Noether's theorem in this setting: and therefore the momentum is conserved along solutions.
Accordingly, we think of a "momentum" µ as an The construction of a momentum map for the action on W PG is standard [18]: for each ξ ∈ k, is an Ad * -equivariant momentum map for the GL (m)-action on W PG .
for all ξ ∈ k. It means that the unique allowed momentum level set for this symmetry is J = 0; accordingly, the isotropy group of this level set is K, and The other ingredient in Routh reduction is the factorization of the restriction bundle I m PG induced by a connection ω K on the underlying bundle p LM K : LM → Σ, where τ Σ : Σ := LM/K → M is the bundle of metrics of signature η. We will carry out this task in Section 8; here we will construct this connection. To this end, consider the decomposition associated to the matrix η (see Section 2). The connection ω K on the bundle p LM K : LM → Σ is induced by this decomposition, namely where π k : gl (m) → k is the canonical projector onto the k-factor in the Cartan decomposition and ω 0 is a connection form on the principal bundle τ : LM → M. The K-invariance of the factor p, ensures us that it has the expected properties of a connection. Finally, let us identify a candidate for the reduced bundle. In order to proceed, consider the adjoint bundle τ k : k → Σ; then, the following result holds.
is a bundle isomorphism.
The inverse of ϒ ω is given by where (·) H e , e ∈ LM, is the horizontal lift associated to ω K . The map ϒ ω enjoys a useful property: under this identification, the action of K on J 1 τ is simply This is a direct consequence of the equivariance of the principal connection ω K . As a result, we get the following corollary.
Remark 4.6. The choice of a connection on the bundle p LM K is allowing us to establish a relationship between the quotient space J 1 τ/K and the jet bundle of the metric bundle J 1 τ Σ , the latter being the relevant bundle in the Einstein-Hilbert approach to relativity. It will be studied in detail in Section 6.
Motivated by these considerations, we are in position to define what is the Lagrangian quotient bundle for Palatini gravity. In the next Sections we will explore a further simplification for this bundle, as well as a reduction for the Lagrangian responsible of the dynamics on these bundles.

LOCAL COORDINATES EXPRESSIONS
Here we will obtain some identities allowing us to write down the isomorphism ϒ −1 ω in local terms. In order to proceed, we fix a coordinate chart on M, inducing coordinates x µ , e µ k on LM. As usual, we will indicate with x µ , e µ k , e µ kσ the coordinates induced on J 1 τ. It can be proved that there exists a set of coordinates x µ , g µν , Γ σ µν on and adapted to this decomposition, namely In terms of these coordinates, we have It means in particular that On the other hand, a principal connection on LM can be written as

Proof. See Appendix C.
This proposition has the following consequence, that will be important to work with the reduction of the Palatini variational problem.

(5.2)
Proof. According to Proposition 5.1, we have that as required.
Let us now introduce coordinates on the vector bundle k. In order to do this, let us suppose that (φ = (x µ ) ,U) is a coordinate chart on M; then it is also a trivializing domain for the principal bundle LM, where Therefore we can define the coordinate chart φ k , τ −1 k (U) [7]. In order to proceed, we use the correspondence between the space of sections of the adjoint bundle Γτ k and the set of it means that, locally, these vector fields are such that In the following we will adopt the usual convention according to which the map Tt U is not explicitly written, namely, where are identified. We can write down any p LM K -vertical K-invariant vector field Z on LM as Z = A ρ σ E σ ρ ; then, using Equation (5.1), we obtain the following result. Proof. In fact, we have that and the identity follows.
Therefore, we will have that . In order to relate the coordinates A defines a set of coordinates on τ −1 k (U). Proof. According to the previous discussion, it is only necessary to prove that for any B ∈ k, i.e. such that η ik B j k + η jk B i k = 0, the corresponding element on T u LM, as required.

METRICITY AND CONTACT STRUCTURES ON THE QUOTIENT SPACE
In this section we will use the local expressions obtained in the Section 5 in order to study the relationship between the quotient bundle and the bundle J 1 τ Σ × Σ Lin τ * Σ T M, k . So far, we have the diagram (6.1) defining the diffeomorphism G ω ; here ϒ ω is the map induced by ϒ ω . In short, we will prove that the introduction of a connection on the bundle p LM K : LM → Σ allows us to split a principal connection on τ : LM → M into horizontal and vertical degrees of freedom. Moreover, this splitting will be powerful enough to relate the metricity forms ω p and the contact structure on the quotient bundle τ Σ : Σ → M.
First, let us stress that Lemma 5.4 allows us to set coordinates on the bundle p : Lin τ * Σ T M, k → Σ.
In fact, any element (g x , α) ∈ Lin τ * Σ T M, k admits coordinates x µ , g µν , A µ σ ρ if and only if (x µ , g µν ) are the corresponding coordinates for g x ∈ Σ and where e x ∈ LM is any element in p LM It is important to see the isomorphism ϒ ω restricted to T 0 . In order to properly set this result, let us construct the pullback bundles The zero torsion submanifold T 0 has some nice properties regarding the decomposition induced by the connection ω K . Proof. The proof of this proposition will be local. Using Equation (5.2) and the coordinates introduced above, we have that Then it follows that, for the K-invariant functions Γ It means that the set T ′ 0 is locally given by the equation

Let us define the set of quantities
A µνσ := g µρ A ρ νσ ; then using this equation and the fact that we can conclude, from Proposition A.1, that the elements A µ νσ are uniquely determined by the fact that they belong to T ′ 0 . In other words, the set consists into a single element.
Restricting G ω to T 0 (see Diagram (6.1)), we obtain the following result, that permits us to reconstruct Levi-Civita connection from a section of the reduced bundle.
such that pr 1 • σ : M → J 1 τ Σ is a holonomic section and is the Levi-Civita connection associated to the metric g σ := pr 1 • G ω • σ .
Because pr 1 • σ is holonomic, we have that Let us define T ′′ 0 := G −1 ω (Σ × C 0 (LM)) = ϒ ω p J 1 τ K (T 0 ) ; then, we need to draw our attention to the diagram in Figure 1. As a consequence of Proof. Locally, composite map ϒ ω • p J 1 τ K is given by where coordinates g µν σ are calculated using Equation (6.3).
For the last result of the Section, we need any of the composite maps As we mentioned above, the splitting induced by the connection form ω K allows us to relate the metricity forms with a contact structure on the quotient bundle.
Proposition 6.4. The metricity forms are (pr 2 • pr Σ • ϒ ω )-horizontal (also pr 1 • ϒ ω • p J 1 τ K horizontal). In fact, Proof. In local coordinates, we have that where g µν σ is calculated using Equation (6.2). On the other hand, the metricity forms have the following local expression [3] (6.5) Using Equation (6.2), it follows that namely, the metricity condition is horizontal with respect to the projection and the form in the base manifold is nothing but the generator of the contact structure.

FIRST ORDER VARIATIONAL PROBLEM FOR EINSTEIN-HILBERT GRAVITY
We have enough background to define a variational problem on J 1 τ Σ for Einstein-Hilbert gravity. The Einstein-Hilbert Lagrangian form will be defined as the unique 2horizontal m-form L (1) EH on J 1 τ Σ such that Recall also that in local terms, Palatini Lagrangian (2.2) can be written as We are pursuing here to establish the equivalence between the classical variational problem associated to the Lagrangian density L EH : J 2 τ Σ → ∧ m M (see [16]) and the variational problem J 1 τ Σ , L (1) EH , I Σ con . As we have said above, the main difference between these variational problems is related to the nature of the Lagrangian form: In the latter, this form is not a horizontal form on J 1 τ Σ , meanwhile in the former case the Lagrangian form on J 2 τ Σ is specified through a Lagrangian density, that gives rise to a horizontal form on this jet bundle. The following lemma tells us how these Lagrangians are related.

Proof. Recall that the horizontalization operator is defined by the map
The result follows from a (rather lenghty) calculation, using expression (7.1) and the formula for the Christoffel symbols (6.4).
The occurrence of the horizontalization operator in this lemma is crucial for our purposes, as the following proposition shows. Proof. It follows from the formula x s π k+1 • T x j k+1 s, that holds for every x ∈ Mand s ∈ Γπ.
It is immediate to prove the desired equivalence. if and only if it is an extremal for the action integral as required.
As usual [17], the equations of motion of this variational problem can be lifted to a space of forms on J 1 τ Σ . Let us define the affine subbundle x s , is the corresponding fiber for the contact subbundle on J 1 τ Σ . The canonical map will be denoted by τ EH : W EH → J 1 τ Σ .
We will indicate with λ EH the pullback of the canonical m-form on ∧ m J 1 τ Σ to W EH . Then we have a result analogous to Proposition 3.2 in the context of (first order) Einstein-Hilbert formulation.
Remark 7.5. This proposition provides us with a unified formalism for Einstein-Hilbert gravity, based on the first order formulation. For the corresponding formalism associated to the second order formulation, see [16].

CONTACT BUNDLE DECOMPOSITION FOR PALATINI GRAVITY
Now we will recall some general facts regarding the decomposition induced for the connection ω K [6] on the bundle of forms W PG defined in Equation (2.3). The contact structure on J 1 τ gives rise to the contact subbundle on T 0 given by where L indicates linear closure. There is an splitting of I m con,2 induced by the choice of a connection on the principal bundle p LM K : LM → Σ. Its construction is as follows. We denote by ω K ∈ Ω 1 (LM, k) the chosen connection and consider the following splitting of the cotangent bundle: . The identification is obtained as follows: . Accordingly, we have an splitting of contact bundle (8.1) x s . The symbol · ∧ , · denotes the natural contraction, defined as follows: For elements of the form α 1 ⊗ ν, α 2 ⊗ η with ν, η ∈ k and α 1 , α 2 forms, we have α 1 ⊗ ν ∧ , α 2 ⊗ η = ν, η α 1 ∧ α 2 . For a general element in the linear closure, extend linearly.
We can split our metricity subbundle I m PG using the inclusion I m PG ⊂ I m con,2 , namely I m PG = I m PG ∩ I m con,2 ⊕ I m PG ∩ I m k * ,2 . But we have the following fact. For every j 1 x s ∈ T 0 I m PG ⊂ I m con,2 . Proof. Let us work in the coordinates considered above; therefore, we have Equation (5.1) for the projector T p LM K and also Then that will be rise to a set of generators of the bundle I m PG (see Equation (6.5)).
This result is compatible with the fact that the whole subbundle W PG is in the zero level set for the momentum map. We will return on that below.

FIRST ORDER EINSTEIN-HILBERT LAGRANGIAN AS ROUTHIAN
Because J ≡ 0 the Routhian density will coincide with the Lagrangian L PG . This density will induce a density on the reduced bundle, which we will define next.
First, we write p : Lin τ * Σ T M, k → Σ for the obvious projection. In principle, the bundle Lin τ * Σ T M, k would be the field bundle for the reduced system; nevertheless, we will show next (see Lemma 9.2 below) that the Routhian, namely, the Lagrangian form for this reduced system, will be horizontal for the projection onto the jet space of the base bundle Σ.
In particular, one can consider the map: projecting onto the quotient bundle for Palatini gravity. So, we can formulate the reduced system as a first order field theory by taking the bundle τ Σ • p : Lin τ * Σ T M, k → M as the basic field bundle. Nevertheless, there are some identifications that will permit us to simplify further this basic bundle.
In order to proceed, let use the connection ω K to define the maps fitting in the following diagram: The definitions are as follows: The map g ω is the identification from Corollary 4.5. Since the Lagrangian density L PG is invariant under K, it defines a reduced density on T 0 /K which can be seen as a density on J 1 τ Σ × Lin τ * Σ T M, k . We will denote it by L PG : is the projection onto the first factor of the fibred product.
Proof. It follows from Equations (6.2) and (6.4) that In short, Routhian L PG does not depend on the fiber coordinates A σ µρ of the bundle p : Lin τ * Σ T M, k → Σ; it is just the pullback along pr 1 of the first order Lagrangian for Einstein-Hilbert gravity.
In the usual Routh reduction, the reduced Routhian is a m-form on J 1 (τ Σ • p); in this case, Lemma 9.2 allows us to consider the form L (1) EH on J 1 τ Σ as the Routhian. Therefore, we can forget about the degrees of freedom associated to the factor Lin τ * Σ T M, k in the quotient bundle, and take as the quotient bundle for Palatini gravity the jet bundle J 1 τ Σ ; this is the way in which we will proceed from this point.

EINSTEIN-HILBERT GRAVITY AS ROUTH REDUCTION OF PALATINI GRAVITY
We will devote the present section to establish the two main results of the article, namely, Theorem 10.3 regarding reduction of Palatini gravity and Theorem 10.5 dealing with reconstruction of metrics verifying Einstein equations of gravity. The strategy, as we mention in the introduction, is to compare the equations of motion (lifted to the corresponding spaces of forms W EH and W PG ) in a bundle containing every relevant degree of freedom; this role is played below by the pullback bundle F * ω (W EH ). So, let us define x s ∈ T 0 . Then we have the diagram pr ω 2 : F * ω (W EH ) −→ W EH are the canonical projections of the pullback bundle.
Lemma 10.1. The bundle map F ω is an affine bundle isomorphism on T 0 between W PG and F * ω (W EH ). Proof. It is consequence of Equation (9.1) and Proposition 6.4.
We will use Diagram (10.1) as a mean to compare the equations of motion of Palatini gravity and Einstein-Hilbert gravity; the idea is to use Propositions 3.2 and 7.4 in order to represent these equations in terms of the spaces of forms W PG and W EH respectively, and to pull them back to the common space F * ω (W EH ). Crucial to this strategy is the following result.

Proposition 10.2. It is true that
be an arbitrary element in this pullback bundle; then using Diagram (10.1) we will have that where it was used that π EH : W EH → J 1 τ Σ is the restriction of the canonical projection τ m Proof. The idea of the proof is encoded in the following diagram Using Proposition 3.2, we construct Γ : U → W PG out of Z; the Palatini gravity equations of motion will become Γ * (Z dλ PG ) = 0 for any Z ∈ X V (τ 1 •π PG ) (W PG ). Using Lemma 10.1 we can define then the Palatini equations of motion translate into . Then using Proposition 10.2 and the fact that pr ω 2 : F * ω (W EH ) → W EH is a submersion, we can conclude that the section Γ := pr ω 2 • Γ ′ : U → W EH obeys the equations of motion is an arbitrary vertical vector field on W EH . Also, using Diagram (10.2), we have that The theorem then follows from Proposition 7.4. 10.2. ...and reconstruction. It is time now to give a (somewhat partial) converse to Theorem 10.3. That is, given a section ζ : U ⊂ M → Σ such that j 1 ζ : U → J 1 τ Σ is extremal for the Einstein-Hilbert variational problem, find a section Z : U → T 0 such that F ω • Z = j 1 ζ and Z is an extremal for the Palatini variational problem. From Figure 1 it is clear that we need to lift the section j 1 ζ through the quotient map p J 1 τ K T 0 : T 0 → T 0 /K, which has the structure of a principal bundle on T 0 /K. It is clear that any principal bundle can be trivialized by a convenient restriction of the base space. As discussed in [6], it is not the way in which this kind of reconstruction problems are solved. Rather, the problem of lifting sections along the projection in a principal bundle is reduced to the problem of deciding if certain connection is flat; moreover, it is expected that this connection is related to the connection used to define the Routhian. We will present in this section a theory of reconstruction along these lines. With this goal in mind, we will recall here some of the details developed in [6]; for proofs we refer to the original article. We begin with a pair of diagrams ( Using that ζ * P is a principal bundle, being trivial can be characterized in terms of a flat connection [22]:

be a G-principal bundle with M simply connected. Then P is trivial if and only if there exists a flat connection on P.
If M is not simply connected, then one can ask for a flat connection with trivial holonomy and obtain a similar result. For the sake of simplicity, we will assume that M is simply connected to apply Theorem 10.5 when needed. For later use, we also observe that the section constructed in the proof of Theorem 10.5 has horizontal image w.r.t. the given connection.
We now wish to apply the previous discussion to the case of the bundle p So, in order to find a lift for the section ζ , it is sufficient to construct a flat connection on the K-principal bundle ζ * (LM).
To this end, we will define where ω 0 ∈ Ω 1 (LM, gl (m)) is a principal connection on LM and π k : gl (m) → k is the canonical projection onto k. Lemma 10.6 allows us to establish the following definition, inspired in the analogous concept from regular Routh reduction.
Definition 10.7 (Flat condition for Palatini gravity). We will say that a metric ζ : M → Σ satisfies the flat condition respect the principal connection ω 0 ∈ Ω 1 (LM, gl (m)) if and only if the associated connection ω ζ is flat.
Remark 10.8. This condition yields to a relationship between the metric ζ : M → Σ and the principal connection ω 0 ; the physical relevance of this relationship remains unclear for the author. Mathematically, it means that, even in the case that the bundle p LM K : LM → Σ is nontrivial, it could be the case when restricted to the image of the section ζ .
Also, it is necessary to establish the following result regarding the map F ω .
Lemma 10.9. The following diagram commutes Proof. In fact, for j 1 and also τ ′ 10 p T 0 K j 1 x s = τ ′ 10 j 1 x s K = [s (x)] K , and the lemma follows.
With this in mind, we are ready to formulate the reconstruction side of this version of Routh reduction for Palatini gravity.
On the other hand, by Lemma 10.6 we have a lift We will define the map and show that it is a section of pr ω 1 : F * ω (W EH ) → T 0 ; namely, we have to show that It is important to this end to note that the conclusion of Proposition 6.4 can be translated to this context into T p LM K • ω p = F * ω ω; moreover, by Lemma 10.6 we know that Z * ω p = 0, so Then the section F ω • Z : M → J 1 τ Σ is holonomic; finally, from Equation (10.7) we must conclude that But j 1 ζ = π EH • Γ by construction of the section Γ; then and Γ ′ is a section of F * ω (W EH ), as required. Now define the section because Γ obeys Equation (10.6).

CONCLUSIONS AND OUTLOOK
We were able to adapt the Routh reduction scheme developed in [6] to the case of affine gravity with vielbeins. It suggests that this formalism could be fit to deal with Griffiths variational problems more general than the classical, at least with cases when the differential restrictions are a subset of those imposed by the contact structure. Extensions of this scheme to gravity interacting with matter fields will be studied elsewhere.

APPENDIX A. AN IMPORTANT ALGEBRAIC RESULT
First we want to state the following algebraic proposition.
is the unique solution for this linear system.
Proof. From first equation we see that The trick now is to form the following combination where in the permutation of indices was used the remaining condition.
APPENDIX B. PROOF OF PROPOSITION 3.2 In order to do this proof, it will be necessary to bring some facts from [5]. First, we have the bundle isomorphism on T 0 with S * (m) := (R m ) * ⊙ (R m ) * the set of symmetric forms on R m , and : γ is horizontal respect to the projection τ ′ 1 : T 0 → M . The bundle E 2 is a bundle of forms with values in a vector space; therefore, it has a canonical (m − 1)-form Θ := Θ i j e i ⊙ e j .
The equations of motion are obtained by choosing a convenient set of vertical vector fields; because of the identification given above, it is sufficient to give a set of vertical vector fields on T 0 and on E 2 . It results that a global basis of vertical vector fields on T 0 is Given that E 2 is a vector bundle on T 0 , any section β : T 0 → E 2 gives rise to a vertical vector field; the equations of motion associated to these kind of vector fields are the metricity conditions ω p = 0.
Therefore, fixing an Ehresmann connection on the bundle p ′ 2 : E 2 → T 0 , we can produce the set of vertical vector fields on