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Griffiths Variational Multisymplectic Formulation for Lovelock Gravity

This work is mainly devoted to constructing a multisymplectic description of Lovelock's gravity, which is an extension of General Relativity. We establish a Griffiths variational problem for the Lovelock Lagrangian, obtaining the geometric form of the corresponding field equations. We give the unified Lagrangian--Hamiltonian formulation of this model and we study the correspondence between the unified formulations for the Einstein--Hilbert and the Einstein--Palatini models of gravity.


Introduction
The development of the geometric description of classical field theories using multisymplectic [7,19,26,28,42,45] or polysymplectic and k-(co)symplectic manifolds [18,30,31] has rekindled the interest in doing a totally covariant description of many theories in physics and, in particular, General Relativity and other derived from it. Many general aspects as well as specific problems and characteristics of the theory have been studied in this way (see, for instance, [9,12,23,28,32,33,35,36,43,44]).
In particular, the multisymplectic techniques have been applied to describe the most standard models of General Relativity: the Einstein-Hilbert [24] and the Einstein-Palatini (or metricaffine) [6] models (see, for instance, [5,6,25,46]). In some of these applications, a unified formalism which joins the Lagrangian and Hamiltonian formalisms into a single one has been used. This unified Lagrangian-Hamiltonian formalism is especially useful in mechanics and field theories [3,15,20,41] when the Lagrangian that describes the system is singular. For this reason, such formalism finds immediate application in the study of both, the Einstein-Hilbert and the Einstein-Palatini models of gravity. In the first case, and following the formulation for second order field theories developed in [41], the symmetrized jet-multimomentum bundle is used as framework, which turns out to be a premultisymplectic bundle and therefore admits the use of the premultisymplectic constraint algorithm [16,17] for the study of the field equations. In the second case, an indirect path for the construction of unified formalism is invoked: first, the field theory corresponding to the Einstein-Palatini model is formulated in [5] as a Griffiths variational problem [29]. Subsequently, and inspired by the work of Gotay [27], a unified formalism is constructed as a Lepage-equivalent problem related to the latter [6]. Although it is known that the Einstein-Palatini and the Hilbert-Einstein Lagrangians essentially lead to the same field equations [13] (the Einstein equations), the way in which the unified formulations correspond to each other is unknown.
In the last decades, new models that extend General Relativity have emerged in theoretical physics [4,10,21]. In particular, Lovelock's gravity is a generalization of General Relativity (in vacuum) introduced by D. Lovelock [37,38]. His idea was to characterize all the symmetric tensors of order 2, without divergence, that can be constructed from the metric tensor and its derivatives up to second order. In dimension 4, it turns out that the only tensors that verify these properties are the metric and the Einstein tensor. In addition Lovelock proved that this tensor encodes the Euler-Lagrange equations of a Lagrangian density that is a polynomial in the (pseudo) Riemannian curvature. An interesting characterization for the Lovelock Lagrangian is provided in [14]: it is the only Lagrangian that is a polynomial in the (pseudo) Riemannian curvature and is also stable under a procedure called consistent Levi-Civita truncation. Similar considerations can be found in [39,40], where the idea consists in considering the Lagrangian as a function independent of the metric and the curvature, and to find relations between the partial derivatives of the Lagrangian with respect to these variables, induced by the geometry of the problem.
The objectives of this paper are to state and prove the most general and precise results on the following aspects: to study the correspondence between the unified formulations for the Einstein-Hilbert and the Einstein-Palatini models of gravity, to define the Lovelock Lagrangian in the context of multisymplectic geometry, to characterize geometrically its properties, to establish the Griffiths variational problem for this Lagrangian and to develop the corresponding Lagrangian-Hamiltonian unified formalism.
The organization of the paper is as follows: First, in Section 2, we set the basic definitions, notation and canonical structures of the frame bundle, which is widely used in the work. In Section 3, the Lovelock Lagrangian is presented and the corresponding variational problem is stated and analyzed. Section 4 is devoted to introduce the infinitesimal symmetries of the system described by the Lovelock Lagrangian and to obtain the field equations that derive from the Griffiths variational problem for this system. Finally, in Section 5, the premultisymplectic description of the Lovelock system is carried out using the unified Lagrangian-Hamiltonian formalism. After the conclusions of Section 6, an Appendix is included where different notations are set and several geometric constructions and definitions used throughout the work are collected.
All manifolds are finite-dimensional, real, paracompact, connected and C ∞ . All maps are C ∞ . Sum over crossed repeated indices is understood.

The frame bundle and its canonical forms 2.1 Basic definitions and notation
Consider a space-time manifold M of dimension m. The corresponding bundle of frames (see for example [34]) τ : LM → M is the set 1 It is well-known that the general linear group G := GL(m) acts naturally on R m by automorphisms. This action in turn induces a G-right action on LM , according to the formula u · A := u • A, for every u ∈ LM, A ∈ G, endowing LM with a G-principal bundle structure.
Let us fix a matrix (any signature can be used in these considerations; the one chosen here follows closely the signature found in General Relativity) η := diag (−1, 1, · · · , 1); it can be considered either a map η : R m → (R m ) * or a bilinear form η : R m × R m → R.
Associated with the bundle τ , we have the fiber bundle of 1-jets J 1 τ of sections of τ . Given a section s ∈ Γ(τ ), the 1-jet of s at x, denoted j 1 x s, is the class of local sections being contact equivalent up to first order at x. This space has natural bundle projections τ 10 : J 1 τ → LM and τ 1 : For every element A ∈ G, right translation R A : LM → LM is a bundle isomorphism over the identity and so it can be lifted to a bundle isomorphism of J 1 τ by taking the 1-jet j 1 R A : J 1 τ → J 1 τ . Accordingly, this defines a right action of G on J 1 τ and it can be checked that the quotient C(LM ) := J 1 τ /G is a smooth manifold, making q : J 1 τ → C(LM ) into a 1 Alternatively, we can think of each of the fibers LMx as the set of ordered bases of the tangent space TxM .
If ρ ∈ τ −1 10 (u), then ρ can also be thought of as a linear map ρ : The interpretation goes as follows: given a local section s ∈ Γ(τ ) and a tangent vector where the action of G in T LM is naturally induced by the action of G on LM , and [T τ ] G : Coordinates in M will be denoted using greek indices (x µ ) and the related fiber coordinates in LM will be denoted (x µ , e ν k ), where u(e k ) = e ν k ∂/∂x ν and {e 1 , . . . , e m } is the canonical basis in R m . Accordingly, fiber coordinates in J 1 τ will be denoted (x µ , e ν k , e ν kσ ). Using these coordinates, it can be seen that (x µ , Γ µ νσ := −e k ν e µ kσ ) define fiber coordinates in C(LM ).

The universal principal connection
Now, we define a principal connection on the bundle q : J 1 τ → C(LM ) fulfilling a universal property. First, observe that every element [ρ] G ∈ C(LM ) can be viewed as a pointwise connection, i.e. every [ρ] G defines a unique family of projections Γ u : T u LM → V u τ for every u ∈ τ −1 (τ (ρ)). Indeed, if ρ is any representative of the class [ρ] G , set u = τ 10 (ρ) and define It is immediate to see that Γ u (X) ∈ V u τ , for every X ∈ T u LM . For any other element u ′ ∈ τ −1 (τ (u)), we use right translation as follows where u ′ = u · g. It is readily seen that this construction is independent of the choice of the representative ρ.
Remark 1. When treating principal connections, we will use the symbol Γ to refer to the family of vertical projections. Furthermore, each principal connection carries a Lie algebra-valued differential form called connection form, which we denote by the symbol ω. In the sequel, we refer to principal connections either through the projections Γ or through its connection form ω.
Remark 2. Taking into account the previous observation, it is clear that the set of principal connections Γ on LM is in one-to-one correspondence with the sections of the bundle of connections. We have just seen the correspondence [ρ] G → Γ. The inverse correspondence is given by Now denote by g the Lie algebra of G and define ω ∈ Ω 1 (J 1 τ, g) as where we are using the identification LM × g → V τ . The fact that this Lie algebra-valued differential form is indeed the connection form of a principal connection can be easily checked.
To introduce the universal property associated with ω, let us observe that, if Γ is a principal connection on LM and σ Γ is the related section of the bundle of connections, then we can define a sectionσ Γ ∈ Γτ 10 by using the identification J 1 τ ≃ C(LM )× M LM as (see the diagram above) Then, if ω Γ is the connection form associated with Γ, it turns out that ω Γ =σ * Γ (ω). In other words, any connection form of a principal connection can be recovered as a pullback of ω by the sectionσ Γ . In this sense we say that ω is a universal connection. Accordingly, the universal curvature is given by (see Appendix 6) and it can be seen that the curvature form associated with Γ is Ω Γ =σ * Γ (Ω).
If {E i j } denotes the canonical basis of g and ω = ω i j E j i , then the coordinate expression of the forms ω i j using fiber coordinates is ω i j = e i µ (de µ j − e µ jσ dx σ ) and Ω i j = dω i j + ω i k ∧ ω k j .

The canonical form θ
In LM we can define a canonical R m -valued 1-formθ as follows. If X ∈ T u LM , then This allows us to define a similar form in J 1 τ , denoted θ, as the pullback τ * 10θ .
In terms of the canonical basis of R m , if we write θ = θ k e k , it can be seen that the coordinate expression of the forms θ k is given by θ k = e k µ dx µ , where e k µ is such that e k µ e µ j = δ k j and e j ν e µ j = δ µ ν .
The form θ turns out to be a tensorial 1-form (for details you can check [34]). We can use the local expressions for θ k and ω i j in a trivializing open set U ⊂ J 1 τ to prove that these forms are linearly independent.
The exterior covariant derivative of θ with respect to ω gives rise to another differential form fulfilling a new universal property, called the universal torsion form T , i.e. (see Appendix 6) The universal property in this case arise as follows: if Γ is a principal connection on LM , then its related torsion form T Γ is recovered as the pullback As we did before, we can express T in terms of the canonical basis of R m by writing T = T k e k with T k = dθ k + ω k i ∧ θ i . A local expression for T in a trivializing open set U ⊂ J 1 τ can be obtained using those for ω and θ. In fact This last expression shows that on the set e µ iν e i σ −e µ iσ e i ν = 0 of each trivializing neighbourhood the torsion form T vanishes identically. It turns out that all of these sets can be smoothly glued together and define a submanifold T 0 ⊂ J 1 τ , as the next proposition shows Proposition 1. There exists a submanifold ι 0 : T 0 ֒→ J 1 τ such that ι * 0 T ≡ 0.
Proof. As we anticipated, the manifold T 0 is given locally by the conditions e µ iν e i σ − e µ iσ e i ν = 0, (2.1) for every µ, ν, σ. To see that this is independent of the choice of coordinates, consider another trivializing neighbourhood (having nonempty intersection with the first) with fibered coordinates (x µ ,ē µ k ,ē µ kσ ). Change of coordinates between these two sets is given bȳ which implies that the vanishing of the expression (2.1) is independent of the particular trivializing set.
Remark 3. It is possible to prove that the action of G preserves the manifold T 0 , i.e. T 0 · A ⊂ T 0 for every A ∈ G. This allows us to define a G-action on T 0 , making T 0 → T 0 /G into a principal G-bundle. Moreover, denoting C 0 (LM ) := T 0 /G and using the identification J 1 τ ≃ C(LM ) × LM , we get the identification T 0 ≃ C 0 (LM ) × LM . If we pullback the universal connection ω through ι 0 , we get a new universal property concernig torsionless connections, instead of arbitrary connections. We will use this fact to write the Griffiths variational principle for Lovelock gravity. For each p ≤ n, define It is readily seen that θ i 1 ...ip is completely antisymmetric in its indices. Additionaly we have the following characterization: Proof. Let us proceed by induction on p. First, observe that and renaming the indices which proves the case p = 1. Inductively again renaming indices We will use the Sparling forms to write down local expressions for the Lovelock Lagrangian and the equations of motion. To ease all the related computations it will be necessary to know some properties of these forms, so we collect some of them in the next proposition. In the proof (and in the rest of the paper) we use the properties of the Levi-Civita symbol and the generalized Kronecker delta listed in Appendix A.1.

Proposition 2.
The following properties hold (the hat on an index indicates that this index has been suppressed): Proof.

Let us compute
2. From the first point of the proposition and taking r = 1 and s = p, We may also prove this by induction on p. The case p = 1 is just Eq. (2.2). Assuming that the formula holds for p − 1, then 3. Let us compute now the differential of θ i 1 ...ip Renaming the indices and using the first part of the proposition

Variational problem for Lovelock gravity
Griffiths variational problems [29] are posed on triples (Λ is an m-form (referred to as the Lagrangian form) and an exterior differential system I ⊂ Ω • (Λ) [2] characterizing the admissible sections of the problem. Remember that σ is integral for I if and only if σ * α = 0 for every α ∈ I. In particular, this implies that the variations of S must be performed in such a way that the transformed sections remain integrals of I. Hence, we define Definition 2. Let I ⊂ Ω • (Λ) be an exterior differential system. A local vector field X ∈ X(Λ) is an infinitesimal symmetry of I if and only if L X I ⊂ I.

The set infinitesimal symmetries of I is denoted Symm(I).
With this definition, it can be proved that the solutions to the variational problem associated with the variational triple (Λ π − → M, λ, I) are those sections σ integral for I such that which are, in turn, the field equations for this problem.
Remark 4. Here we are assuming that M is a manifold without boundary. Also, in order for S to be well-defined, the form σ * λ must be compactly supported. In the sequel, we will assume that all the integrals we work with exist.

The Lovelock Lagrangian
Now we are ready to define a Griffiths variational problem for Lovelock gravity [38]. To do that we have to define the corresponding triple introduced in the previous section. The bundle chosen is Here we are writing τ 1 instead of τ 1 | T 0 only to simplify the notation (we will do the same with the pullbacks of ω and θ through ι 0 ).
As the exterior differential system restricting the admissible sections we take (see Appendix A.3) The subscript dif f indicates the smallest exterior differential system containing the form ω p .
Using the canonical basis of R m , we can alternatively describe I L as the exterior differential system generated as follows It is also useful to define the forms ω ij := η ik ω j k , in terms of which I L = ω ij + ω ji dif f . Then we look for sections σ ∈ Γ(τ 1 ) fulfilling the condition It is customary to refer to this condition as the metricity condition.
Remark 5. Using the identification T 0 ≃ C 0 (LM ) × M LM , every (local) section σ ∈ Γ(τ 1 ) over U ⊂ M that is integral for I L can be thought of as a couple of sections σ 1 := q • σ and σ 2 := τ 10 • σ. As we saw in the previous section, if Γ is the principal connection induced by σ 1 on τ , then ω Γ =σ * 1 ω. Hence, the metricity condition implies that ω Γ is a torsionless (pseudo) metric connection.
Following the constructions of Appendix A.4 about vector-valued differential forms, for every k ≤ n, we can define a k-form with values in Λ k R n given by Using the canonical basis of R m we can write Now we can take the Hodge star operator in the second factor (see Appendix A.2), namely where we have used the properties of the Levi-Civita symbol (see Appendix A.1). Now, renaming indices and using the definition of the forms θ i 1 ...ip , we find denotes the integral part.

Definition 3.
Let V be an m-dimensional real vector space. We define as the unique linear map whose action on elementary 2r-vectors is given by whereη is the of extension of η to Λ r V defined on Appendix A.2. It is readily seen that it is in fact well-defined and linear.
Using the linear map A r , we can construct an (m − 2r)-form with values in (End(Λ r R m )) * as We can think of Ξ r as taking values in (Λ r g) * rather than (End(Λ r R m )) * because the latter can be viewed as a subspace of the former. That is which is a consequence of the inclusion Λ r g ⊂ End(Λ r R m ) given by the monomorphism Γ : We can use these considerations to introduce the Lovelock Lagrangian: where a r are constants and Ω r = Ω ∧ · · · ∧ Ω r times .

Expressions in terms of the canonical basis of R m
If we denote by {e 1 , . . . , e m } the dual basis of the canonical basis in R m , we can write Also, using (3.1), and, as all are dummy indices, we finally get so that, in view of the inclusion (3.2), we obtain where Ω ab = η bq Ω a q and all the possible multiplicative constants have been absorbed in the constants a r . From now on, we will work with each homogeneous component Remark 6. To simplify the computations, it will be convenient to introduce the following multiindex notation. We use capital letters I, J to denote multi-indices I = (i 1 , . . . , i p ), J = (j 1 , . . . , j p ). An apostrophe denotes a multi-index formed by removing the first index of a given multi-index, i.e. I ′ = (i 2 , . . . , i p ) if I = (i 1 , . . . , i p ). In this case, we use concatenation of indices and multiindices and write I = i 1 I ′ . Also, we will write Ω IJ = Ω i 1 j 1 ∧ · · · ∧ Ω irjr and θ i 1 ...irj 1 ...jr = θ IJ . Thus, the Lovelock Lagrangian can be written

Relation with the metric-affine Lagrangian
To relate λ L with the metric-affine formalism, remember that every principal connection Γ gives rise to a linear connection in T M (as an associated vector bundle with fiber R m and canonical action of G). Let ω Γ is the corresponding connection form (obtained as the pullback of the universal connection ω by a suitable section) and Ω Γ its related curvature.
where R στ µν = g ρτ R σ ρµν are the components of the curvature tensor with respect to the linear connection induced by Γ, i.e.
and g µν := e a µ η ab e b ν is the corresponding metric.
Thus, we can compute locally the pullback of λ L by a section as follows . . . e ir αr e αr cr e lr βr e βr dr Ω c 1 d 1 where we have used Eq. Then, as det(e) = √ −g, we recover the usual Lovelock Lagrangian, i.e.

Field equations
As we have said in the previous section, to compute the field equations associated with the Lovelock problem we need to characterize the infinitesimal symmetries of the exterior differential system I L . We devote the following section to this task.

Infinitesimal symmetries of I L
To give a characterization of the infinitesimal symmetries of I L , it is useful to introduce a convenient basis of vector fields for the vertical bundle V τ 1 . Using the identification J 1 τ ≃ C(LM ) × M LM we can construct this basis using q-vertical and τ 10 -vertical vector fields.
First, consider the infinitesimal generators associated with the action of G on J 1 τ . If E i j denote a vector of the canonical basis of g, we denote these vector fields by (E i j ) J 1 τ . It can be seen that T τ 10 (E i j ) J 1 τ = (E i j ) LM and hence they are τ 1 -vertical vector fields. From the principal bundle structure of q : J 1 τ → C(LM ) it is immediate to see that they are also q-vertical.
Furthermore, as J 1 τ → LM is an affine bundle, we can construct vertical lifts of τ -vertical vector fields. Given a differential form α ∈ Ω(M ) and a τ -vertical vector field X, the vertical lift (α, X) V is defined as the vector field whose flow is given by where the sign "+" must be understood as the affine action of τ * π M ⊗ LM V τ on J 1 τ . 2 In other words, We can adapt this definition replacing the differential forms α by differential forms along In particular, we can use the forms θ r and the infinitesimal generators (E s t ) LM , which we denote It is clear that these vector fields are τ 10 -vertical, and in consequence they are also τ 1 -vertical.
It can be proved that the set of vector fields (E s t ) J 1 τ , (θ r , (E s t ) LM ) V form a basis of the vertical bundle V τ 1 (see [6]). 14 Using these expressions we can check that the of vector fields (E s t ) form a basis of the bundle V ( τ 1 | T 0 ).
Remark 8. If we fix a principal connection (that may be chosen torsionless) on τ , it is possible to complete this basis to a full basis of T J 1 τ by considering the prolongations of the standard horizontal vector fields on LM (see [34]).

Proposition 3. The following contractions hold
Before moving on, remember that given a vector field U ∈ X(LM ) the first prolongation of U is the unique vector field j 1 U ∈ X(J 1 τ ) that is projectable to U and is an infinitesimal symmetry of the contact exterior differential system. The next lemma shows that prolongations of G-invariant vector fields are infinitesimal symmetries of the universal connection [8]: Proof. We know that U is G-invariant if, and only if, its flow Ψ U t : LM → LM is an automorphism of LM . Furthermore, for every automorphism F : LM → LM , we have that and the lemma follows from this fact.
is an infinitesimal symmetry of I L tangent to T 0 .
Proof. Let U i j be the basis of G-invariant local vector fields generating X V τ (V ). Since the set of infinitesimal generators (E k l ) LM form another basis, there exist smooth functions M il jk , N il jk ∈ C ∞ (V ) such that U i j = M il jk (E k l ) LM and M ip jq N ql pk = δ i k δ l j . Now, from the formula we obtain ∂x µ e µ r . In consequence, in order for Z to be tangent to T 0 , we must take with the functions G i kj fulfilling G i kj = G i jk . To compute the Lie derivative, let us write V ω pq p and compute separately. First, On the other hand Thus, it is sufficient to take functions G i kj fulfilling the equation This assures us that L Z (ω p ) pq = 0.
In order to look for a solution to (4.3), consider the decomposition of the set of 0 3 -tensors of a vector space V , i.e.
where A ∈ S 12 V if, and only if, A(u, v, w) = A(v, u, w), and B ∈ S 23 V if, and only if, B(u, v, w) = B(u, w, v), for every u, v, w ∈ V (here Sym denotes the symmetrization projector). Such decomposition is given by or using a basis for V Now, we want to solve the equation with A ijk = G q ij η kq + G q ik η jq and for symmetric tensors G k ij = G k ji . We will use the above mentioned decomposition. It is readily seen that and for B we get Thus, equating each term, we find But using the initial equation (4.4), we get and In conclusion, Eq.

Field equations for Lovelock gravity from its Griffiths variational problem
Let us compute the differential of the Lovelock Lagrangian: The first term was computed in Proposition 2, now let us work out the second term. If σ ∈ S r is the permutation that transpose 1 and a, then and we can reorder every summand as follows Let us analize the two sums in the brackets. As above, suppose that σ is the permutation that transpose 1 and s; then Furthermore, using a similar argument, Now let us simplify the terms r Ω i 1 q ∧ ω qj 1 − ω i 1 q ∧ Ω qj 1 ∧ θ i 1 ...irj 1 ...jr ∧ Ω i 2 j 2 ∧ · · · ∧ Ω irjr . First, renaming the dummy indices, which cancels out the first sum. Second and consequently These computations amounts to the Lagrangian form on J 1 τ , so we have to take its pullback to T 0 , i.e.
Nevertheless, we will omit the pullback ι 0 to simplify notation. Now we are ready to find the field equations associated with the Griffiths problem J 1 τ, λ L , I L . First we state a lemma we will use later on

Lemma 4. If Ω takes values in k, then on T
Proof. Using the structure equation, we have dT q + ω q l ∧ T l = Ω q l ∧ θ l , and since the torsion form annihilates on T 0 , we have Ω q l ∧ θ l = 0. Then multiplying both sides by θ qi 1 ...irj 1 ...jr ∧ Ω IJ Ω q l ∧ θ l ∧ θ qi 1 ...irj 1 ...jr ∧ Ω IJ = 0, hence, using Proposition 2, Let us study both terms in the sum. If σ ∈ S r is the permutation transposing 1 and a, where we use the skew-symmetry of the Sparling form and the fact that the curvature is a two-form. On the other hand Thus, since Ω l l = 0, we get As we did in the proof of Lemma 3, consider a basis U i j of G-invariant local vector fields generating X V τ (V ) on some trivializing open set V ⊂ LM and let M kl ij be smooth functions on V such that Let us compute the contraction j 1 U t r dλ L by using (4.1). First Let us study the terms in brackets. First and using Lemma 4 and renaming indices, Now let us compute the other contraction: where Ω jk, So, gathering both terms together: Thus if Σ : M → T 0 is a section such that Σ * ω p = 0, we can conclude that Proof. We consider infinitesimal symmetries Z ∈ X V τ 1 (T 0 ) of I L as in Lemma 3, namely and the result follows from the fact that the functions f i k are arbitrary and the matrix M il jk is invertible.
Remark 11. It is useful to compare this with the Einstein case. In [6] it is seen that the Einstein equations in vacuum are together with the constraints T = 0 = ω p ; the previous theorem, on the other hand, gives us the set of equations under the same constraints. Nevertheless, it can be proved (see Corollary 21 in [6]) that under the constraints T = 0 = ω p , it is true that ω ik ∧ Ω k p − ω pk ∧ Ω k i = 0 as consequence of a Bianchi identity. Therefore, these sets of equations are equivalent.
Theorem 1 gives us a set of necessary conditions for a section Σ : M → T 0 to be extremal of the Griffiths variational problem associated with the triple (T 0 , λ L , I L ). Our next task is to set the sufficiency of these conditions.
Proof. It is a consequence of the fact that every Z ∈ X V τ 1 (T 0 ) can be written in terms of the vector fields Thus, in particular, for every τ 1 -vertical infinitesimal symmetry Z of the exterior differential system I L and any section Σ : M → J 1 τ fulfilling the hypotheses in the previous proposition, we have that Σ * (Z dλ L ) = 0 ; therefore, Σ is an extremal for the Griffiths variational problem (T 0 , λ L , I L ), as required.

Tautological form on a bundle of forms
The next definitions are quoted from [6]. Let π : P → N be a principal fiber bundle with structure group H and assume that π and χ are surjective submersions fitting the diagram Let V be a finite dimensional real vector space V and define the bundleτ k n,π : k n,π T * P ⊗V → P of V -valued k-forms that annihilates when contracted with n π-vertical vectors. This bundle has a canonical V -valued k-form Θ k n,π defined through the formula Θ k n,π α (Z 1 , . . . , Z k ) := α(T ατ k n,π (Z 1 ), . . . , T ατ k n,π (Z k )).
Given a H-representation (V, ρ), it is readily seen that k n,π T * P ⊗ V is a H-space with action given by Φ k g (α)(X 1 , . . . , X k ) := ρ(α) · (α(T u·h R h −1 X 1 , . . . , T u·h R h −1 X k )) , where R is the right action in P and h ∈ H. It can be proved that the tautological form Θ k n,π is then a H-equivariant map.
We point out two instances that will be used in the next section. If H = G = GL(m), P = J 1 τ , N = C(LM ), ψ = τ 1 , χ = p and π = q (that is, the left triangle in the diagram of Section 2.1), we have 1. Set k = m − 2 and n = r + 1, and consider V 1 = (R m ) * and ρ 1 the natural representation of G on this vector space. Then, we denote the space E 1 := m−2 2,τ 1 J 1 τ ⊗ (R m ) * and the projection p 1 : E 1 → J 1 τ .
2. Set k = m − 1 and n = r, and consider V 2 = (R m ) * ⊙ (R m ) * and ρ 2 the natural representation of G on this vector space. (The symbol ⊙ denotes the symmetrized tensor product). Then, we denote the space E 2 := m−1 1,τ 1 J 1 τ ⊗ ((R m ) * ⊙ (R m ) * ) and the projection To simplify notation we denote by Θ 1 and Θ 2 the corresponding tautological forms on these bundles and, when using the component forms with respect to the canonical bases, we simply write Θ 1 = Θ l e l and Θ 2 = Θ ij e i ⊙ e j , indicating that a single index corresponds to Θ 1 and two indices to Θ 2 .

The multimomentum bundle
The unified formalism for a Griffiths variational problem is built from the idea of a Lepage equivalent problem. Briefly, the construction goes as follows (this idea is inspired in the work of [27] and was proposed in [11]): assume that the differential ideal is locally generated by a subbundle I ⊂ Λ • T * J 1 τ (this means that there is an open cover {U λ } such that every α ∈ I can be generated by sections of I| U λ when pulled back to U λ ). Consider an integer k such that λ L (u) ∈ Λ m k (T * u J 1 τ ) and I m u : is the bundle of m-forms that annihilate when contracted with k τ 1 -vertical vectors. Then define the multimomentum bundle W λ by the equation In the case of Lovelock gravity for a polynomial Lagrangian of degree r in the curvature, it suffices to take k = r + 1. (Notice that the form θ i 1 j 1 ...irjr is τ 1 -horizontal whereas the form Ω i 1 j 1 ∧ · · · ∧ Ω irjr is only q-horizontal, which implies that more than r τ 1 -vertical vectors are needed to annihilate λ L ). In this case we can write any Observe that an element ρ in W λ is completely determined by an element j 1 x s ∈ J 1 τ and the forms γ l and β ij projecting onto j 1 x s. Hence we have the following identification: where j 1 x s := p 1 (γ l e l ) = p 2 (β ij e i ⊙ e j ), induces an isomorphism of the bundles τ λ : W λ → J 1 τ and pr 0 : To build the canonical form on W λ we need the tautological forms Θ 1 and Θ 2 , as well as the forms θ, ω, T and Ω. We use the three projections pr 0 , pr 1 , and pr 2 to pull these forms back to W λ , but we do not change the symbols so as to keep notation as simple as possible, e.g. if ρ ∈ W λ , then ω| ρ and Θ 2 | ρ must be understood as pr * 0 (ω)| ρ = ω| u=pr 0 (ρ) (T ρ pr 0 (·), . . . , T ρ pr 0 (·)) and pr * 2 (Θ 2 )| ρ = Θ 2 | β=p 2 (ρ) (T ρ pr 2 (·), . . . , T ρ pr 2 (·)), respectively.
Let us now denote by Θ λ the pullback of the tautological form Θ ∈ Ω m (Λ m (J 1 τ )) to W λ . Then, at any ρ = λ L | j 1 or, omitting the symbol ρ and recalling the expression for λ L , In consequence, the differential is given by , and recalling the expression of dλ L , we can rewrite

Field equations
To find the field equations we have to find the contraction of Θ λ with vertical vectors. We divide this task considering vertical vectors of the form X + Y + Z, where X is pr 0 -projectable and τ 1 -vertical, Y is p 1 -vertical, and Z is p 2 -vertical. Let us now give a useful description for the vectors Y and Z.
Going back to the notation of Section 5.1, consider a cross section ξ ∈ Γ(τ k n,q ). Then, since this is a vector bundle, we have the associated vertical lift, which is aτ k n,q -vertical vector field δξ ∈ X(Λ k n,q T * P ⊗ V ) given by In other words, this is the vector field whose flow is given by ϕ ξ t (α u ) = α u + tξ(u), for every α u ∈ E := Λ k n,q T * u P ⊗V . It is clear that δξ isτ k n,q -vertical, therefore it annihilates the tautological form Θ k n,q (because this is a horizontal form by definition). Now, let us see the contraction of this type of vectors with the differential dΘ k n,q . Lemma 6. The following identity holds δξ dΘ k n,q = (τ k n,q ) * (ξ) .
Proof. Let us compute δξ dΘ k n,q = L δξ Θ k n,q − dδξ Θ k n,q = L δξ Θ k n,q = lim t→0 Now, if X 1 , . . . , X k ∈ T αu E, Let us start with the p 2 -vertical vector fields. Let Z = δβ for some section β ∈ Γ(p 2 ). Observe that the unique non-vanishing contraction for this kind of vectors is with the form dΘ ij , so is a cross section, in order to be an extremal for our variational problem it must fulfill for every section β. Therefore, it must fulfill the equation which is none other than the metricity condition.
Remark 12. This can be seen locally as follows. Since Eq. (5.1) must hold for every β ij , in particular it must hold for horizontal β ij . Then, writing σ * (ω p ) i l η jl = f ij λ dx λ and β ij = p µ ij d m−1 x µ , for some functions p µ ij ∈ C ∞ (J 1 τ ) and f ij λ ∈ C ∞ (M ) (both symmetric in the indices ij), we have ij we conclude f ij λ = 0 which means σ * (ω p ) i l η jl = 0 and then σ * (ω p ) i l = 0.
Now consider Y = δγ for some section γ ∈ Γ(p 1 ). Similarly to the previous case, the unique non-vanishing contraction for this kind of vectors is that with the form dΘ l , so δγ Ω λ = T l ∧ (δγ dΘ l ) = T l ∧ pr * 0 (γ l ) = pr * 0 (T l ∧ γ l ).
If σ ∈ Γ(τ λ ) is a cross section, then in order to be an extremal for the variational problem, it must fulfill for every section γ. Therefore, it must fulfill the equation which is in turn the zero torsion condition.
Moving on, we separate several cases for the vector fields X. Every vector field can be written as a sum of a τ 10 -vertical vector field plus a τ 10 -projectable τ -vertical vector field. The first kind is generated by the vector fields (θ a , (E b c ) LM ) V , while the second is generated by the set of infinitesimal generators of the action. Consider now a vector field X = g c ab (θ a , (E b c ) LM ) V , for some functions g c ab ∈ C ∞ (J 1 τ ). We only consider those contractions that do not vanish when taking the pullback by a section fulfilling the previous conditions we have found so far. Then Therefore, taking the pullback by a solution to the previous equations and varying the functions g c ab , we conclude that it must also fulfill and given the symmetry of Θ ct , we have which in turn gives us σ * (Θ c ) = 0 .
Now assume that X = A J 1 τ . If A ∈ k, then it is easy to see that so we have no new equations.
This was introduced by Lovelock [37,38] as a generalization of the Einstein tensor for dimensions higher than 4. In this sense, the form Ψ is a global expression for such a tensor.
Gathering all the equations found so far, we conclude that the solutions to the field equations are those cross sections that are integral for the following exterior differential system Notice that the forms Ω k l ∧ θ l and (Ω p ) i j come as a consequence of the first Bianchi identity and the structure equation for the connection ω, respectively. Remark 14. It is interesting to note that the forms Ψ ij can be written in terms of the (m − 1)forms θ lIJ ∧ Ω IJ . Indeed where the symbol I ′ a has been used to indicate that the index i a has been removed from the multiindex I ′ . Then rearranging terms and using the symmetry of the product Ω st ∧ Ω I ′ J ′ , and renaming some indices we get Remark 15. In particular, in [5,6], two exterior differential systems where introduced to describe Palatini gravity, the first as a Griffiths variational problem takes the forms θ lIJ ∧ Ω IJ among its generators, and the second one by means of the unified formalism associated with the first one, where the forms Ψ ij are used instead.

Conclusions and outlook
We have defined the Lovelock Lagrangian in the context of the multisymplectic framework for classical field theories, and we have used this geometric formulation to characterize the properties of this Lagrangian, to establish its Griffiths variational problem and derive the corresponding field equations, and to study the infinitesimal symmetries of the system. As this Lagrangian is singular, this is a (pre-multisymplectic) field theory with constraints and then we have developed the Lagrangian-Hamiltonian unified formalism which is very suitable for its analysis.
Furthermore, if a variational problem has constraints, one can consider applying the constraints before or after performing the variations. In general, these two procedures lead to different sets of equations [1]. When both sets of equations are equal, we say that the variational problem has a consistent truncation (by the constraints). In [14] the authors claim that the Lovelock Lagrangians can be characterized by the consistency of the Levi-Civita truncation; that is, replacing the arbitrary connection by the Levi-Civita connection associated to the metric. We hope that the formalism presented in this paper could be very appropriate to analyze this property of gravitational theories, and to study the concept of consistent truncation for variational principles in a geometrical way. This would be a topic for further research.
Finally, the methods and results obtained in this paper are suitable to describe other gravity theories, such as the f (R) and f (T ) [4,21] models or the BF-gravity [10]. Thus, the multisymplectic formulation of these theories and, eventually, the development of their Lagrangian-Hamiltonian unified formalism are lines of further research.

A.1 Levi-Civita symbol and generalized Kronecker delta
We denote the Levi-Civita symbol in k indices by ε i 1 ...i k and ε i 1 ...i k if (i 1 , . . . , i k ) is an even permutation of (1, . . . , k), −1 if (i 1 , . . . , i k ) is an odd permutation of (1, . . . , k), 0 in other case On the other hand, the generalized Kronecker delta [22] in k indices δ i 1 ...i k j 1 ...j k is given by if (i 1 , . . . , i k ) is an even permutation of (j 1 , . . . , j k ), −1 if (i 1 , . . . , i k ) is an odd permutation of (j 1 , . . . , j k ), 0 in other case and can also be expressed as Both tensors are completely antisymmetric in their indices and they are seen to fulfill the following properties 3. If A is a matrix with entries a i j , ε i 1 ...im a i 1 j 1 . . . a im jm = det(A)ε j 1 ...jm .
Given that Λ m V is one-dimensional, it follows that there are exactly two m-vectors v fulfillinĝ η(v, v) = 1. Let ω be a preferred unit m-vector (observe that fixing such a vector amounts to choosing an orientation for V ). Then, the star Hodge operator ⋆ : Λ k V → Λ n−k V related to η is defined by requiring α ∧ (⋆β) =η(α, β)ω.
It is customary to refer to ⋆β as the Hodge dual of β.

A.3 Cartan decomposition of gl(m)
Let us use the matrix η in order to decompose gl (m); in order to get it, consider the involution Denoting s := θ |gl(m) , we have that k (resp. p) is the eigenspace corresponding to the eigenvalue +1 (resp. −1) for s. The projectors in every of these eigenspaces become Given N a manifold and γ ∈ Ω p (N, gl (m)), we define γ k := π k • γ, γ p := π p • γ.
If γ = γ i j E j i is the expression of γ in terms of the canonical basis of gl (m), then we have These previous considerations are useful when dealing with gl (m)-valued forms.
We are interested in a series of particular instances of this definition: Pairing. Consider U = V * , W = R and B = ·, · the natural pairing between V * and V . In this particular case we denote B (α ∧ , β) = α ∧ , β .
Linear representation. Consider U = End(V ), W = V , and denote B the natural action of End(V ) on V . In this particular case we denote Lie bracket. Consider U = V = W = g, where g is a Lie algebra and B = [·, ·] is the related Lie bracket. In this particular case we denote Wedge product. Consider U = V , W = Λ 2 V and let B = · ∧ · be the usual wedge product. In this particular case we denote B (α ∧ , β) = α ∧ β.

Constant linear map.
Consider a zero form α assigning to each x ∈ M the same linear map A ∈ Lin(V, W ), in this particular case we denote B (α ∧ , β) = A(β).
Similar definitions can be given if V is a vector bundle over M and α and β are vector bundle valued differential forms.