Routh Reduction of Palatini Gravity in Vacuum

Routh Reduction of Palatini Gravity in Vacuum

Santiago CAPRIOTTI

S. Capriotti

Departamento de Matemática, Instituto de Matemática de Bahía Blanca (INMABB),
CONICET, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca,
Argentina<EMAIL_ADDRESS>
Received September 24, 2019, in final form May 11, 2020; Published online May
30, 2020

An interpretation of Einstein–Hilbert gravity equations as Lagrangian
reduction of Palatini gravity is made. The main technique involved in this
task consists in representing the equations of motion as a set of differential
forms on a suitable bundle. In this setting Einstein–Hilbert gravity can be
considered as a kind of Routh reduction of the underlying field theory for
Palatini gravity. As a byproduct of this approach, a novel set of conditions
for the existence of a vielbein for a given metric is found.

symmetry reduction; Palatini gravity; frame bundle

53C80; 53C05; 83C05; 70S05; 70S10

##  1 Introduction

The relationship between Einstein–Hilbert and Palatini formulation of gravity
has been studied by several authors. It could be established in the Lagrangian
formulation by comparing the action functionals for each case – see for
example [37, 40] and references therein. From the Hamiltonian point of view,
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, [15] and [37] use ADM formalism [1] in order to
establish the connection; it has also been explored in [13, 27], 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
exist two ways to implement reduction at the Lagrangian level, namely
_Lagrange–Poincaré reduction_ [8, 11, 12, 17] and _Routh reduction_ [4, 6, 14,
18, 31, 32]. Moreover, there are physical considerations that can be said in
support of this kind of reduction: They deal with not only the reduction, but
also the reconstruction problem, and it is argued in [33] that reconstruction
can be relevant from the physical point of view.

Routh reduction was originally designed as a technique to get rid of cyclic
variables in a Lagrangian function by using a fixed value of the corresponding
momentum [34]. The mechanical system thus obtained has a configuration space
where the cyclic variables are absent and whose Lagrangian has a term forcing
the momentum to have the prescribed value; the new Lagrangian function is
called _Routh function_ or _Routhian_. This reduction scheme can be further
generalized to symmetries characterized by a non abelian Lie group – see [32]
and references therein. In the general case, the equations of motion of the
system obtained by quotient out the symmetry are modified by the addition of a
new term, called _force term_ , which is related to the curvature of a
principal bundle associated to the symmetry. It should be noted that the
Routhian reduces to the Lagrangian and the force term annihilates when the
momentum value is set to zero.

These versions of Routh reduction require some sort of regularity to the
Lagrangian in order to construct the Routhian; this regularity is used to
translate the constraints into the momentum variables to constraints into the
velocities. Therefore, it is possible to avoid this regularity condition by
working in a unified setting, where variables associated to both velocities
and momenta are taken into account; this approach can be used to set a Routh
reduction procedure not suffering this limitation [18]. This viewpoint proves
its usefulness in [6], where unified formalism was employed in the
generalization of Routh reduction to the field theory realm. In fact, in order
to improve the understanding of the tools and strategies used in the present
work, it could be helpful to give an account of both the problem addressed in
this reference and the techniques used to solve it:

  * •

Result in [6]: For the moment, we will represent a variational problem with a
pair $(\pi\colon E\to M,{\cal L})$, where $\pi\colon E\to M$ is a bundle and
${\cal L}\colon J^{1}\pi\to\wedge^{m}(T^{*}M)$ is a Lagrangian density; in
this setting, we can define the action

$\Gamma\pi\ni s\mapsto\int_{U}{\cal L}\circ j^{1}s$

for any compact $U\subset M$, and the variational problem becomes

$\delta\int_{U}{\cal L}\circ j^{1}s=0$

for arbitrary variations $\delta s$. Let us suppose that there exists a Lie
group acting freely and vertically on $E$, such that the Lagrangian density is
invariant by this action, yielding to a momentum map $J\colon
J^{1}\pi\to\wedge^{m-1}\big{(}T^{*}J^{1}\pi\big{)}\otimes\mathfrak{g}^{*}$. In
order to construct the reduced Lagrangian theory, it is necessary to fix a
value for the momentum map through the specification of a
$\mathfrak{g}^{*}$-valued $(m-1)$-form

$\mu\in\Omega^{m-1}(M,\mathfrak{g}^{*}),$

and also to choose a connection $\omega$ on the principal bundle
$p_{G}^{E}\colon E\to\Sigma:=E/G$. Let us indicate by $G_{\mu}\subset G$ the
stabilizer of the momentum $\mu$. Then the main result in [6] established a
correspondence between the extremals of the variational problem $(\pi\colon
E\to M,{\cal L})$ and

$\big{(}\overline{\pi}\colon E/G_{\mu}\times_{M}\mathop{{\rm
Lin}}{(\overline{\pi}^{*}TM,\widetilde{\mathfrak{g}})}\to
M,\mathcal{R}_{\mu}\big{)}.$

This variational problem is the reduced variational problem for Routh
reduction in field theory. As we warned before, when dealing with variational
problems obtained through Routh reduction, it could be necessary to add a
_force term_ , in order to take care of the non trivial nature of the bundles
involved [28]. In short, the local Routhian functions constructed through
Routh original procedure cannot be pasted into a global Routhian function
unless the bundle obtained by the symmetry quotient is trivial. It is thus
necessary to use a connection in order to define this global function; as a
by-product, a force term associated to the curvature of the chosen connection
should be taken into account. In any case, the force term is in this context
represented by a $(m+1)$-form $\phi_{\mu}$ on $J^{1}\overline{\pi}$
constructed from $\omega$ and the momentum $\mu$. Therefore, the underlying
variational problem becomes modified by an external force

$\delta\int_{U}\mathcal{R}_{\mu}\circ
j^{1}\overline{s}=\int_{U}\langle\phi_{\mu},\delta\overline{s}\rangle.$

  * •

Techniques: As was said above, it is useful to add momentum variables in order
to avoid regularity issues when dealing with Routh reduction. It leads us to
work with the unified formulation of field theory [16], where the space of
velocities $J^{1}\pi$ is enlarged with the space of multimomentum – see
equation (1.5) below for an explanation of the symbols involved in this
formula:

$J^{1}\pi^{\circ}:=\wedge^{m}_{2}(E)/\wedge^{m}_{1}(E)$

to form a bundle

$\displaystyle W_{\text{cl}}:=J^{1}\pi\times_{M}J^{1}\pi^{\circ}.$ (1.1)

In this space, a variational problem can be set. A word of caution should be
given here: The variational problem on $W_{\text{cl}}$ is different to those
described by a pair $(\pi\colon E\to M,{\cal L})$ above. Concretely, in this
new setting we have a Lagrangian $\Theta_{\cal L}$ that is not a density, but
a $m$-form on $W_{\text{cl}}$; the associated action becomes

$\Gamma(W_{\text{cl}}\to M)\ni\sigma\mapsto\int_{U}\sigma^{*}\Theta_{{\cal
L}}.$

A crucial difference with the variational problems of Lagrangian field theory
arises from this action: When performing variations, they should be free
because in $W_{\text{cl}}$ there are no variables that can be identified as
velocities. It claims for a definition of variational problem capable to
associate geometrical structures to these differences; this is captured by the
notion of _Griffiths variational problem_ considered in [22, 26] and will be
crucial in our approach to reduction – see Definition 2.1 below.

Beyond better regularity properties, the unified setting has several
advantages stemming from the linear structures that come equipped with it, and
the formulation of relevant data in terms of differential forms, on which
pullback maps allow for straightforward comparisons of equations and other
quantities.

Now, we are looking for a Routh reduction scheme similar to the one described
above, in order to relate the variational problems for Palatini and
Einstein–Hilbert gravity. This forces us to discuss how we will represent the
flavors of gravity to work with. For Palatini gravity we will use the
formulation given in [3], where it was interpreted as an example of the
concept of Griffiths variational problem. In order to describe this
formulation, let us summarize the main characteristics of a Griffiths
variational problem. It consists into three kind of data: A bundle $p\colon
W\to M$, whose sections will be the fields of the theory, a Lagrangian form
$\lambda\in\Omega^{m}(W)$ setting the dynamics, and a set of forms
$\mathcal{I}\subset\Omega^{\bullet}(W)$ (more precisely, an _exterior
differential system_) describing the set of differential restrictions on the
fields. This additional geometric structure replaces the contact structure of
the jet bundle; therefore, the usual constraints

$u^{A}_{\mu}=\frac{\partial u^{A}}{\partial x^{\mu}}$

are absent. It should be stressed that this change has profound consequences
on the equations governing the extremals of the underlying variational
problem: In general, when performing variations, it results that

$\delta u^{A}_{\mu}\not=\big{(}\delta u^{A}\big{)}_{\mu}.$

In fact, because the set of sections verifying the differential restrictions
represented by $\mathcal{I}$ must be invariant by variations, this
relationship should be replaced by conditions describing the infinitesimal
symmetries of this set.

Accordingly, a Griffiths variational problem will be described with the symbol

$\displaystyle(p\colon W\to M,\lambda,\mathcal{I}),$ (1.2)

where the additional data $\mathcal{I}$ has been added. The variational
problem underlying such triple consists in finding the extremals of the action

$S[s]:=\int_{M}s^{*}\lambda,$

where the sections $s\colon M\to W$ of the bundle $p$ must be integral for the
set of forms in $\mathcal{I}$, namely,

$s^{*}\alpha=0$

for every $\alpha\in\mathcal{I}$.

We can use this approach to represent Palatini gravity as a first order field
theory. Recall that for this type of gravity theory, the degrees of freedom
are a vielbein and a connection; it suggests taking the bundle of frames
$\tau\colon LM\to M$ (whose sections are the _vielbein_) as the field bundle.
Moreover, its jet space $J^{1}\tau$ can be decomposed as

$J^{1}\tau=LM\times_{M}C(LM),$

where $C(LM)\to M$ is the connection bundle, that is, a bundle whose sections
are the principal connections of $LM$ – see Section 3 for details. It means
that sections of $\tau_{1}\colon J^{1}\tau\to M$ are composed by a pair
vielbein $+$ connection and so, they are suitable for the description of the
fundamental degrees of freedom in Palatini gravity. Now, in Palatini
formulation, the vielbein and the connection are not independent; they are
related by two constraints: on the one side, the connection must be
torsionless; on the other, the metric associated to a given vielbein should be
invariant respect to the parallel translation associated to the given
connection. So, instead of working with the total jet space $J^{1}\tau$, we
will work in a submanifold $\mathcal{T}_{0}\subset J^{1}\tau$, namely, the
submanifold corresponding to the _torsion zero constraint_ ; also, the contact
structure is changed by a set of differential constraints implementing the
_metricity conditions_ – see equation (3.4) for a geometrical definition of
these conditions, and Section 3.2 for a discussion of its physical meaning. It
should be stressed that the zero torsion condition in this approach becomes
part of the bundle of fields, rather than appearing as part of the equations
of motion, as is usually the case (see for example [24, 35]).

The underlying variational problem is given by the triple

$(\tau_{1}^{\prime}\colon\mathcal{T}_{0}\to M,\mathcal{L}_{\rm
PG},\langle\omega_{\mathfrak{p}}\rangle),$

where $\mathcal{L}_{\rm PG}$ is a Lagrangian $m$-form on $\mathcal{T}_{0}$ and
$\langle\omega_{\mathfrak{p}}\rangle$ is an exterior differential system on
$\mathcal{T}_{0}$ encoding the metricity conditions.

It should be stressed that, when working with such variational problem, the
allowed variations are not those fullfilling the commutativity property

$\delta u^{A}_{\mu}=\big{(}\delta u^{A}\big{)}_{\mu};$

instead, they must be thought as infinitesimal symmetries of the metricity
conditions. We will not explore further this topic here; the interested reader
can find a detailed study of these questions in [7]. On the other hand, the
usual variational problem for Einstein–Hilbert gravity is given by the triple

$\big{(}(\tau_{\Sigma})_{2}\colon J^{2}\tau_{\Sigma}\to M,{\cal L}_{\rm
EH},\mathcal{I}_{{\rm con}}^{\Sigma}\big{)}$

where $\tau_{\Sigma}\colon\Sigma\to M$ is the bundle of metrics with a given
signature, ${\cal L}_{\rm EH}$ is the (second order) Lagrangian density [19]
and $\mathcal{I}_{{\rm con}}^{\Sigma}$ is the contact structure on the second
order bundle $J^{2}\tau_{\Sigma}$. Although it would be possible in principle
to work with a second order jet, it is easier to deal with first order jets;
so, we are forced to find a first order variational problem describing
Einstein–Hilbert gravity. In fact, one of the results of the present article
is to prove that there exists an $m$-form (not a density) ${\cal L}_{\rm
EH}^{(1)}$ on $J^{1}\tau_{\Sigma}$ such that

$\big{(}(\tau_{\Sigma})_{1}\colon J^{1}\tau_{\Sigma}\to M,{\cal L}_{\rm
EH}^{(1)},\mathcal{I}_{{\rm con}}^{\Sigma}\big{)}$

also describes Einstein–Hilbert gravity – see Section 8, where a
correspondence between the extremals for these problems will be established.

In summary, a large part of the present article consists in repeating for
Palatini gravity what was done for first order field theory in [6]; more
concretely and in line with the previous discussion, we want to answer an
specific problem through a particular set of techniques:

  * •

Problem: To establish correspondences between the extremals of the variational
problem

$\big{(}\tau_{1}^{\prime}\colon\mathcal{T}_{0}\to M,\mathcal{L}_{\rm
PG},\langle\omega_{\mathfrak{p}}\rangle\big{)},$

describing Palatini gravity, with the extremals of the variational problem

$\big{(}(\tau_{\Sigma})_{2}\colon J^{2}\tau_{\Sigma}\to M,{\cal L}_{\rm
EH},\mathcal{I}_{{\rm con}}^{\Sigma}\big{)}$

for Einstein–Hilbert gravity.

  * •

Techniques: The following diagram could help us to outline the strategy
involved in proving the correspondences:

${(W_{\rm PG},\lambda_{\rm PG},0)}$${(W_{\rm EH},\lambda_{\rm
EH},0)}$${\big{(}J^{1}\tau_{\Sigma},{\cal L}_{\rm EH}^{(1)},\mathcal{I}_{{\rm
con}}^{\Sigma}\big{)}}$${(\mathcal{T}_{0},\mathcal{L}_{\rm
PG},\langle\omega_{\mathfrak{p}}\rangle)}$${\big{(}J^{2}\tau_{\Sigma},{\cal
L}_{\rm EH},\mathcal{I}_{{\rm
con}}^{\Sigma}\big{)}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{3}$ReconstructionReduction
(1.3)

The main idea is to replace the Griffiths variational problems representing
Palatini and Einstein–Hilbert gravity with other kind of variational problems
playing the role, in this generalized context, of the unified formalism. This
operation is indicated schematically by arrows 1 and 2. As we described above,
formulation of the unified version of field theory requires to pass to a new
phase space where multivelocities and multimomentum variables are represented.
Nevertheless, the product bundle construction (1.1) is not versatile enough to
establish the correspondences between the variational problems mentioned above
because it depends on the interpretation of the bundle $J^{1}\pi^{\circ}$ as a
dual of the jet space $J^{1}\pi$. Instead, these correspondences could be
studied by using a very general setting described in [20], called _Lepage-
equivalent variational problem_. It consists in lifting the variational
problem $\big{(}\pi_{1}\colon J^{1}\pi\to M,{\cal L},\mathcal{I}_{{\rm
con}}\big{)}$ to a variational problem on an affine subbundle of the bundle of
$m$-forms on $J^{1}\pi$, and this subbundle becomes isomorphic to (a quotient
of) $W_{\text{cl}}$. Concretely, let us fix an adapted coordinate chart
$\big{(}\phi=\big{(}x^{\mu},u^{A}\big{)},U\big{)}$ on $E$; then we have a set
of $(m-1)$-forms defined as follows

$\eta:={\rm d}x^{1}\wedge\cdots\wedge{\rm
d}x^{m},\qquad\eta_{\nu}:=\frac{\partial}{\partial x^{\nu}}\lrcorner\eta.$

In this setting, the Lepage-equivalent problem of Gotay is given by the data

$(\pi_{0}\colon W_{0}\to M,\lambda_{0})$

where

$\displaystyle W_{0}:={\cal L}+{I}^{m}_{{\rm
con}}\subset\wedge^{m}\big{(}T^{*}J^{1}\pi\big{)}$ (1.4)

and, locally, the bundle of forms $I^{m}_{{\rm con}}$ is given by

$I^{m}_{{\rm
con}}\big{|}_{(x^{\mu},u^{A},u^{A}_{\nu})}:=\big{\\{}p_{A}^{\mu}\big{(}{\rm
d}u^{A}-u^{A}_{\nu}{\rm d}x^{\nu}\big{)}\wedge\eta_{\mu}\colon
p_{A}^{\mu}\in\mathbb{R}\big{\\}}\subset\wedge^{m}\big{(}T^{*}_{(x^{\mu},u^{A},u^{A}_{\nu})}\big{(}J^{1}\pi\big{)}\big{)}.$

The Lagrangian form $\lambda_{0}$ arises from the fact that $W_{0}$ is a
subbundle of the bundle of $m$-forms $\wedge^{m}\big{(}T^{*}J^{1}\pi\big{)}$;
it has a canonical $m$-form $\lambda$, and $\lambda_{0}$ is its pullback to
$W_{0}$. The underlying action becomes

$\Gamma\pi_{0}\ni\sigma\mapsto\int_{U}\sigma^{*}\lambda_{0}$

and the variations must be performed in an entirely free way; that is, in this
case, the variations $\delta u^{A}$ and $\delta u^{A}_{\mu}$ on $J^{1}\pi$ are
independent. As we said above, the coordinates along the fibers of the map

$q\colon\ W_{0}\to J^{1}\pi$

correspond to multimomentum variables, and we can see that

$W_{0}\simeq J^{1}\pi\times_{M}J^{1}\pi^{\circ},$

namely, the bundle $W_{0}$ is isomorphic to the total bundle $W_{\text{cl}}$
in the unified formulation of first order field theory. Thus, it can be
readily seen that Lepage-equivalent variational problems serve as a
generalization of the unified formalism to variational problems of Griffiths
type; the interested reader is referred to [20] for details.

Part of the techniques we will use to establish the relationship (reduction
and reconstruction) between the extremals of these variational problems are
analogous to those that were employed in [6]: To transform the variational
problems into its Lepage-equivalent problems, represented by the two-way
arrows 1 and 2 in diagram (1.3), and to use the operations available there in
order to set the correspondences. As we indicated above, in order to apply
this construction to Einstein–Hilbert gravity, it will be necessary to
consider an equivalent first-order variational problem instead; this operation
is shown as arrow 3 in the previous diagram. The rest of the techniques
involved in the proof of reduction/reconstruction are represented by the
dashed arrow: The basic idea is to take advantage of the fact that the
variational problems we are trying to connect are formulated on an affine
subbundles of a bundle of forms. As we said above, the naturality of forms
respect to the pullback operation is crucial to compare the equations
governing the extremals in every variational problem. Although in our present
article this characteristic will not be used, it is still true that the vector
bundle structure enhances this comparison by allowing the translation along a
given form.

These considerations set the purposes of the following article: On the 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 short, the
results achieved in the article could help to put the relationship between
Einstein–Hilbert and Palatini gravity in a precise geometrical framework,
which can in principle be extended to more sophisticated field theoretic
analyses.

In this vein, it should be mentioned that among the objectives in mind while
planning the article there was one of mathematical nature; in short, to
contribute to the improvement of the language in a geometrical sense around
the general question on how Einstein–Hilbert and Palatini versions of gravity
are related. In this regard, it appears that the subject has continuously been
developed since these formulations were found, but the use of a more precise
geometrical language in describing the new achievements in the area has grown
at a slower pace – see [10, 19, 38, 41] for examples dealing with the
geometrical perspective in gravity. To some extent, this imbalance could place
some obstacles to the natural development of some aspects of the subject; in
these cases, the geometrical viewpoint might be more useful than a mere
calculation in their understanding. Hence, the present article intends to
contribute to a better comprehension of the correspondence between
Einstein–Hilbert and Palatini gravity, through the formulation of this
correspondence in a geometrical framework.

The rest of the paper is organized as follows: In Section 2 fundamentals of
Griffiths variational problems and their relationship with the unified
formalism in field theory are given. In Section 3 geometrical tools necessary
for the construction of the variational problem for Palatini gravity we will
use in this article are reviewed. In Section 4 the actual construction of this
variational problem, as well as the associated unified problem, is done. The
symmetry considerations necessary to carry out the reduction are discussed in
Section 5. Section 6 is rather technical, containing some calculations used in
the reduction and reconstruction theorems. In Section 7 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 with
the contact structure of a jet bundle after projection onto the quotient.
Construction of the first order formalism for Einstein–Hilbert gravity (and
its correspondence with the usual second order formalism) is delayed until
Section 8; also, a 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 9 the effects of
this splitting in the variational formulation of Palatini gravity are
analyzed. In Section 10 the Routhian is constructed, showing that the Routhian
for Palatini gravity is the (first order) Einstein–Hilbert Lagrangian.
Finally, in Section 11 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. Also, it is proved there that this condition is
equivalent to the parallelizability of the spacetime manifold $M$; therefore,
the reconstruction scheme can be carried out only locally.

#### Notations

We are adopting here the notational conventions from [39] when dealing with
bundles and its associated jet spaces. Also, if $Q$ is a manifold,
$\Lambda^{p}(Q)=\wedge^{p}(T^{*}Q)$ denotes the $p$-th exterior power of the
cotangent bundle of $Q$. Moreover, for $k\leq l$ the set of $k$-horizontal
$l$-forms on the bundle $\pi\colon P\to N$ is

$\displaystyle\wedge^{l}_{k}(P):=\big{\\{}\alpha\in\wedge^{l}(P)\colon
v_{1}\lrcorner\cdots v_{k}\lrcorner\alpha=0\text{ for any
}v_{1},\dots,v_{k}\text{ }\pi\text{-vertical vectors}\big{\\}}.$ (1.5)

For the same bundle, the set of vectors tangent to $P$ in the kernel of $T\pi$
will be represented with the symbol $V\pi\subset TP$. In this regard, the set
of vector fields which are vertical for a bundle map $\pi\colon P\to N$ will
be indicated by $\mathfrak{X}^{V\pi}(P)$. The space of differential $p$-forms,
sections of $\Lambda^{p}(Q)\to Q$, will be denoted by $\Omega^{p}(Q)$. We also
write $\Lambda^{\bullet}(Q)=\bigoplus_{j=1}^{\dim Q}\Lambda^{j}(Q)$. If
$f\colon P\to Q$ is a smooth map and $\alpha_{x}$ is a $p$-covector on $Q$, we
will sometimes use the notation $\alpha_{f(x)}\circ T_{x}f$ to denote its
pullback $f^{*}\alpha_{x}$. If $P_{1}\to Q$ and $P_{2}\to Q$ are fiber bundles
over the same base $Q$ we will write $P_{1}\times_{Q}P_{2}$ for their fibred
product, or simply $P_{1}\times P_{2}$ if there is no risk of confusion.
Unless explicitly stated, the canonical projections onto its factor will be
indicated by

${\rm pr}_{i}\colon\ P_{1}\times P_{2}\to P_{i},\qquad i=1,2$

or, in general

${\rm pr}_{i_{1}\cdots i_{k}}\colon\ P_{1}\times\cdots\times P_{L}\to
P_{i_{1}}\times\cdots\times P_{i_{k}}$

for any collection of different indices $i_{1},\dots,i_{k}\in\\{1,\dots,L\\}$.

Given a manifold $N$ and a Lie group $G$ acting on $N$, the symbol $[n]_{G}$
for $n\in N$ will indicate the $G$-orbit in $N$ containing $n$; the canonical
projection onto its quotient will be denoted by

$p_{G}^{N}\colon\ N\to N/G.$

Also, if $\mathfrak{g}$ is the Lie algebra for the group $G$, the symbol
$\xi_{N}$ will represent the infinitesimal generator for the $G$-action
associated to $\xi\in\mathfrak{g}$. Finally, Einstein summation convention
will be used everywhere.

##  2 Variational problems and unified formalism

As we mentioned in Section 1, the scheme used for Routh reduction relies on
the notion of unified formulation of a variational problem. So, we will devote
the present section to describe the construction of a unified formalism for a
particular family of variational problem, the so called _Griffiths variational
problems_.

######  Definition 2.1 (Griffiths variational problem).

A _Griffiths variational problem_ is a triple

$(p\colon W\to M,\lambda,\mathcal{I})$

where $p\colon W\to M$ is a bundle, $\lambda\in\Omega^{m}(W)$ is a form on $W$
(here $m$ is the dimension of the base manifold $M$) and $\mathcal{I}$ is an
exterior differential system. The underlying variational problem consists in
finding the extremals of the action

$\Gamma p\ni\sigma\mapsto\int_{U}\sigma^{*}\lambda,$

where $U\subset M$ is any compact submanifold and $\sigma$ is a section
integral for $\mathcal{I}$, namely,

$\sigma^{*}\alpha=0$

for any $\alpha\in\mathcal{I}$.

######  Remark 2.2.

A particular instance of a Griffiths variational problem is provided by the so
called _classical variational problem_ , which is the variational problem
underlying the first order classical field theory [21]: In it, the underlying
bundle is the jet space $\pi_{1}\colon J^{1}\pi\to M$ associated to a bundle
$(E,\pi,M)$, the Lagrangian form is induced on
$\Omega^{m}\big{(}J^{1}\pi\big{)}$ by the Lagrangian density

${\cal L}\colon\ J^{1}\pi\to\wedge^{m}(T^{*}M)$

and the set of differential restrictions is imposed by the so called _contact
structure on the jet bundle_. Particularly, it means that the sections to be
evaluated in the action should be holonomic.

In view of the previous discussion, there are two crucial differences between
a classical variational problem and a more general Griffiths variational
problem that we would like to point out:

  * •

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
any bundle in principle.

  * •

More important is the fact that the sections are integral for the set of forms
$\mathcal{I}$, that in the general case (as in the present article) could be
different of the set of forms belonging to the contact structure.

A construction used in [20] becomes relevant to this article: Given a
Griffiths variational problem $(p\colon W\to M,\lambda,\mathcal{I})$, it is
possible to build another bundle $p^{\prime}\colon W^{\prime}\to M$ and a
bundle map $q\colon W^{\prime}\to W$ covering the identity on $M$. The new
bundle becomes part of a new variational problem $(p^{\prime}\colon
W^{\prime}\to M,\lambda^{\prime},0)$, and the extremals of the new variational
problem can be set into a one-to-one correspondence with the extremals of the
original variational problem through the map $q$. The variational problem
$(p^{\prime}\colon W^{\prime}\to M,\lambda^{\prime},0)$ is called a _Lepage-
equivalent variational problem_.111There are some subtleties regarding the use
of the word “equivalent” in this context; they will not be discussed here, so
we are referring to the readers interested in these questions to the original
article of Gotay. The coordinates along the fibers of the map
$p^{\prime}\colon W^{\prime}\to W$ can be seen as a generalization of momentum
variables in the context of these variational problems. An important feature
of a Lepage-equivalent variational problem is that it imposes no differential
restrictions on the sections of $p^{\prime}\colon W^{\prime}\to M$, which
could be useful when variations are performed.

A particular instance of this construction called _canonical Lepage-equivalent
problem_ will be relevant for the present article. It requires that the
differential constraints encoded by the exterior differential system
$\mathcal{I}$ be generated by the sections of a subbundle
$I\subset\wedge^{\bullet}(T^{*}W)$; under such conditions, we can define

$W^{\prime}|_{w}:=\big{\\{}\lambda(w)+\beta\colon\beta\in
I\cap\wedge^{m}(T^{*}_{w}W)\big{\\}}\subset\wedge^{m}(T^{*}W)$

for all $w\in W$. The projection $p^{\prime}\colon W^{\prime}\to W$ is the
restriction of the canonical map
$\overline{\tau}^{m}_{W}\colon\wedge^{m}(T^{*}W)\to W$ to this subbundle, and
the $m$-form $\lambda^{\prime}$ is the pullback of the canonical $m$-form
$\Theta\in\Omega^{m}(\wedge^{m}(T^{*}W))$ to $W^{\prime}$.

In order to describe the relevance of this construction, it will be necessary
to refer to the concept of _unified formalism for classical first order field
theory_ , as defined in [2, 16, 36] and references therein. This formulation
becomes useful when dealing with variational problems whose Lagrangian
densities have singular Legendre transformations. The trick is to lift the
variational problem to an space where both velocities and momenta are included
and to forget about the differential restrictions on the fields imposed by the
contact structure of the jet bundle. Therefore, variations can be performed
without having to identify the independent degrees of freedom, and the
formulas defining Legendre transform become part of the equations of motion;
on the downside, these equations of motion are in general of the algebraic-
differential type, and thus more difficult to deal with. As we mentioned in
the Introduction, this underlying bundle $W_{\text{cl}}$ for the unified
formulation of field theory is given by equation (1.1); the Lagrangian
functional ${\cal L}$ gives rise to an $m$-form $\Theta_{\cal L}$ on
$W_{\text{cl}}$, and the associated action becomes

$\Gamma(W_{\text{cl}}\to M)\ni\sigma\mapsto\int_{U}\sigma^{*}\Theta_{\cal L}.$

Finally, let us discuss briefly the relationship between Lepage-equivalent
problems and the unified formalism. It can be seen that, when restricted to
the particular case of the classical variational problem, the canonical
Lepage-equivalent variational problem devised by Gotay reduces to the
variational problem associated to the unified formalism. In fact, if the
fields are sections of a bundle $\pi\colon E\to M$, the triple describing the
associated Griffiths variational problem is

$\big{(}\pi_{1}\colon J^{1}\pi\to M,{\cal L},\mathcal{I}_{{\rm con}}\big{)},$

where $\mathcal{I}_{{\rm con}}\subset\Omega^{\bullet}\big{(}J^{1}\tau\big{)}$
is the exterior differential system induced by the contact structure. In this
case, we have the correspondences

$W\leftrightsquigarrow J^{1}\pi,\qquad\lambda\leftrightsquigarrow{\cal
L},\qquad\mathcal{I}\leftrightsquigarrow\mathcal{I}_{{\rm con}},\qquad
W^{\prime}\leftrightsquigarrow W_{\text{cl}},$

allowing us to think on Lepage-equivalent problems as generalizations of the
unified formalism for field theory.

##  3 Geometrical tools for Palatini gravity

We choose to focus on variational problems of Griffiths type because there
exists a description of Palatini gravity in terms of this kind of variational
problems [5]. The present section is devoted to give a brief account of the
geometrical ingredients involved in this construction.

###  3.1 Geometry of the jet space for the frame bundle

The basic bundle is the frame bundle $\tau\colon LM\rightarrow M$ on the
spacetime manifold $M$ ($\dim M=m$); because it is a principal bundle with
structure group ${\rm GL}(m)$, we can lift this action to the jet bundle
$J^{1}\tau$, so that we obtain a commutative diagram

${J^{1}\tau}$${LM}$${C(LM)}$${M}$$\scriptstyle{\tau_{10}}$$\scriptstyle{p_{{\rm
GL}(m)}^{J^{1}\tau}}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\tau}$$\scriptstyle{\overline{\tau}}$

where $C(LM):=J^{1}\tau/{\rm GL}(m)$ is the so called _connection bundle of
$LM$_, whose sections can be naturally identified with the principal
connections of the bundle $\tau$ – for details, see [9] and references
therein. It is interesting to note that there exists an affine isomorphism

$F\colon\ J^{1}\tau\rightarrow LM\times_{M}C(LM)\colon\
j_{x}^{1}s\mapsto\big{(}s(x),\big{[}j_{x}^{1}s\big{]}_{{\rm GL}(m)}\big{)}$

and under this correspondence, the ${\rm GL}(m)$-action is isolated to the
first factor in the product, namely

$\displaystyle F\big{(}j_{x}^{1}s\cdot g\big{)}=\big{(}s(x)\cdot
g,\big{[}j_{x}^{1}s\big{]}_{{\rm GL}(m)}\big{)}.$ (3.1)

It means that a section of the bundle $\tau_{1}$ is equivalent to a connection
on $LM$ plus a moving frame $(X_{1},\dots,X_{m})$ on $M$; although this moving
frame has no direct physical interpretation, we can associate a metric to it,
namely, in contravariant terms,

$g:=\eta^{ij}X_{i}\otimes X_{j}$

for some nondegenerate symmetric matrix $\eta$ – see equation (3.2) below. It
is the same to declare that the metric $g$ is the unique metric on $M$ making
the moving frame $(X_{1},\dots,X_{m})$ (pseudo)orthonormal, with the signature
given by $\eta$.

The tautological form $\widetilde{\theta}\in\Omega^{1}(LM,\mathbb{R}^{m})$ can
be pulled back along $\tau_{10}$ to a $1$-form
$\theta:=\tau_{10}^{*}\widetilde{\theta}$ on $J^{1}\tau$; moreover, the Cartan
form $\widetilde{\omega}\in\Omega^{1}\big{(}J^{1}\tau,V\tau\big{)}$, given by
the formula

$\widetilde{\omega}|_{j_{x}^{1}s}:=T_{j_{x}^{1}s}\tau_{10}-T_{x}s\circ
T_{j_{x}^{1}s}\tau_{1},$

gives rise to a $\mathfrak{gl}(m)$-valued $1$-form $\omega$ on $J^{1}\tau$, by
using the identification

$V\tau\simeq LM\times\mathfrak{gl}(m).$

By means of the canonical basis $\\{e_{i}\\}$ on $\mathbb{R}^{m}$ and
$\big{\\{}E^{i}_{j}\big{\\}}$ on $\mathfrak{gl}(m)$, where

$\big{(}E_{j}^{i}\big{)}^{q}_{p}:=\delta^{q}_{j}\delta^{i}_{p},$

we can define the collection of $1$-forms
$\big{\\{}\theta^{i},\omega^{i}_{j}\big{\\}}$ on $J^{1}\tau$ such that

$\theta=\theta^{i}e_{i},\qquad\omega=\omega^{j}_{i}E^{i}_{j}.$

We also have the formula

$\widetilde{\omega}=\omega^{j}_{i}\big{(}E^{i}_{j}\big{)}_{J^{1}\tau},$

where $A_{J^{1}\tau}\in\mathfrak{X}^{Vp_{{\rm
GL}(m)}^{J^{1}\tau}}\big{(}J^{1}\tau\big{)}$ is the infinitesimal generator
associated to $A\in\mathfrak{gl}(m)$ for the lifted action. It can be proved
that $\omega$ is a connection form for a principal connection on the bundle

$p_{{\rm GL}(m)}^{J^{1}\tau}\colon\ J^{1}\tau\rightarrow C(LM).$

######  Remark 3.1 (coordinates on the jet space $J^{1}\tau$ and the
connection bundle $\overline{\tau}\colon C(LM)\to M$).

Let $(\phi=(x^{\mu}),U)$ be a coordinate chart on $M$; for
$u=(X_{1},\dots,X_{m})\in\tau^{-1}(U)$, we define the maps $u\mapsto
e_{i}^{\mu}(u)$ such that

$X_{i}=e_{i}^{\mu}(u)\frac{\partial}{\partial x^{\mu}}$

for $1\leq i\leq m$. Then

$u\mapsto\big{(}x^{\mu}(\tau(u)),e^{\mu}_{i}(u)\big{)}$

defines a set of adapted coordinates on $\tau^{-1}(U)$; let
$\big{(}x^{\mu},e^{\mu}_{i},e^{\nu}_{k\sigma}\big{)}$ be the associated
coordinates on $\tau_{1}^{-1}(U)\subset J^{1}\tau$. We can also define the
functions

$\Gamma_{\rho\sigma}^{\mu}:=-e^{k}_{\rho}e^{\mu}_{k\sigma},$

on $\tau_{1}^{-1}(U)$, and we can prove that they are ${\rm GL}(m)$-invariant;
so, they define a set of coordinates on $\overline{\tau}^{-1}(U)\subset
C(LM)$.

Let us define

$\theta_{0}:=\theta^{1}\wedge\cdots\wedge\theta^{m};$

as every $u\in LM$ is a collection $u=(X_{1},\dots,X_{m})$ of vectors on
$\tau(u)\in M$, and $\theta^{i}$ is a $\tau_{1}$-horizontal $1$-form on
$J^{1}\tau$, we can define the set of forms

$\theta_{i_{1}\cdots i_{k}}|_{j_{x}^{1}s}:=X_{i_{1}}\lrcorner\cdots
X_{i_{k}}\lrcorner\theta_{0}|_{j_{x}^{1}s}$

for $1\leq i_{1},\dots,i_{k}\leq m$, where $j_{x}^{1}s\in J^{1}\tau$ is any
element such that $u=\tau_{10}\big{(}j_{x}^{1}s\big{)}$.

Although the reduction scheme we will develop in this article would work for
any signature, let us fix it using the matrix

$\displaystyle\eta:=\begin{bmatrix}-1&0&\cdots&0\\\ 0&1&&0\\\
\vdots&&\ddots&\vdots\\\ 0&\cdots&0&1\end{bmatrix}\in{\rm GL}(m).$ (3.2)

Accordingly, let $\eta_{ij}$ be its $(i,j)$-entry; we will represent with the
symbol $\eta^{ij}$ the $\left(i,j\right)$-entry of its inverse. With these
ingredients we can construct the _Palatini Lagrangian_

$\displaystyle{\cal L}_{\rm PG}:=\eta^{ip}\theta_{ik}\wedge\Omega^{k}_{p},$
(3.3)

where $\Omega:=\Omega^{i}_{j}E^{j}_{i}$ is the curvature of the canonical
connection $\omega$. This $m$-form will determine the dynamics of the vacuum
gravity in this formulation.

Finally, let us describe a decomposition of $\mathfrak{gl}(m)$ induced by
$\eta$. In fact, this matrix yields to a real form $\mathfrak{u}$ in
$\mathfrak{gl}\left(m,\mathbb{C}\right)$, given by

$\mathfrak{u}=\big{\\{}\xi\in\mathfrak{gl}(m,\mathbb{C})\colon\xi^{\dagger}\eta+\eta\xi=0\big{\\}}$

and thus we have a Cartan decomposition

$\mathfrak{gl}(m,\mathbb{C})=\mathfrak{u}\oplus\mathfrak{s}.$

Given the inclusion

$\mathfrak{gl}(m)\subset\mathfrak{gl}(m,\mathbb{C}),$

we obtain the decomposition

$\mathfrak{gl}(m)=\mathfrak{k}\oplus\mathfrak{p}.$

The subalgebra $\mathfrak{k}$ is the Lie algebra of the subgroup $K\subset{\rm
GL}(m)$, composed of the linear transformations keeping invariant the matrix
$\eta$,

$K:=\big{\\{}A=\big{(}A_{i}^{j}\big{)}\colon\eta_{ij}A^{i}_{k}A^{j}_{l}=\eta_{kl}\big{\\}}.$

The canonical action of ${\rm GL}(m)$ on $LM$ restricts to an free action of
$K$ on this bundle; accordingly, we have the $K$-principal bundle

$p_{K}^{LM}\colon\ LM\to\Sigma:=LM/K.$

The bundle $\tau_{\Sigma}\colon\Sigma\to M$ induced by this quotient has an
immediate physical meaning: Any section $\zeta\colon M\to\Sigma$ is a metric
on $M$ with $\eta$-signature. In fact, let us define the map $q\colon LM\to
TM\otimes_{M}TM$ via

$q(X_{1},\dots,X_{m}):=\eta^{ij}X_{i}\otimes X_{j};$

indicating by $R_{k}\colon LM\to LM$ the action of an element $k\in K$ on
$LM$, it can be proved that $q\circ R_{k}=q$, so that there exists a map
$\overline{q}\colon LM/K\to TM\otimes_{M}TM$ making the following diagram
commutative

${LM}$${\Sigma}$${TM\otimes_{M}TM}$$\scriptstyle{p_{K}^{LM}}$$\scriptstyle{q}$$\scriptstyle{\overline{q}}$

When restricted to the subbundle of nondegenerate symmetric $2$-tensors (with
$\eta$-signature) on $M$, this map becomes a bundle isomorphism on $M$,
allowing us to interpret $\Sigma$ as the bundle of metrics with
$\eta$-signature on $M$.

######  Remark 3.2 (adapted coordinates for the principal bundle
$p_{K}^{LM}\colon LM\to\Sigma$).

It will be useful to introduce a set of coordinates on $LM$ adapted to the
quotent map $p_{K}^{LM}$. In order to proceed, we need a theorem on
generalized polar decomposition [25]. So, let $U\subset M$ be a parallelizable
open set in $M$, let $\\{Z_{1},\dots,Z_{m}\\}\subset\mathfrak{X}(U)$ be a
moving basis on $U$, and $\\{e_{1},\dots,e_{n}\\}$ the canonical basis on
$\mathbb{R}^{m}$; for every $x\in U$ denote $V=T_{x}M$ and $W=\mathbb{R}^{m}$.
The matrix $\eta$ defines a linear map

$\eta\colon\ W\to W^{*};$

another operator that will be important for the formulation of the polar
decomposition theorem is

$I\colon\ V\to W\colon\ Z_{\mu}\mapsto\delta_{\mu}^{i}e_{i}.$

Also, set

$\widetilde{\eta}:=I^{*}\circ\eta\circ I.$

It is clear that, in this setting, a metric is a map

$g_{x}\colon\ V\to V^{*}$

for every $x\in U$, and a vielbein becomes a linear map

$E\colon\ V\to W.$

Then, the following can be proved.

######  Theorem 3.3 (generalized polar decomposition).

Let $g\colon V\to V^{*}$ be an invertible linear map such that
$\widetilde{\eta}\circ g^{-1}$ has no eigenvalues on the nonpositive real
axis. Then $g$ can be uniquely descomposed as

$g=Q\circ s,$

where $Q\colon W\to V^{*}$ satisfies

$Q^{*}\circ\widetilde{\eta}\circ Q=\eta$

and for $s\colon V\to W$ the following property

$I^{*}\circ\eta\circ s=(I^{*}\circ\eta\circ s)^{*}$

holds.

Therefore, the condition on the factor $Q$ tells us that the composite map

$I^{*}\circ Q\colon\ W\to W^{*}$

belongs to $K$. Thus, let $V\subset\tau_{\Sigma}^{-1}(U)$ be an open set in
$\Sigma$ such that $\widetilde{\eta}\circ g^{-1}$ has no eigenvalues on the
nonpositive real axis for every $g\in V$. Using this theorem, we can define
the $K$-bundle isomorphism

$u\in\big{(}p_{K}^{LM}\big{)}^{-1}(V)\mapsto\big{(}p_{K}^{LM}(u),I^{*}\circ
Q\big{)}\in V\times K,$

where $Q$ is the corresponding factor in the polar decomposition of
$g=p_{K}^{LM}(u)$. Finally, given a set of coordinates
$\big{(}x^{\mu},g^{\mu\nu}\big{)}$ on $\tau_{\Sigma}^{-1}(U)$ and
$\big{(}k_{i}^{j}\big{)}$ on $K$, we can introduce the set of coordinates

$u\mapsto\big{(}x^{\mu},g^{\mu\nu},k_{i}^{j}\big{)}$

on $\tau_{1}^{-1}(U)\subset LM$, where $\big{(}x^{\mu},g^{\mu\nu}\big{)}$ are
the coordinates for $p_{K}^{LM}(u)$ and $\big{(}k_{i}^{j}\big{)}$ are the
coordinates on $K$ for $I^{*}\circ Q$.

###  3.2 Restrictions in Palatini gravity: Zero torsion submanifold
and metricity forms

It is time to discuss the restrictions we must impose on the sections of
$\tau_{1}$ 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; this approach has been extensively discussed in the
references [3, 5].

Recall that, according to the identification performed by the map $F$ – see
equation (3.1) above – a section of $J^{1}\tau$ can be seen as a pair composed
by a frame plus a connection. Correspondingly, there are two types of
conditions to be imposed to a section of $J^{1}\tau$, each of them motivated
on physical grounds which we will not discuss here:

  1. 1)

the connection which is a solution for the field equations of Palatini gravity
must be torsionless, and

  2. 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_

$T:=\big{(}{\rm d}\theta^{j}+\omega^{j}_{k}\wedge\theta^{k}\big{)}\otimes
e_{j}\in\Omega^{2}\big{(}J^{1}\tau,\mathbb{R}^{m}\big{)}.$

Now, every connection $\Gamma\colon M\rightarrow C(LM)$ gives rise to a
section $\sigma_{\Gamma}\colon LM\rightarrow J^{1}\tau$ of the bundle
$\tau_{10}\colon J^{1}\tau\rightarrow LM$, as the equivariance of the
following diagram shows

${J^{1}\tau}$${LM}$${C(LM)}$${M}$$\scriptstyle{\tau_{10}}$$\scriptstyle{p_{{\rm
GL}(m)}^{J^{1}\tau}}$$\scriptstyle{\tau}$$\scriptstyle{\sigma_{\Gamma}}$$\scriptstyle{\overline{\tau}}$$\scriptstyle{\Gamma}$

The interesting fact is that the pullback form $\sigma_{\Gamma}^{*}T$
coincides with the torsion of the connection $\Gamma$. Additionally, it can be
proved that $T$ is a $1$-horizontal form on $\tau_{1}\colon J^{1}\tau\to M$,
so that there exists a maximal (respect to the inclusion) submanifold
$i_{0}\colon\mathcal{T}_{0}\hookrightarrow J^{1}\tau$ such that

  1. 1)

$\mathcal{T}_{0}$ is transversal to the fibers of $\tau_{1}\colon
J^{1}\tau\rightarrow M$ (namely, $T_{j^{1}_{x}s}(\mathcal{T}_{0})\oplus
V_{j_{x}^{1}s}\tau_{1}=T_{j_{x}^{1}s}\left(J^{1}\tau\right)$), and

  2. 2)

it annihilates the torsion, i.e.,

$i_{0}^{*}T=0.$

The transformation properties of the form $T$ allow us to conclude that
$\mathcal{T}_{0}$ is ${\rm GL}(m)$-invariant. The connections associated to
sections of $J^{1}\tau$ taking values in $\mathcal{T}_{0}$ are torsionless, so
that the zero torsion restriction can be achieved through the requirement that
these sections take values in this submanifold. Accordingly, we can use the
affine isomorphism $F\colon J^{1}\tau\to LM\times C(LM)$, to define the
_bundle of torsionless connections_ as the bundle
$\overline{\tau}^{\prime}\colon C_{0}(LM)\to M$ obtained by restricting $F$ to
$\mathcal{T}_{0}$

$C_{0}(LM):={\rm pr}_{2}(F(\mathcal{T}_{0})).$

Moreover, the following lemma can be proved using standard facts about
principal bundles [29].

######  Lemma 3.4.

The submanifold $\mathcal{T}_{0}\subset J^{1}\tau$ is a subbundle of the ${\rm
GL}(m)$-bundle $p_{{\rm GL}(m)}^{J^{1}\tau}\colon J^{1}\tau\to C(LM)$,
associated to the isomorphism ${\rm id}\colon{\rm GL}(m)\to{\rm GL}(m)$. Also,
it is a ${\rm GL}(m)$-principal bundle with respect to the restriction of the
${\rm GL}(m)$-action.

These considerations give rise to the commutative diagram

${\mathcal{T}_{0}}$${LM}$${C_{0}(LM)}$${M}$$\scriptstyle{\tau_{10}^{\prime}}$$\scriptstyle{p_{{\rm
GL}(m)}^{\mathcal{T}_{0}}=p_{{\rm
GL}(m)}^{J^{1}\tau}\big{|}_{\mathcal{T}_{0}}}$$\scriptstyle{\tau^{\prime}}$$\scriptstyle{\overline{\tau}^{\prime}}$

######  Remark 3.5 (local description for $\mathcal{T}_{0}$).

Let $\left(x^{\mu},e_{k}^{\nu}\right)$ be a set of adapted coordinates for
$LM$ induced on $\tau^{-1}(U)$ by a set of coordinates $(x^{\mu})$ on
$U\subset M$; as was shown in Remark 3.1, it induces coordinates
$(x^{\mu},e_{k}^{\nu},e^{\nu}_{k\sigma})$ on $\tau_{1}^{-1}(U)$. On this open
set we have

$T=e_{\sigma}^{i}\big{(}e^{k}_{\mu}e_{k\nu}^{\sigma}dx^{\mu}\wedge
dx^{\nu}\big{)}\otimes e_{i},$

where $(e^{k}_{\mu})$ is the inverse matrix of $(e_{k}^{\mu})$, so that the
set $\mathcal{T}_{0}\cap\tau_{1}^{-1}(U)$ is described by the constraints

$e^{k}_{\mu}e_{k\nu}^{\sigma}=e^{k}_{\nu}e_{k\mu}^{\sigma}.$

On the other hand, the metricity condition has a differential nature: As we
mentioned above, matrix $\eta$ determines a factorization of
$\mathfrak{gl}(m)$ in a subalgebra $\mathfrak{k}$ (the subalgebra of
$\eta$-Lorentz transformations) and an invariant subspace $\mathfrak{p}$. The
explicit formulas for this decomposition are given by the projectors

$A_{\mathfrak{k}}:=\frac{1}{2}\big{(}A-\eta A^{T}\eta\big{)},\qquad
A_{\mathfrak{p}}:=\frac{1}{2}\big{(}A+\eta A^{T}\eta\big{)}$

for every $A\in\mathfrak{gl}(m)$. The metricity condition is imposed on a
section $s\colon M\rightarrow J^{1}\tau$ by requiring that

$\displaystyle s^{*}\omega_{\mathfrak{p}}=0,$ (3.4)

where $\omega_{\mathfrak{p}}$ is the $\mathfrak{p}$-component of the canonical
connection $\omega$ respect to this decomposition. Taking into account the
affine isomorphism $F\colon J^{1}\tau\to LM\times_{M}C(LM)$, this constraint
means that the parallel transport of the connection ${\rm pr}_{2}\circ F\circ
s\colon M\to C(LM)$ leaves invariant the metric associated to the vielbein
${\rm pr}_{1}\circ F\circ s\colon M\to LM$ (see equation (7.4) below).

######  Remark 3.6 (local expression for the metricity conditions).

Let us provide a description of the metricity conditions (3.4) in terms of the
adapted coordinates $(x^{\mu},e^{\mu}_{k},e^{\mu}_{k\nu})$ on
$\tau_{1}^{-1}(U)\subset J^{1}\tau$. It results that for a section

$s\colon\
(x^{\mu})\mapsto\big{(}x^{\mu},e^{\mu}_{k}(x),e^{\mu}_{k\nu}(x)\big{)},$

these conditions can be written as [3]

${\rm
d}{g}^{\mu\nu}+\big{(}{g}^{\mu\sigma}\Gamma^{\nu}_{\gamma\sigma}+{g}^{\nu\sigma}\Gamma^{\mu}_{\gamma\sigma}\big{)}{\rm
d}x^{\gamma}=0,$ (3.5)

where ${g}^{\mu\nu}(x):=\eta^{kl}e_{k}^{\mu}(x)e_{l}^{\nu}(x)$ is the metric
associated to this section and

$\Gamma^{\mu}_{\rho\sigma}(x):=-e^{k}_{\rho}(x)e^{\mu}_{k\sigma}(x)$

are the Christoffel symbols for the associated connection. As it is well-
known, these conditions plus zero torsion imply the metric and the symbols
become related by Levi-Civita formula

$\Gamma_{\mu\nu}^{\sigma}=\frac{1}{2}g^{\sigma\rho}\left(\frac{\partial
g_{\rho\mu}}{\partial x^{\nu}}+\frac{\partial g_{\rho\nu}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\rho}}\right).$

######  Remark 3.7 (on the nature of the metricity conditions).

In some approaches to Palatini gravity, it is customary to work with a ${\rm
SO}(3,1)$ connection, without a clear reference to the bundle to which this
connection should be associated. It gives rise to an ambiguity that could be
problematic in some cases. Namely, when in local terms it is referred to a
$\mathfrak{so}(3,1)$-valued $1$-form

$\alpha\colon\ TM\to\mathfrak{so}(3,1),$

it is not clear whether it is a local version of a connection on a principal
bundle with structure group ${\rm SO}(3,1)$ or instead, a local version of a
connection on a principal bundle with a larger structure group $G\supset{\rm
SO}(3,1)$ taking values in the smaller Lie algebra
$\mathfrak{so}(3,1)\subset\mathfrak{g}$. It is not uncommon to find articles
in the literature where the information provided to the reader to decide in
which case one is working is not enough, because in general, the underlying
principal bundle is not associated to quantities of physical nature. It should
remain clear that in general, these connections are not equivalent, unless a
reduction of the $G$-bundle to a subbundle with structure group ${\rm
SO}(3,1)$ is admitted. When $G={\rm GL}(4)$, it is equivalent to have a metric
with signature $3+1$; in this case, the ${\rm SO}(3,1)$-bundle is the bundle
of pseudo orthogonal frames. It could be argued that to have a metric at our
disposal is a feature of working with metric-affine theories of gravitation;
but then it should be pointed out in this case that variations of the metric
have the undesired side effect of changing the ${\rm SO}(3,1)$-bundle. In
order to avoid this behavior, we have adopted metricity conditions (3.4): A
connection verifying these conditions will be a connection on the frame
bundle, whose structure group is ${\rm GL}(m)$, and will have a local version
taking values in the subalgebra $\mathfrak{k}\subset\mathfrak{gl}(m)$
associated to the matrix $\eta$.

##  4 Griffiths variational problem for Palatini gravity

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 in Section 2:

  * •

The relevant bundle is not the first order jet $J^{1}\tau$; instead, it is the
subset $\mathcal{T}_{0}$ consisting in 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 $J^{1}\tau$ to $\mathcal{T}_{0}$;
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 $\tau_{10}$ and $\tau_{1}$, which will be indicated as

$\tau_{10}^{\prime}\colon\ \mathcal{T}_{0}\to
LM\qquad\text{and}\qquad\tau_{1}^{\prime}\colon\ \mathcal{T}_{0}\to M.$

  * •

The forms we will use for the restriction of the sections of
$\tau_{1}^{\prime}\colon\mathcal{T}_{0}\rightarrow M$ are not the whole set of
contact forms $\big{\\{}\omega_{j}^{i}\big{\\}}$, but a geometrically relevant
subset, namely, the components of the metricity forms $\omega_{\mathfrak{p}}$.

Using these considerations, we will introduce the following definition.

######  Definition 4.1 (Griffiths variational problem for Palatini gravity).

The variational problem for Palatini gravity is given by the action

$S[\sigma]:=\int_{U}\sigma^{*}{\cal L}_{\rm PG},$

where $\sigma\colon U\subset M\rightarrow\mathcal{T}_{0}$ is any section of
$\tau_{1}^{\prime}$ such that $\sigma^{*}\omega_{\mathfrak{p}}=0$. According
to equation (1.2), it is described by the triple
$(\tau_{1}^{\prime}\colon\mathcal{T}_{0}\to M,{\cal L}_{\text{\rm
PG}},\langle\omega_{\mathfrak{p}}\rangle)$, where $\langle\cdot\rangle$
indicates the exterior differential system generated by the set of forms
enclosed by the brackets.

######  Remark 4.2.

The considerations made in Section 3 allow us to compare the variational
problem given by Definition 4.1 with the classical variational problem, as in
[8]. In fact, Proposition 4 in this reference tells us that when holonomic
sections are factorized using the isomorphism $F$ (in [8] it is called
$\Psi$), the connection

${\rm pr}_{2}\circ F\circ s\colon\ M\to C(LM)$

is flat, a restriction far more stringent than those imposed by the metricity
conditions, that only require the associated connection form to have values in
$\mathfrak{k}$. For a description of these admissible variations in the case
of the metricity constraints, see [7, Section 4.1].

It is interesting to point out the effect that the replacement of the contact
structure with the differential system $\langle\omega_{\mathfrak{p}}\rangle$
has on the way that Euler–Lagrange equations are calculated. The key change
has to do with the characterization of the admissible infinitesimal
variations: Given that the metricity constraints are a part of the constraints
imposed by the contact structure, it results that the set of admissible
infinitesimal variations considered in [8] contain them as a subset.

In order to establish the unified version of the equations of motion for
Palatini gravity, it will be necessary to represent the metricity constraints
as a subbundle of the set of $m$-forms on $\mathcal{T}_{0}$. To this end, let
us define the _metricity subbundle_ $I_{\rm PG}^{m}$ on $\mathcal{T}_{0}$,

$I_{\rm
PG}^{m}:=\big{\\{}\eta^{ik}\beta_{kp}\wedge\omega_{k}^{p}\colon\beta_{ij}\in\wedge^{m-1}_{1}(T^{*}\mathcal{T}_{0}),\beta_{ij}-\beta_{ji}=0\big{\\}}\subset\wedge^{m}_{2}(T^{*}\mathcal{T}_{0}),$

where $\wedge^{m-1}_{1}(T^{*}\mathcal{T}_{0})$ indicates the set of
$\tau_{1}^{\prime}$-horizontal $(m-1)$-covectors on $\mathcal{T}_{0}$.

######  Remark 4.3 (local description for the metricity subbundle).

Given a nondegenerate symmetric matrix $(g^{\mu\nu})$, there exists a matrix
$\left(e^{\mu}_{k}\right)$ such that

$g^{\mu\nu}=\eta^{kl}e_{k}^{\mu}e_{l}^{\nu};$

also, any two matrices $(e^{\mu}_{k})$, $(f^{\mu}_{k})$ giving rise to the
same symmetric matrix $\left(g^{\mu\nu}\right)$ differ in an element
$(k^{i}_{j})\in{\rm GL}(m)$ such that

$k^{i}_{j}k^{l}_{m}\eta^{jm}=\eta^{il}.$

Using the factorization

$\mathcal{T}_{0}=LM\times_{M}C_{0}(LM),$

we can set the coordinates
$\big{(}x^{\mu},g^{\mu\nu},k^{i}_{j},\Gamma^{\mu}_{\rho\sigma}\big{)}$ on
$\mathcal{T}_{0}$; the functions $\Gamma^{\mu}_{\rho\sigma}$ are not
independent; rather, they are constrained by the relations

$\Gamma^{\mu}_{\rho\sigma}=\Gamma^{\mu}_{\sigma\rho}.$

On the other hand, metricity constraints are given by equation (3.5));
therefore

$I_{\rm
PG}^{m}\big{|}_{(x^{\mu},g^{\mu\nu},k^{i}_{j},\Gamma^{\mu}_{\rho\sigma})}=\big{\\{}p_{\mu\nu}^{\rho}\big{[}{\rm
d}{g}^{\mu\nu}+\big{(}{g}^{\mu\sigma}\Gamma^{\nu}_{\gamma\sigma}+{g}^{\nu\sigma}\Gamma^{\mu}_{\gamma\sigma}\big{)}{\rm
d}x^{\gamma}\big{]}\wedge\eta_{\rho}\colon
p_{\mu\nu}^{\rho}=p_{\nu\mu}^{\rho}\big{\\}},$

where, as before

$\eta_{\rho}:=\frac{\partial}{\partial x^{\rho}}\lrcorner\eta,\qquad\eta={\rm
d}x^{1}\wedge\cdots\wedge{\rm d}x^{m}.$

It is a description of the fibers of $I_{\rm PG}^{m}$ in local terms; the
independence of these forms respect to the variables $\big{(}k_{i}^{j}\big{)}$
is due to the fact that the group $K$ describes a fundamental symmetry of this
description of gravity.

######  Remark 4.4 (metricity conditions and the annihilation of the metricity
tensor).

It could be useful to give an explanation of the relevance of this subbundle.
To this end, recall the notion of _metricity tensor of a connection respect to
a metric_ [23]; in short, given a manifold with a metric and a connection in
it, this tensor is essentially the covariant derivative of the metric. When
working in the jet space of the frame bundle, where a section induced both a
metric and a connection, this tensor can be encoded in terms of the canonical
connection. Concretely, let us suppose that $s\colon M\to\mathcal{T}_{0}$ is a
section with the following property

$s^{*}\big{(}\eta^{ik}\beta_{ip}\wedge\omega^{p}_{k}\big{)}=0$

for any collection of $l$-forms $\beta_{ip}$ such that

$\beta_{ip}-\beta_{pi}=0.$

Then the connection on $LM$ induced by $s$ has zero metricity tensor;
therefore, the metricity subbundle allows us to introduce the set of
restrictions imposed by the annihilation of the metricity tensor into the
Lagrangian. From this viewpoint, forms $\beta_{ij}$ in the definition of the
metricity subbundle play the role of Lagrange multipliers.

With the metricity subbundle in mind, and following the prescriptions made in
[20] for the construction of the canonical Lepage-equivalent variational
problem, we define the affine subbundle

$\displaystyle W_{\rm PG}:={\cal L}_{\rm PG}+I_{\rm
PG}^{m}\subset\wedge^{m}_{2}(T^{*}\mathcal{T}_{0}),$ (4.1)

which comes with the projection

$\tau_{\rm PG}\colon\ W_{\rm PG}\rightarrow\mathcal{T}_{0}\colon\
\alpha\in\wedge^{m}_{2}\big{(}T^{*}_{j^{1}_{x}s}\mathcal{T}_{0}\big{)}\mapsto
j_{x}^{1}s.$

We are referring to the interested reader to the article of Gotay for details;
for the purposes of the present work, it should be enough to mention that the
bundle defined by equation (4.1) will become the underlying bundle for the
Lepage-equivalent problem associated to the Griffiths variational problem
describing Palatini gravity (see Definition 4.1 above).

Because this is a subbundle in the set of $m$-forms on $\mathcal{T}_{0}$, it
has a canonical $m$-form $\lambda_{\rm PG}$ on it given by

$\lambda_{\rm PG}|_{\alpha}(w_{1},\dots,w_{m}):=\alpha(T_{\alpha}\tau_{\rm
PG}(w_{1}),\dots,T_{\alpha}\tau_{\rm PG}(w_{m})),\qquad w_{1},\dots,w_{m}\in
T_{\alpha}(W_{\rm PG}).$

This $m$-form will be the Lagrangian form for the Lepage-equivalent problem
associated to the Griffiths variational problem for Palatini gravity, namely

$(\tau_{1}^{\prime}\circ\tau_{\rm PG}\colon W_{\rm PG}\to M,\lambda_{\rm
PG},0).$ (4.2)

Because no differential restrictions must be considered when performing
variations of the variational problem

$\Gamma\left(\tau_{1}^{\prime}\circ\tau_{\rm
PG}\right)\ni\Gamma\mapsto\int_{U}\Gamma^{*}\lambda_{\rm PG},$

and assuming that the variations annihilate at the boundary, we obtain that
the Euler-Lagrange equations for the Lepage-equivalent variational problem
(4.2) are

$\Gamma^{*}(X\lrcorner{\rm d}\lambda_{\rm PG})=0\qquad\text{for all}\qquad
X\in\mathfrak{X}^{V(\tau_{1}^{\prime}\circ\tau_{\rm PG})}(W_{\rm PG}).$

The relevance of the unified formalism in dealing with the variational
problems posed by Definition 4.1 is guaranteed by the following result [5].

######  Proposition 4.5.

A section $s\colon U\subset M\rightarrow\mathcal{T}_{0}$ is critical for the
variational problem established in Definition 4.1 if and only if there exists
a section $\Gamma\colon U\subset M\rightarrow W_{\rm PG}$ such that

  1. $1)$

$\Gamma$ covers $s$, i.e., $\tau_{\rm PG}\circ\Gamma=s$, and

  2. $2)$

$\Gamma^{*}(X\lrcorner{\rm d}\lambda_{\rm PG})=0$, for all
$X\in\mathfrak{X}^{V(\tau_{1}^{\prime}\circ\tau_{\rm PG})}(W_{\rm PG})$.

$\Gamma$ is called a _solution_ of the Palatini gravity equations of motion.

######  Remark 4.6.

Although the proof in [5] refers to sections of $\tau_{1}\colon J^{1}\tau\to
M$, it can be also readily adapted to cover this case; in this regard, see
Appendix B.

The situation described by Proposition 4.5 is summarized in the following
diagram:

${W_{\rm
PG}}$${\mathcal{T}_{0}}$${M}$$\scriptstyle{\tau_{1}^{\prime}\circ\tau_{\rm
PG}}$$\scriptstyle{\tau_{\rm
PG}}$$\scriptstyle{\tau_{1}^{\prime}}$$\scriptstyle{\Gamma}$$\scriptstyle{s}$

Accordingly, any section $s$ that is extremal for the Griffiths variational
problem

$(\tau_{1}^{\prime}\colon\mathcal{T}_{0}\to M,{\cal L}_{\text{\rm
PG}},\langle\omega_{\mathfrak{p}}\rangle)$

can be lifted to a section $\Gamma$ that is extremal for the Lepage-equivalent
variational problem

$(\tau_{1}^{\prime}\circ\tau_{\rm PG}\colon W_{\rm PG}\to M,\lambda_{\rm
PG},0)$

and viceversa.

######  Remark 4.7 (local version of Proposition 4.5).

According to Remark 4.3, section $\Gamma$ will be obtained from section $s$ by
providing the set of functions

$(p_{\mu\nu}^{\sigma})\colon\ U\to\mathbb{R}^{\frac{m^{2}(m+1)}{2}}.$

These momentum variables $p_{\mu\nu}^{\sigma}$ are determined by the equations
of motion in Proposition 4.5, concretely, through the contractions

$\Gamma^{*}\left(\frac{\partial}{\partial\Gamma^{\sigma}_{\mu\nu}}\lrcorner{\rm
d}\lambda_{\rm PG}\right)=0.$

We will see below (Section 8) that an unified formalism for (first order)
Einstein–Hilbert variational problem can also be given; the reduction and
reconstruction theorems (see Section 11) will be proved using these lifted
systems.

##  5 Symmetry and reduction

As we described in the introduction, a crucial ingredient in Routh reduction
is the restriction of the dynamics to a level set of the momentum map. It
forces us to discuss the presence of natural symmetries in our formulation of
gravity, and to construct their momentum maps. Also, this procedure requires
the choice of a connection on a bundle obtained by quotient out the symmetries
of the variational problem. This section is devoted to these tasks.

###  5.1 Momentum map and connection

As we said above (see Lemma 3.4), there exists a ${\rm GL}(m)$-action on
$\mathcal{T}_{0}$; nevertheless, the Lagrangian ${\cal L}_{\rm PG}$ is
preserved by the action of the subgroup $K\subset{\rm GL}(m)$ composed of the
linear transformations keeping invariant the matrix $\eta$,

$K:=\big{\\{}A=\big{(}A_{i}^{j}\big{)}\colon\eta_{ij}A^{i}_{k}A^{j}_{l}=\eta_{kl}\big{\\}}.$

We can lift the ${\rm GL}(m)$-action to $\wedge^{m}(\mathcal{T}_{0})$; it
results that the subbundle $I^{m}_{\rm PG}$ is also preserved by the action of
$K$, and so

$K\cdot W_{\rm PG}\subset W_{\rm PG}.$

Our aim is to find a momentum map for this action, in the sense of the
following definition.

######  Definition 5.1.

A _momentum map_ for the action of $K$ on $W_{\rm PG}$ is a map

$J\colon\ W_{\rm PG}\to\Lambda^{m-1}(T^{*}W_{\rm PG})\otimes\mathfrak{k}^{*}$

over the identity in $W_{\rm PG}$ such that

$\xi_{W_{\rm PG}}\lrcorner{\rm d}\lambda_{\rm PG}=-{\rm d}J_{\xi},$

where $J_{\xi}$ is the $(m-1)$-form on $W_{\rm PG}$ whose value at $\alpha\in
W_{\rm PG}$ is $J_{\xi}(\alpha)=\langle J(\alpha),\xi\rangle$.

A momentum map is $Ad^{*}$-_equivariant_ if it satisfies

$\langle J(g\alpha),{\rm Ad}_{g^{-1}}\xi\rangle=g\langle
J(\alpha),\xi\rangle.$

Also, it is said that $J$ is _conserved along a section $\Gamma\colon M\to
W_{\rm PG}$_ if and only if $\Gamma^{*}({\rm d}J)=0$.

######  Remark 5.2.

A clarification about the nomenclature seems necessary here: Suppose
$M=\mathbb{R}\times N$ for some $(m-1)$-dimensional manifold $N$, we can take
$U\subset N$ and $\Gamma\colon[T_{1},T_{2}]\times U\to W_{\rm PG}$ such that
$J$ is conserved along $\Gamma$ and moreover

$J(\Gamma(t,q))=0\qquad\text{for all}\quad T_{1}\leq t\leq
T_{2}\quad\text{and}\quad q\in\partial U;$

then, denoting $\Gamma_{t}\colon U\to W_{\rm PG}\colon r\mapsto\Gamma(t,r)$
for any $t\in[T_{1},T_{2}]$, Stokes’ theorem will tell us that

$\int_{U}\Gamma_{T_{1}}^{*}J=\int_{U}\Gamma_{T_{2}}^{*}J.$

It is in this sense that the momentum map is “conserved”, namely, the integral
of the $(m-1)$-form $\Gamma_{t}^{*}J$ is independent of $t$.

Thus, we obtain Noether’s theorem in this setting:

######  Proposition 5.3.

The momentum map $J$ is conserved along solutions of the Palatini gravity
equations of motion.

###### Proof.

Recall that $\Gamma\colon U\subset M\rightarrow W_{\rm PG}$ is a solution for
the Palatini gravity equations of motion if and only if

$\Gamma^{*}(Z\lrcorner{\rm d}\lambda_{\rm PG})=0$

for any $\tau_{1}^{\prime}\circ\tau_{\rm PG}$-vertical vector field $Z$. Then
for each $\xi\in\mathfrak{k}$ we have

${\rm d}(\Gamma^{*}J_{\xi})=\Gamma^{*}({\rm d}J_{\xi})=\Gamma^{*}(-\xi_{W_{\rm
PG}}\lrcorner{\rm d}\lambda_{\rm PG})=0,$

and therefore the momentum is conserved along solutions. ∎

Accordingly, we think of a “momentum” $\widehat{\mu}$ as an element
$\widehat{\mu}\in\Omega^{m-1}(W_{\rm PG},\mathfrak{gl}(m)^{*})$, i.e., as a
$\mathfrak{gl}(m)^{*}$-valued $(m-1)$-form on $W_{\rm PG}$; a conserved value
$\widehat{\mu}$ of the momentum map is a closed one, i.e., ${\rm
d}\widehat{\mu}=0$.

The construction of a momentum map for the action on $W_{\rm PG}$ is standard
[21]:

######  Lemma 5.4.

The map $J\colon W_{\rm PG}\to\Lambda^{m-1}(T^{*}W_{\rm
PG})\otimes\mathfrak{k}^{*}$ defined by

$\langle J(\alpha),\xi\rangle=\xi_{W_{\rm PG}}(\alpha)\lrcorner\lambda_{\rm
PG}|_{\alpha},$

for each $\xi\in\mathfrak{k}$, is an ${\rm Ad}^{*}$-equivariant momentum map
for the ${\rm GL}(m)$-action on $W_{\rm PG}$.

Now, because

$T\tau_{\rm PG}\circ\xi_{W_{\rm PG}}=\xi_{\mathcal{T}_{0}}\circ\tau_{\rm PG},$

then for every $\alpha\in W_{\rm PG}$ it results that

$\displaystyle\langle J(\alpha),\xi\rangle$ $\displaystyle=\xi_{W_{\rm
PG}}(\alpha)\lrcorner\lambda_{\rm
PG}|_{\alpha}=\xi_{\mathcal{T}_{0}}\lrcorner\alpha$
$\displaystyle=i_{0}^{*}\big{[}\xi_{J^{1}\tau}\lrcorner\big{(}\eta^{ip}\theta_{pk}\wedge\Omega^{k}_{i}+\eta^{ip}\beta_{pq}\wedge\omega^{q}_{i}\big{)}\big{]}=0$

for all $\xi\in\mathfrak{k}$. 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

$J^{-1}(0)=W_{\rm PG}.$

The other ingredient needed in Routh reduction is the factorization of the
metricity bundle $I^{m}_{\rm PG}$ induced by a connection $\omega_{K}$ on the
underlying bundle $p_{K}^{LM}\colon LM\rightarrow\Sigma$, where

$\tau_{\Sigma}\colon\ \Sigma:=LM/K\to M$

is the _bundle of metrics of signature $\eta$_. We will carry out this task in
Section 9; here we will construct this connection. To this end, consider the
decomposition associated to the matrix $\eta$ (see Section 3). The connection
$\omega_{K}$ on the bundle $p_{K}^{LM}\colon LM\rightarrow\Sigma$ is induced
by this decomposition, namely

$\omega_{K}:=\pi_{\mathfrak{k}}\circ\omega_{0},$

where $\pi_{\mathfrak{k}}\colon\mathfrak{gl}(m)\to\mathfrak{k}$ is the
canonical projector onto the $\mathfrak{k}$-factor in the Cartan decomposition
and $\omega_{0}$ is a connection form on the principal bundle $\tau\colon
LM\to M$. The $K$-invariance of the factor $\mathfrak{p}$,

${\rm Ad}_{A}\mathfrak{p}\subset\mathfrak{p}\qquad\forall\,A\in K$

ensures us that it has the expected properties of a connection.

###  5.2 Reduced bundle for Palatini gravity

We have singled out the symmetries of our formulation of Palatini gravity;
they are described by the Lie group $K$. On the other hand, Palatini gravity
can be formulated through the variational problem

$\big{(}\tau_{1}^{\prime}\circ\tau_{\rm PG}\colon W_{\rm PG}\to M,\lambda_{\rm
PG},0\big{)}.$

The relevant bundles in this triple fit in the following diagram

${W_{\rm
PG}}$${\mathcal{T}_{0}}$${M}$$\scriptstyle{\tau_{1}^{\prime}\circ\tau_{\rm
PG}}$$\scriptstyle{\tau_{\rm PG}}$$\scriptstyle{\tau_{1}^{\prime}}$

It should be stressed that the link between this diagram and the original
variational problem depends on the facts that $\mathcal{T}_{0}\subset
J^{1}\tau$ and $W_{\rm
PG}\subset\wedge^{m}\big{(}T^{*}\big{(}J^{1}\tau\big{)}\big{)}$: The first
inclusion allows us to interpret some fields as derivatives, while the second
one tells us that other degrees of freedom behave like (multi)momenta. These
correspondences could be lost when symmetries are singled out; by performing
the quotients, the corresponding diagram becomes

${W_{\rm PG}/K}$${\mathcal{T}_{0}/K}$${M}$

The immediate problem is to find a variational problem associated to this
diagram. As we said above, this is difficult to achieve, because neither
$\mathcal{T}_{0}/K$ is a subset of a jet bundle nor $W_{\rm PG}/K$ is a
subbundle of a space of forms. One of the objectives of Routh reduction is to
identify in these quotient spaces the degrees of freedom that can be described
in this way, and to deal with those that cannot be fitted in this
classification. In order to proceed with this identification, the connection
$\omega_{K}$ defined in Section 5.1 is used. Here we will carry out this
operation on the quotient space $\mathcal{T}_{0}/K$, leaving the discussion of
the splitting of $W_{\rm PG}/K$ for later (see Section 9). Now, using the
adjoint bundle $\tau_{\mathfrak{k}}\colon\widetilde{\mathfrak{k}}\to\Sigma$,
the following result holds.

######  Proposition 5.5.

The map

$\displaystyle\Upsilon_{\omega}\colon\ J^{1}\tau$
$\displaystyle\longrightarrow\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}{\rm
Lin}\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}\big{)},$
$\displaystyle j_{x}^{1}s$
$\displaystyle\longmapsto\big{(}s(x),j^{1}_{x}[s]_{K},[s(x),\omega_{K}\circ
T_{x}s]_{K}\big{)}.$

is a bundle isomorphism.

The inverse of $\Upsilon_{\omega}$ is given by

$\displaystyle\Upsilon_{\omega}^{-1}\big{(}e,j_{x}^{1}\overline{s},\big{[}e,\widehat{\xi}\big{]}_{K}\big{)}=\big{[}v_{x}\in
T_{x}M\xmapsto{\hskip
19.91684pt}(T_{x}\overline{s}(v_{x}))_{e}^{H}+\big{(}\widehat{\xi}(v_{x})\big{)}_{LM}(e)\big{]},$

where $(\cdot)^{H}_{e}$, $e\in LM,$ is the horizontal lift associated to
$\omega_{K}$.

The map $\Upsilon_{\omega}$ enjoys a useful property: under this
identification, the action of $K$ on $J^{1}\tau$ is simply

$g\cdot\big{(}e,j_{x}^{1}\overline{s},[e,\widehat{\xi}]_{K}\big{)}=\big{(}g\cdot
e,j_{x}^{1}\overline{s},[e,\widehat{\xi}]_{K}\big{)}.$

This is a direct consequence of the equivariance of the principal connection
$\omega_{K}$. As a result, we get the following corollary, which is well-known
in the literature on Lagrangian reduction [11, 12, 17].

######  Corollary 5.6.

There is an identification

$J^{1}\tau/K\simeq J^{1}\tau_{\Sigma}\times_{\Sigma}{\rm
Lin}\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}.$

######  Remark 5.7.

The choice of a connection on the bundle $p_{K}^{LM}$ allows us to establish a
relationship between the quotient space $J^{1}\tau/K$ and the jet bundle of
the metric bundle $J^{1}\tau_{\Sigma}$, the latter being the relevant bundle
in the Einstein–Hilbert approach to relativity, which will be studied in
detail in Section 7.

Motivated by these considerations, we are in position to split the quotient
bundle $J^{1}\tau/K$ into a jet bundle part and a set of complementary degrees
of freedom. It suggests the following definition.

######  Definition 5.8 (quotient bundle for Palatini gravity).

The bundle $J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}$ is the
_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.

###  5.3 Routh reduction scheme for Palatini gravity

Our aim is to interpret usual Einstein–Hilbert variational problem as a Routh
reduction of the Griffiths variational problem for Palatini gravity. As far as
I know, there is no formulation of this type of reduction that could be used
in dealing with a Griffiths variational problem, so it is necessary to
generalize the techniques employed in [6] for Routh reduction in field theory
to cover this case. The variational problems to be related by this procedure
are the Griffiths variational problem for Palatini gravity, described in
Definition 4.1, and a variational problem which has not been determined yet,
but whose underlying bundle would be related to the quotient bundle given by
Definition 5.8. Now, in Routh reduction the equivalence between the extremals
is restricted to those having a particular value of the momentum map;222In our
case this consideration is superfluous, as the momentum map assumes just one
value, but it is the way the method proceeds in the general case. hence, it
has some advantages to work in the unified formalism, where momentum level
sets have a straightforward meaning (this fact was first recognized in [18]).
Moreover, as the setting of the unified formalism is an affine subbundle of a
bundle of forms (see equation (1.4)), the proof of the equivalence between the
equations of motion is less involved. A partially filled diagram could clarify
these considerations:

${\left(W_{\rm PG},\lambda_{\rm PG},0\right)}$${\left(W_{\rm
EH}^{\prime},\lambda_{\rm EH}^{\prime},0\right)}$${(W_{\rm EH},\lambda_{\rm
EH},0)}$${(\mathcal{T}_{0},{\cal L}_{\rm
PG},\langle\omega_{\mathfrak{p}}\rangle)}$${\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)},\,\bigstar\,,\,\bigstar\,\big{)}}$${\big{(}J^{1}\tau_{\Sigma},\,\bigstar\,,\,\bigstar\,\big{)}}$$\scriptstyle{A}$$\scriptstyle{1}$$\scriptstyle{\leavevmode\hbox
to7.21pt{\vbox to7.21pt{\pgfpicture\makeatletter\hbox{\hskip
3.60283pt\lower-3.60283pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.22499pt}{-1.57889pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,+\,\leavevmode\hbox
to7.21pt{\vbox to7.21pt{\pgfpicture\makeatletter\hbox{\hskip
3.60283pt\lower-3.60283pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.40283pt}{0.0pt}\pgfsys@curveto{3.40283pt}{1.87935pt}{1.87935pt}{3.40283pt}{0.0pt}{3.40283pt}\pgfsys@curveto{-1.87935pt}{3.40283pt}{-3.40283pt}{1.87935pt}{-3.40283pt}{0.0pt}\pgfsys@curveto{-3.40283pt}{-1.87935pt}{-1.87935pt}{-3.40283pt}{0.0pt}{-3.40283pt}\pgfsys@curveto{1.87935pt}{-3.40283pt}{3.40283pt}{-1.87935pt}{3.40283pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.22499pt}{-1.57889pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{2}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$$\scriptstyle{B}$$\scriptstyle{2}$$\scriptstyle{C}$$\scriptstyle{3}$

Stars ($\bigstar$) refer to geometrical structures (Lagrangians forms and
differential constraints) not identified yet. Arrows $\leavevmode\hbox
to13.34pt{\vbox to13.34pt{\pgfpicture\makeatletter\hbox{\hskip
6.66801pt\lower-6.66801pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
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}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox
to13.04pt{\vbox to13.04pt{\pgfpicture\makeatletter\hbox{\hskip
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the equivalence given by the Lepage-equivalent construction detailed at the
end of Section 2; Theorems 11.1 and 11.12 below proved the equivalence
indicated by the composition of the horizontal arrows $\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$. In order to carry
out this operation, we will see in the next sections that the reduced
variational problem on $J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$ can be
further simplified to a variational problem on $J^{1}\tau_{\Sigma}$; this fact
is depicted in the diagram by arrow 3, and will be carried out in Section 7.

##  6 Local coordinates expressions

Here we will obtain some identities allowing us to write down the isomorphism
$\Upsilon_{\omega}^{-1}$ in local terms. In order to proceed, we fix a
coordinate chart on $M$, inducing coordinates $(x^{\mu},e_{k}^{\mu})$ on $LM$.
As usual, we will indicate with $(x^{\mu},e_{k}^{\mu},e^{\mu}_{k\sigma})$ the
coordinates induced on $J^{1}\tau$. As shown above, there exists a set of
coordinates $(x^{\mu},g^{\mu\nu},\Gamma^{\sigma}_{\mu\nu})$ on

$J^{1}\tau/K=\Sigma\times C(LM)$

and adapted to this decomposition, namely

$p_{K}^{LM}\big{(}x^{\mu},e_{k}^{\mu}\big{)}=\big{(}x^{\mu},\eta^{kl}e_{k}^{\mu}e^{\nu}_{l}\big{)}.$

In terms of these coordinates, we have

$p^{J^{1}\tau}_{K}\big{(}x^{\mu},e_{k}^{\mu},e^{\mu}_{k\sigma}\big{)}=\big{(}x^{i},\eta^{ij}e_{i}^{\mu}e_{j}^{\nu},-e^{k}_{\mu}e_{k\nu}^{\sigma}\big{)}.$

It means in particular that

$Tp_{K}^{LM}\left(\frac{\partial}{\partial
x^{\mu}}\right)=\frac{\partial}{\partial x^{\mu}}$

and

$Tp_{K}^{LM}\left(\frac{\partial}{\partial
e^{\mu}_{k}}\right)=\big{(}\eta^{kq}e^{\rho}_{q}\delta^{\sigma}_{\mu}+\eta^{kp}e^{\sigma}_{p}\delta_{\mu}^{\rho}\big{)}\frac{\partial}{\partial
g^{\sigma\rho}}.$ (6.1)

On the other hand, a principal connection on $LM$ can be written as

$\omega_{0}=-e^{l}_{\mu}\big{(}{\rm d}e^{\mu}_{k}-f^{\mu}_{k\sigma}{\rm
d}x^{\sigma}\big{)}E^{k}_{l},$

where $(f^{\mu}_{k\sigma})$ is a collection of local functions on $M$; its
Christoffel symbols will be

$\overline{\Gamma}^{\sigma}_{\rho\mu}=-e_{\rho}^{k}f_{k\mu}^{\sigma}.$

Given our definition of the connection $\omega_{K}$ on the $K$-bundle
$p_{K}^{LM}\colon LM\rightarrow\Sigma$, its components become

$[(\omega_{0})_{\mathfrak{k}}]_{k}^{l}=-\eta_{kp}\big{(}\eta^{pq}e_{\mu}^{l}-\eta^{lq}e^{p}_{\mu}\big{)}\big{(}{\rm
d}e_{q}^{\mu}-f^{\mu}_{q\sigma}{\rm d}x^{\sigma}\big{)}.$

Now we will find the horizontal lift defined by $\omega_{K}$ for vector fields
on $\Sigma$:

######  Proposition 6.1.

The horizontal lift of vector fields on $\Sigma$ associated to the connection
$\omega_{K}$ is locally given by

$\displaystyle\left(\frac{\partial}{\partial
x^{\mu}}\right)^{H}=\frac{\partial}{\partial
x^{\mu}}+\frac{1}{2}g_{\beta\rho}e^{\rho}_{k}\big{(}g^{\alpha\sigma}\overline{\Gamma}^{\beta}_{\alpha\mu}-g^{\alpha\beta}\overline{\Gamma}^{\sigma}_{\alpha\mu}\big{)}\frac{\partial}{\partial
e^{\sigma}_{k}},$ $\displaystyle\left(\frac{\partial}{\partial
g^{\mu\nu}}\right)^{H}=\frac{1}{4}g_{\rho\beta}e^{\beta}_{k}\big{(}\delta_{\mu}^{\alpha}\delta^{\rho}_{\nu}+\delta_{\nu}^{\alpha}\delta^{\rho}_{\mu}\big{)}\frac{\partial}{\partial
e^{\alpha}_{k}}.$

###### Proof.

See Appendix C. ∎

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

######  Corollary 6.2.

Let $(x^{\mu},g^{\mu\nu},g^{\mu\nu}_{\sigma})$ be the induced coordinates on
$J^{1}\tau_{\Sigma}$. Then

$\displaystyle\left(\frac{\partial}{\partial
x^{\sigma}}+g^{\mu\nu}_{\sigma}\frac{\partial}{\partial
g^{\mu\nu}}\right)^{H}=\frac{\partial}{\partial
x^{\sigma}}+\frac{1}{2}g_{\beta\rho}e^{\rho}_{k}\big{[}g^{\kappa\beta}_{\sigma}+\big{(}g^{\alpha\kappa}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\kappa}_{\alpha\sigma}\big{)}\big{]}\frac{\partial}{\partial
e^{\kappa}_{k}},$ (6.2)

where $\left(\cdot\right)^{H}$ is the horizontal lift in the $K$-principal
bundle $p_{K}^{LM}\colon LM\to\Sigma$ for the connection $\omega_{K}$.

###### Proof.

According to Proposition 6.1, we have that

$\displaystyle\left(\frac{\partial}{\partial
x^{\sigma}}+g^{\mu\nu}_{\sigma}\frac{\partial}{\partial
g^{\mu\nu}}\right)^{H}$ $\displaystyle\qquad{}=\frac{\partial}{\partial
x^{\sigma}}+\frac{1}{2}g_{\beta\rho}e^{\rho}_{k}\big{(}g^{\alpha\kappa}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\kappa}_{\alpha\sigma}\big{)}\frac{\partial}{\partial
e^{\kappa}_{k}}+\frac{1}{4}g^{\mu\nu}_{\sigma}g_{\rho\beta}e^{\rho}_{k}\big{(}\delta_{\mu}^{\alpha}\delta^{\beta}_{\nu}+\delta_{\nu}^{\alpha}\delta^{\beta}_{\mu}\big{)}\frac{\partial}{\partial
e^{\alpha}_{k}}$ $\displaystyle\qquad{}=\frac{\partial}{\partial
x^{\sigma}}+\frac{1}{2}g_{\beta\rho}e^{\rho}_{k}\big{[}g^{\kappa\beta}_{\sigma}+\big{(}g^{\alpha\kappa}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\kappa}_{\alpha\sigma}\big{)}\big{]}\frac{\partial}{\partial
e^{\kappa}_{k}},$

as required. ∎

Let us now introduce coordinates on the vector bundle
$\widetilde{\mathfrak{k}}$. In order to do this, let us suppose that
$\left(\phi=\left(x^{\mu}\right),U\right)$ is a coordinate chart on $M$; then
it is also a trivializing domain for the principal bundle $LM$, where

$t_{U}\colon\ \tau^{-1}(U)\to U\times{\rm GL}(m)\colon\
u=(X_{1},\dots,X_{m})\mapsto\big{(}x^{\mu}(\tau(u)),e_{k}^{\mu}(u)\big{)}$

if and only if

$X_{k}=e_{k}^{\mu}(u)\frac{\partial}{\partial x^{\mu}}.$

Hence, we can define the coordinate chart
$\big{(}\phi_{\mathfrak{k}},\tau_{\mathfrak{k}}^{-1}(U)\big{)}$ [9]. In order
to proceed, we use the correspondence between the space of sections of the
adjoint bundle $\Gamma\tau_{\mathfrak{k}}$ and the set of
$p_{K}^{LM}$-vertical $K$-invariant vector fields on $LM$.

Therefore, taking the base $\big{\\{}E_{\sigma}^{\rho}\big{\\}}$ on
$\mathfrak{gl}(m)$ such that

$\big{(}E_{\sigma}^{\rho}\big{)}_{\alpha}^{\beta}=\delta^{\beta}_{\sigma}\delta_{\alpha}^{\rho},$

we can define the set of ${\rm GL}(m)$-invariant $\tau$-vertical vector fields
$\widetilde{E}_{\sigma}^{\rho}$ whose flow
$\Phi^{\widetilde{E}_{\sigma}^{\rho}}_{t}\colon\tau^{-1}(U)\to\tau^{-1}(U)$ is
given by

$\Phi^{\widetilde{E}_{\sigma}^{\rho}}_{t}(u):=t_{U}^{-1}\big{(}\tau(u),\big{[}\exp{\big{(}tE_{\sigma}^{\rho}\big{)}}\big{]}^{\alpha}_{\beta}e_{i}^{\beta}(u)\big{)};$

it means that, locally, these vector fields are such that

$\displaystyle
T_{u}t_{U}\big{(}\widetilde{E}_{\sigma}^{\rho}(u)\big{)}=e^{\rho}_{i}\frac{\partial}{\partial
e_{i}^{\sigma}}.$ (6.3)

In the following, we will adopt the usual convention according to which the
map $Tt_{U}$ is not explicitly written, namely, where

$\frac{\partial}{\partial e_{i}^{\sigma}}\qquad\text{and}\qquad
Tt_{U}^{-1}\left(\frac{\partial}{\partial e_{i}^{\sigma}}\right)$

are identified. We can write down any $p_{K}^{LM}$-vertical $K$-invariant
vector field $Z$ on $LM$ as

$Z=A^{\rho}_{\sigma}\widetilde{E}^{\sigma}_{\rho};$

then, using equation (6.1), we obtain the following result.

######  Lemma 6.3.

The vector field on $\tau^{-1}(U)$ given by

$Z=A^{\rho}_{\sigma}\widetilde{E}^{\sigma}_{\rho}$

is $p_{K}^{LM}$-vertical if and only if

$g^{\sigma\alpha}A_{\alpha}^{\rho}+g^{\rho\alpha}A_{\alpha}^{\sigma}=0.$

###### Proof.

In fact, we have that

$\displaystyle 0$
$\displaystyle=T_{u}p_{K}^{LM}\big{(}A_{\sigma}^{\rho}\widetilde{E}_{\rho}^{\sigma}(u)\big{)}=A_{\sigma}^{\rho}T_{u}p_{K}^{LM}\big{(}\widetilde{E}_{\rho}^{\sigma}(u)\big{)}=A_{\sigma}^{\rho}e_{i}^{\sigma}(u)T_{u}p_{K}^{LM}\left(\frac{\partial}{\partial
e_{i}^{\rho}}\right)$
$\displaystyle=A_{\sigma}^{\rho}e_{i}^{\sigma}(u)\eta^{iq}\big{[}e^{\alpha}_{q}(u)\delta^{\beta}_{\rho}+e^{\beta}_{q}(u)\delta^{\alpha}_{\rho}\big{]}\frac{\partial}{\partial
g^{\alpha\beta}},$

and the identity follows. ∎

Therefore, we will have that

$\phi_{\mathfrak{k}}([u,B]_{K}):=\big{(}x^{\mu}(u),\eta^{kl}e_{k}^{\mu}(u)e_{l}^{\nu}(u),A_{\sigma}^{\rho}([u,B]_{K})\big{)}$

if and only if
$g^{\sigma\alpha}A_{\alpha}^{\rho}+g^{\rho\alpha}A_{\alpha}^{\sigma}=0$ and

$[u,B]_{K}=A_{\sigma}^{\rho}\widetilde{E}_{\rho}^{\sigma}(u).$

In order to relate the coordinates $A_{\sigma}^{\rho}$ with the element
$[u,B]_{K}$, we need to look closely to the identification between
$\Gamma\widetilde{\mathfrak{k}}$ and the set of $p_{K}^{LM}$-vertical
$K$-invariant vector fields on $LM$. It uses the correspondence

$V\tau\simeq LM\times\mathfrak{gl}(m)$

given by

$(u,B)\mapsto\frac{\vec{{\rm d}}}{{\rm d}t}\bigg{|}_{t=0}(u\cdot\exp{(-tB)}).$

In coordinates it reads

$\big{(}u=(X_{1},\dots,X_{m}),B=\big{(}B_{i}^{j}\big{)}\big{)}\mapsto-
B_{j}^{i}e_{i}^{\rho}\frac{\partial}{\partial e_{j}^{\rho}},$

and using equation (6.3) it becomes

$\big{(}u=(X_{1},\dots,X_{m}),B=\big{(}B_{i}^{j}\big{)}\big{)}\mapsto-
B_{j}^{i}e_{i}^{\rho}e^{j}_{\sigma}\widetilde{E}^{\sigma}_{\rho}.$

Therefore, it results that

$\widehat{A}_{\rho}^{\sigma}(u,B)=-e_{\rho}^{i}B_{i}^{j}e_{j}^{\sigma}$

is a ${\rm GL}(m)$-invariant function on $LM\times\mathfrak{gl}(m)$ when ${\rm
GL}(m)$ acts on $\mathfrak{gl}(m)$ by the adjoint action; then, it gives us
the set of functions $A_{\rho}^{\sigma}$ on
$\tau_{\mathfrak{k}}^{-1}(U)\subset\widetilde{\mathfrak{k}}$ that completes
the coordinates $\phi_{\mathfrak{k}}$.

######  Lemma 6.4.

The map $\phi_{\mathfrak{k}}\colon\tau_{\mathfrak{k}}^{-1}(U)\to
U\times\mathbb{R}^{m^{2}+\frac{m(m+1)}{2}}$ given by

$\phi_{\mathfrak{k}}([u,B]_{K})=\big{(}x^{\mu}(u),\eta^{kl}e_{k}^{\mu}(u)e_{l}^{\nu}(u),-e_{\rho}^{i}(u)B_{i}^{j}e_{j}^{\sigma}(u)\big{)}$

defines a set of coordinates on $\tau_{\mathfrak{k}}^{-1}(U)$.

###### Proof.

According to the previous discussion, it is only necessary to prove that for
any $B\in\mathfrak{k}$, i.e., such that

$\eta^{ik}B_{k}^{j}+\eta^{jk}B_{k}^{i}=0,$

the corresponding element on $T_{u}LM$,

$Z=-e_{\rho}^{i}(u)B_{i}^{j}e_{j}^{\sigma}(u)\widetilde{E}^{\rho}_{\sigma}$

verifies the constraint

$T_{u}p_{K}^{LM}(Z)=0.$

But it follows that

$\displaystyle
g^{\rho\alpha}A_{\alpha}^{\sigma}+g^{\sigma\alpha}A_{\alpha}^{\rho}$
$\displaystyle=-g^{\rho\alpha}e_{\alpha}^{i}B_{i}^{j}e_{j}^{\sigma}-g^{\sigma\alpha}e_{\alpha}^{i}B_{i}^{j}e_{j}^{\rho}=-\eta^{ik}B_{i}^{j}\big{(}e_{k}^{\sigma}e_{j}^{\rho}+e_{k}^{\rho}e_{j}^{\sigma}\big{)}$
$\displaystyle=-\big{(}\eta^{ik}B_{k}^{j}+\eta^{jk}B_{k}^{i}\big{)}e_{j}^{\rho}e_{k}^{\sigma}=0,$

as required. ∎

Then, let us point out that Lemma 6.4 allows us to set coordinates on the
bundle

$\overline{p}\colon\ \mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\to\Sigma.$

In fact, any element $\left(g_{x},\alpha\right)\in\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$ admits
coordinates $\big{(}x^{\mu},g^{\mu\nu},A_{\sigma\rho}^{\mu}\big{)}$ if and
only if $\big{(}x^{\mu},g^{\mu\nu}\big{)}$ are the corresponding coordinates
for $g_{x}\in\Sigma$ and

$\alpha\left(\frac{\partial}{\partial
x^{\rho}}\right)=A_{\sigma\rho}^{\mu}\widetilde{E}^{\sigma}_{\mu}(e_{x}),$

where $e_{x}\in LM$ is any element in $\big{(}p_{K}^{LM}\big{)}^{-1}(g_{x})$.

##  7 Torsion, metricity and contact structures
on the quotient space

There are two tasks to carry out in order to understand the Routh reduction of
Palatini gravity: We need to characterize the effects produced by the fact
that we are working on the subbundle $\mathcal{T}_{0}\subset J^{1}\tau$
instead of the full jet space $J^{1}\tau$; additionally, we want to find the
differential constraints for the reduced system. Accordingly, in this section
we will prove that

  * •

When restricted to $\mathcal{T}_{0}$, the quotient map

$J^{1}\tau\to J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$

will reduce to

$\mathcal{T}_{0}\to J^{1}\tau_{\Sigma},$

that is, we can forget about the “vertical” degrees of freedom related to the
factor $\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$. This is
achieved in Propositions 7.1 and 7.2, and in Corollary 7.3.

  * •

The metricity conditions are the pullback along the quotient map of the
contact forms on $J^{1}\tau_{\Sigma}$. This result is very interesting because
it tells us that the reduction scheme implemented relates a Griffiths
variational problem (Palatini gravity, see Definition 4.1) with a classical
variational problem (Einstein–Hilbert gravity as described in Section 8
below). This is accomplished in Proposition 7.4.

So, let us use the following diagram

${J^{1}\tau}$${\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\big{)}}$${J^{1}\tau/K}$${J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}}$${\Sigma\times
C(LM)}$$\scriptstyle{\Upsilon_{\omega}}$$\scriptstyle{p_{K}^{J^{1}\tau}}$$\scriptstyle{{\rm
pr}_{23}}$$\scriptstyle{\overline{\Upsilon}_{\omega}}$$\scriptstyle{\sim}$$\scriptstyle{G_{\omega}}$

in order to define the diffeomorphism $G_{\omega}$; here
$\overline{\Upsilon}_{\omega}$ is the map induced by $\Upsilon_{\omega}$.
Therefore, let us construct the pullback bundle ${\rm
pr}_{1}\colon\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\big{)}\to
LM$ by means of the commutative diagram

${\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\big{)}}$${J^{1}\tau_{\Sigma}}$${LM}$${\Sigma}$$\scriptstyle{{\rm
pr}_{2}}$$\scriptstyle{{\rm
pr}_{1}}$$\scriptstyle{(\tau_{\Sigma})_{10}}$$\scriptstyle{p_{K}^{LM}}$

In this setting, we can prove that the zero torsion submanifold
$\mathcal{T}_{0}$ has some nice properties regarding the decomposition induced
by the connection $\omega_{K}$.

######  Proposition 7.1.

The canonical projection

$\displaystyle{\rm pr}_{\Sigma}\colon\ \big{(}p_{K}^{LM}\big{)}^{*}$
$\displaystyle\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}{\rm
Lin}\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}\big{)}\longrightarrow\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\big{)}$
$\displaystyle\big{(}e,j_{x}^{1}\overline{s},\big{[}e,\widehat{\xi}\big{]}_{K}\big{)}\xmapsto{\hskip
62.59596pt}\big{(}e,j_{x}^{1}\overline{s}\big{)}$

restricted to the submanifold

$\mathcal{T}_{0}^{\prime}:=\Upsilon_{\omega}(\mathcal{T}_{0})\subset\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}{\rm
Lin}\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}\big{)}$

is a diffeomorphism between $\mathcal{T}_{0}^{\prime}$ and
$\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\big{)}$.

###### Proof.

The proof of this proposition will be local. Using equation (6.2) and the
coordinates introduced above, we have that

$\displaystyle\frac{\partial}{\partial
x^{\sigma}}+e^{\mu}_{k\sigma}\frac{\partial}{\partial e^{\mu}_{k}}$
$\displaystyle=\left(\frac{\partial}{\partial
x^{\sigma}}+g^{\mu\nu}_{\sigma}\frac{\partial}{\partial
g^{\mu\nu}}\right)^{H}+A_{\rho\sigma}^{\mu}\widetilde{E}^{\rho}_{\mu}(e_{x})$
$\displaystyle=\frac{\partial}{\partial
x^{\sigma}}+\frac{1}{2}g_{\beta\rho}e^{\rho}_{k}\big{[}g^{\mu\beta}_{\sigma}+\big{(}g^{\alpha\mu}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\mu}_{\alpha\sigma}\big{)}\big{]}\frac{\partial}{\partial
e^{\mu}_{k}}-A_{\rho\sigma}^{\mu}e^{\rho}_{k}\frac{\partial}{\partial
e^{\mu}_{k}},$

namely

$e^{\mu}_{k\sigma}=\frac{1}{2}g_{\beta\rho}e^{\rho}_{k}\big{[}g^{\mu\beta}_{\sigma}+\big{(}g^{\alpha\mu}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\mu}_{\alpha\sigma}\big{)}\big{]}-A_{\rho\sigma}^{\mu}e^{\rho}_{k}.$

Then it follows that, for the $K$-invariant functions
$\Gamma^{\mu}_{\nu\sigma}$,

$\displaystyle\Gamma_{\rho\sigma}^{\mu}=-e^{k}_{\rho}e^{\mu}_{k\sigma}=-\frac{1}{2}g_{\beta\rho}\big{[}g^{\mu\beta}_{\sigma}+\big{(}g^{\alpha\mu}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\mu}_{\alpha\sigma}\big{)}\big{]}+A_{\rho\sigma}^{\mu}.$
(7.1)

It means that the set $\mathcal{T}_{0}^{\prime}$ is locally given by the
equation

$\displaystyle\frac{1}{2}g_{\beta\sigma}\big{[}g^{\mu\beta}_{\rho}+\big{(}g^{\alpha\mu}\overline{\Gamma}^{\beta}_{\alpha\rho}-g^{\alpha\beta}\overline{\Gamma}^{\mu}_{\alpha\rho}\big{)}\big{]}-\frac{1}{2}g_{\beta\rho}\big{[}g^{\mu\beta}_{\sigma}+\big{(}g^{\alpha\mu}\overline{\Gamma}^{\beta}_{\alpha\sigma}-g^{\alpha\beta}\overline{\Gamma}^{\mu}_{\alpha\sigma}\big{)}\big{]}+A_{\rho\sigma}^{\mu}-A_{\sigma\rho}^{\mu}=0.$

Let us define the set of quantities

$A_{\mu\nu\sigma}:=g_{\mu\rho}A^{\rho}_{\nu\sigma};$

then using this equation and the fact that

$A_{\mu\nu\sigma}+A_{\nu\mu\sigma}=g_{\mu\rho}A^{\rho}_{\nu\sigma}+g_{\nu\rho}A^{\rho}_{\mu\sigma}=0,$

we can conclude, from Proposition A.1, that the elements $A^{\mu}_{\nu\sigma}$
are uniquely determined by the fact that they belong to
$\mathcal{T}_{0}^{\prime}$. In other words, the set

$({\rm
pr}_{\Sigma})^{-1}\big{(}e,j_{x}^{1}\overline{s}\big{)}\cap\mathcal{T}_{0}^{\prime}$

consists in a single element. ∎

Proposition 7.1 can be geometrically interpreted: Recall that, through
isomorphism

$G_{\omega}\colon\ J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\to\Sigma\times
C(LM),$

any section of the reduced bundle
$J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$ can be seen
as a pair “metric” plus “connection”. With this interpretation in mind, the
previous proposition tells us that, when projected to the quotient, the
“connection part” of the section corresponds to Levi-Civita connection, and
so, it is uniquely determined by its “metric part”. The following result
summarizes it.

######  Proposition 7.2.

Let

$\sigma\colon\ M\rightarrow J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$

be a section of the composite map

$J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}{\longrightarrow}\Sigma\stackrel{{\scriptstyle\tau_{\Sigma}}}{{\longrightarrow}}M$

such that ${\rm pr}_{1}\circ\sigma\colon M\to J^{1}\tau_{\Sigma}$ is a
holonomic section and

$\mathop{{\rm Im}}{\sigma}\subset{\rm pr}_{23}(\mathcal{T}_{0}^{\prime}).$

Then

$\Gamma_{\sigma}:={\rm pr}_{2}\circ G_{\omega}\circ\sigma\colon\ M\rightarrow
C_{0}(LM)$

is the Levi-Civita connection associated to the metric $g_{\sigma}:={\rm
pr}_{1}\circ G_{\omega}\circ\sigma$.

###### Proof.

Locally, the map $G_{\omega}$ is given by equation (7.1). Therefore, from the
proof of the previous Proposition and using Proposition A.1, we will have that
the elements

$\Gamma_{\mu\nu\sigma}:=g_{\mu\rho}\Gamma^{\rho}_{\nu\sigma}$

are uniquely determined by the set of equations

$\displaystyle\Gamma_{\mu\rho\sigma}-\Gamma_{\rho\mu\sigma}=0,\qquad\Gamma_{\mu\rho\sigma}+\Gamma_{\rho\mu\sigma}=-g_{\mu\alpha}g_{\rho\beta}g^{\alpha\beta}_{\sigma}.$
(7.2)

It means that

$\Gamma_{\mu\rho\sigma}=-\frac{1}{2}\big{(}g_{\mu\alpha}g_{\rho\beta}g^{\alpha\beta}_{\sigma}+g_{\rho\alpha}g_{\sigma\beta}g^{\alpha\beta}_{\mu}-g_{\sigma\alpha}g_{\mu\beta}g^{\alpha\beta}_{\mu}\big{)}.$

Now, using the definition

$g_{\mu\nu,\sigma}:=-g_{\mu\alpha}g_{\nu\beta}g^{\alpha\beta}_{\sigma}$

we obtain

$\displaystyle\Gamma_{\rho\sigma}^{\mu}=\frac{1}{2}g^{\mu\alpha}\big{(}g_{\alpha\nu,\sigma}+g_{\rho\sigma,\alpha}-g_{\sigma\alpha,\nu}\big{)}.$
(7.3)

Because ${\rm pr}_{1}\circ\sigma$ is holonomic, we have that

$g^{\mu\nu}_{\sigma}=\frac{\partial g^{\mu\nu}}{\partial x^{\sigma}},$

as required. ∎

Let us define

$\mathcal{T}_{0}^{\prime\prime}:=G_{\omega}^{-1}(\Sigma\times
C_{0}(LM))=\overline{\Upsilon}_{\omega}\big{(}p_{K}^{J^{1}\tau}(\mathcal{T}_{0})\big{)};$

then, we need to draw our attention to the diagram in Fig. 1.

${\mathcal{T}_{0}}$${\mathcal{T}_{0}^{\prime}}$${\mathcal{T}_{0}/K}$${J^{1}\tau}$${\big{(}p_{k}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\big{)}}$${\Sigma\times
C_{0}(LM)}$${\big{(}p_{k}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\big{)}}$${J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}}$${\mathcal{T}_{0}^{\prime\prime}}$${J^{1}\tau_{\Sigma}}$$\scriptstyle{\Upsilon_{\omega}|_{\mathcal{T}_{0}}}$$\scriptstyle{p_{K}^{J^{1}\tau}\big{|}_{\mathcal{T}_{0}}}$$\scriptstyle{{\rm
pr}_{\Sigma}|_{\mathcal{T}_{0}^{\prime}}}$$\scriptstyle{\sim}$$\scriptstyle{\overline{\Upsilon}_{\omega}|_{\mathcal{T}_{0}/K}}$$\scriptstyle{\Upsilon_{\omega}}$$\scriptstyle{\overline{\Upsilon}_{\omega}\circ
p_{K}^{J^{1}\tau}}$$\scriptstyle{{\rm pr}_{23}}$$\scriptstyle{{\rm
pr}_{\Sigma}}$$\scriptstyle{G_{\omega}^{-1}\big{|}_{\Sigma\times
C_{0}(LM)}}$$\scriptstyle{{\rm pr}_{2}}$$\scriptstyle{{\rm
pr}_{1}}$$\scriptstyle{{\rm pr}_{1}|_{\mathcal{T}_{0}^{\prime\prime}}}$

Figure 1: Maps involved in the Routh reduction of Palatini gravity.

As a consequence of formula (7.3), we obtain the following corollary; in
short, it says that in the reduced bundle
$J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$, the
degrees of freedom associated to the factor $\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$ are
superfluous.

######  Corollary 7.3.

The map

${\rm pr}_{1}|_{\mathcal{T}_{0}^{\prime\prime}}\colon\
\mathcal{T}_{0}^{\prime\prime}\to J^{1}\tau_{\Sigma}$

is a bundle isomorphism over the identity on $\Sigma$.

###### Proof.

Locally, composite map $\overline{\Upsilon}_{\omega}\circ p_{K}^{J^{1}\tau}$
is given by

$\overline{\Upsilon}_{\omega}\circ
p_{K}^{J^{1}\tau}\big{(}\big{[}x^{\mu},e_{k}^{\nu},e^{\sigma}_{k\rho}\big{]}_{K}\big{)}=\big{(}x^{\mu},\eta^{ij}e_{i}^{\mu}e_{j}^{\nu},g^{\mu\nu}_{\sigma}\big{)},$

where coordinates $g_{\sigma}^{\mu\nu}$ are calculated using equation (7.2). ∎

For the last result of the section, we need any of the composite maps

${J^{1}\tau}$${\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\big{)}}$${\big{(}p_{K}^{LM}\big{)}^{*}\big{(}J^{1}\tau_{\Sigma}\big{)}}$${J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}}$${J^{1}\tau_{\Sigma}}$$\scriptstyle{\Upsilon_{\omega}}$$\scriptstyle{\overline{\Upsilon}_{\omega}\circ
p^{J^{1}\tau}_{K}}$$\scriptstyle{{\rm pr}_{\Sigma}}$$\scriptstyle{{\rm
pr}_{2}}$$\scriptstyle{{\rm pr}_{1}}$

So far, we have obtained a result allowing us to reduce the bundle
$J^{1}\tau_{\Sigma}\times_{\Sigma}\mathop{{\rm
Lin}}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$ further
down to $J^{1}\tau_{\Sigma}$; thus, we are halfway to connect the reduced
variational problem defined on this bundle with the Einstein–Hilbert
variational problem. Now, as we mentioned above, the splitting induced by the
connection form $\omega_{K}$ allows us to relate the metricity forms with a
contact structure on the quotient bundle; it will make possible to complete
this connection by showing that the contact structure on $J^{1}\tau_{\Sigma}$
is a sort of reduction structure for metricity conditions.

######  Proposition 7.4.

The metricity forms are $({\rm pr}_{2}\circ{\rm
pr}_{\Sigma}\circ\Upsilon_{\omega})$-horizontal $($also $\big{(}{\rm
pr}_{1}\circ\overline{\Upsilon}_{\omega}\circ
p^{J^{1}\tau}_{K}\big{)}$-horizontal$)$. In fact,

$Tp_{K}^{LM}\circ\omega_{\mathfrak{p}}=({\rm pr}_{2}\circ{\rm
pr}_{\Sigma}\circ\Upsilon_{\omega})^{*}\overline{\omega}=\big{(}{\rm
pr}_{1}\circ\overline{\Upsilon}_{\omega}\circ
p^{J^{1}\tau}_{K}\big{)}^{*}\overline{\omega}$

where $\overline{\omega}$ is the contact form on $J^{1}\tau_{\Sigma}$.

###### Proof.

In local coordinates, we have that

$({\rm pr}_{2}\circ{\rm
pr}_{\Sigma}\circ\Upsilon_{\omega})\big{(}x^{\mu},e^{\nu}_{k},e^{\nu}_{k\sigma}\big{)}=\big{(}x^{\mu},g^{\mu\nu},g^{\mu\nu}_{\sigma}\big{)},$

where $g^{\mu\nu}_{\sigma}$ is calculated using equation (7.1). On the other
hand, the metricity forms have the following local expression [3]

$\eta^{ik}\omega_{k}^{j}+\eta^{jk}\omega_{k}^{i}=e^{i}_{\mu}e^{j}_{\nu}\big{[}{\rm
d}g^{\mu\nu}+\big{(}g^{\mu\sigma}\Gamma_{\sigma\rho}^{\nu}+g^{\nu\sigma}\Gamma_{\sigma\rho}^{\mu}\big{)}{\rm
d}x^{\rho}\big{]}.$ (7.4)

Using equation (7.1), it follows that

$\eta^{ik}\omega_{k}^{j}+\eta^{jk}\omega_{k}^{i}=e^{i}_{\mu}e^{j}_{\nu}\big{(}{\rm
d}g^{\mu\nu}-g^{\mu\nu}_{\sigma}{\rm d}x^{\sigma}\big{)},$

namely, the metricity condition is horizontal with respect to the projection

${\rm pr}_{2}\circ{\rm pr}_{\Sigma}\circ\Upsilon_{\omega}\colon\
J^{1}\tau\longrightarrow J^{1}\tau_{\Sigma},$

and the form in the base manifold is nothing, but the generator of the contact
structure. ∎

##  8 First order variational problem for Einstein–Hilbert gravity

Einstein–Hilbert variational problem is a classical second order variational
problem on the bundle of metrics $\tau_{\Sigma}\colon\Sigma\to M$ of signature
$\eta$ on $M$ – see [10, 19] and references therein. It means that its
dynamics is dictated by a Lagrangian density

${\cal L}_{\rm EH}\colon J^{2}\tau_{\Sigma}\to\wedge^{m}(T^{*}M),$

given essentially by the scalar curvature of the Levi-Civita connection
associated to a metric. In terms of our nomenclature regarding variational
problems, this variational problem is prescribed by the triple

$\big{(}(\tau_{\Sigma})_{2}\colon J^{2}\tau_{\Sigma}\to M,{\cal L}_{\rm
EH},\mathcal{I}_{{\rm con}}^{\Sigma}\big{)},$

with $\mathcal{I}_{{\rm con}}^{\Sigma}$ referring to the exterior differential
system on $J^{2}\tau_{\Sigma}$ associated to its contact structure.

The main objective of this section is to provide a first order variational
problem for Einstein–Hilbert gravity. Namely, we have proved in Section 7 that
the quotient bundle given by Definition 5.8 can be further reduced to
$J^{1}\tau_{\Sigma}$; also, it was shown (see Proposition 7.4) that contact
structure on this jet space is related to metricity conditions via reduction
map. Therefore, we will construct a variational problem on this bundle
$J^{1}\tau_{\Sigma}$ and we will prove that this variational problem describes
Einstein–Hilbert gravity. Later on, we will prove that this variational
problem can be interpreted as Routh reduction of the variational problem for
Palatini gravity as defined in Section 4.1.

######  Definition 8.1 (Einstein–Hilbert Lagrangian form).

The _Einstein–Hilbert Lagrangian form_ is the unique $2$-horizontal $m$-form
${\cal L}_{\rm EH}^{(1)}$ on $J^{1}\tau_{\Sigma}$ such that

$\big{(}{\rm pr}_{1}\circ\overline{\Upsilon}_{\omega}\circ
p_{K}^{J^{1}\tau}\big{)}^{*}{\cal L}_{\rm EH}^{(1)}=i_{0}^{*}{\cal L}_{\rm
PG}.$

Recall also that in local terms, Palatini Lagrangian (3.3) can be written as

$\displaystyle{{\cal L}_{\rm
PG}}=\epsilon_{\mu_{1}\cdots\mu_{n-2}\gamma\kappa}\sqrt{|\det{g}|}g^{\kappa\phi}{\rm
d}x^{\mu_{1}}\wedge\\!\cdots\\!\wedge{\rm
d}x^{\mu_{n-2}}\\!\wedge\\!\big{(}{\rm d}\Gamma^{\gamma}_{\rho\phi}\wedge{\rm
d}x^{\rho}+\Gamma^{\sigma}_{\delta\phi}\Gamma^{\gamma}_{\beta\sigma}{\rm
d}x^{\beta}\wedge{\rm d}x^{\delta}\big{)}.\\!\\!\\!$ (8.1)

Using equation (7.3) we see that ${\cal L}_{\rm EH}^{(1)}$ has the same form
than ${\cal L}_{\rm PG}$, but replacing $\Gamma_{\mu\nu}^{\sigma}$ by their
expressions in terms of the jet variables $g_{\sigma}^{\mu\nu}$. Nevertheless,
it is not yet Einstein–Hilbert Lagrangian density because is it neither a
density nor a functional on $J^{2}\tau_{\Sigma}$.

We are pursuing here to establish the equivalence between the classical
variational problem associated to the Lagrangian density ${\cal L}_{\rm
EH}\colon J^{2}{\tau}_{\Sigma}\rightarrow\wedge^{m}\left(T^{*}M\right)$ and
the variational problem $\big{(}J^{1}{\tau}_{\Sigma},{\cal L}_{\rm
EH}^{(1)},\mathcal{I}^{\Sigma}_{{\rm con}}\big{)}$. 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
$J^{1}{\tau}_{\Sigma}$, whereas in the former case, the Lagrangian form on
$J^{2}{\tau}_{\Sigma}$ is specified through a Lagrangian density, giving rise
to a horizontal form on this jet bundle. The following lemma tells us how
these Lagrangians are related. In order to formulate this result, the
definition of the _horizontalization operator_
$h\colon\Omega^{m}\left(J^{1}{\tau}_{\Sigma}\right)\rightarrow\Omega^{m}\big{(}J^{2}{\tau}_{\Sigma}\big{)}$
should be kept in mind [30]; in fact, defining the map

$h_{j_{x}^{2}s}:=T_{x}j^{1}s\circ T_{j^{2}_{x}s}({\tau}_{\Sigma})_{2}\colon\
T_{j^{2}_{x}s}\big{(}J^{2}\tau_{\Sigma}\big{)}\to
T_{j^{1}_{x}s}\big{(}J^{1}\tau_{\Sigma}\big{)},$

where $(\tau_{\Sigma})_{2}\colon J^{2}\tau_{\Sigma}\to M$ is the canonical
projection of the $2$-jet bundle of the metric bundle onto $M$, we have

$h(\alpha)\big{|}_{j_{x}^{2}s}:=\alpha\big{|}_{j_{x}^{1}s}\circ
h_{j_{x}^{2}s}$

for every $\alpha\in\Omega^{m}\big{(}J^{1}\tau_{\Sigma}\big{)}$ and
$j_{x}^{2}s\in J^{2}\tau_{\Sigma}$.

######  Lemma 8.2.

It results that

${\cal L}_{\rm EH}=h\big{(}{\cal L}_{\rm EH}^{(1)}\big{)}$

for
$h\colon\Omega^{m}\big{(}J^{1}{\tau}_{\Sigma}\big{)}\rightarrow\Omega^{m}\left(J^{2}{\tau}_{\Sigma}\right)$
the horizontalization operator.

###### Proof.

In terms of the coordinates
$\big{(}x^{\mu},g^{\mu\nu},g^{\mu\nu}_{\alpha},g^{\mu\nu}_{\alpha\beta}\big{)}$
on $J^{2}{\tau}_{\Sigma}$, we have that

$h\big{(}{\rm d}g^{\mu\nu}\big{)}=g^{\mu\nu}_{\alpha}{\rm d}x^{\alpha},\qquad
h\big{(}{\rm d}g^{\mu\nu}_{\alpha}\big{)}=g^{\mu\nu}_{\alpha\beta}{\rm
d}x^{\beta}.$

The result follows from a (rather lenghty) calculation, using expression (8.1)
and the formula for the Christoffel symbols (7.3). ∎

The occurrence of the horizontalization operator in this lemma is crucial for
our purposes, as the following proposition shows.

######  Theorem 8.3.

Let $\pi\colon E\rightarrow M$ be a bundle on a $($compact$)$ manifold $M$ of
dimension $m$. For any $\alpha\in\Omega^{m}\big{(}J^{k}\pi\big{)}$ and any
section $s\in\Gamma\pi$, we have that

$\int_{M}\big{(}j^{k}s\big{)}^{*}\alpha=\int_{M}\big{(}j^{k+1}s\big{)}^{*}h(\alpha).$

###### Proof.

It follows from the formula

$T_{x}j^{k}s=T_{x}j^{k}s\circ T_{j^{k+1}_{x}s}\pi_{k+1}\circ T_{x}j^{k+1}s,$

that holds for every $x\in M$and $s\in\Gamma\pi$. ∎

It is immediate to prove the desired equivalence.

######  Corollary 8.4.

The classical variational problem specified by the Lagrangian density ${\cal
L}_{\rm EH}$ on $J^{2}{\tau}_{\Sigma}$ and the variational problem
$\big{(}J^{1}{\tau}_{\Sigma},{\cal L}_{\rm
EH}^{(1)},\mathcal{I}^{\Sigma}_{{\rm con}}\big{)}$ have the same set of
extremals.

###### Proof.

From Lemma 8.2 and using Theorem 8.3, we see that $g\colon M\rightarrow\Sigma$
is an extremal for the action integral

$g\mapsto\int_{M}\big{(}j^{2}g\big{)}^{*}{\cal L}_{\rm EH}$

if and only if it is an extremal for the action integral

$g\mapsto\int_{M}\big{(}j^{1}g\big{)}^{*}{\cal L}_{\rm EH}^{(1)},$

as required. ∎

As usual [20], the equations of motion of this variational problem can be
lifted to a space of forms on $J^{1}\tau_{\Sigma}$. Let us define the affine
subbundle

$W_{\rm EH}:={\cal L}_{\rm EH}^{(1)}+{I}^{\Sigma}_{{\rm
con},2}\subset\wedge^{m}\big{(}J^{1}\tau_{\Sigma}\big{)}.$

Here, for every $j_{x}^{1}s\in J^{1}\tau_{\Sigma}$,

$\displaystyle I^{\Sigma}_{{\rm
con},2}\big{|}_{j_{x}^{1}s}=\mathcal{L}\big{\\{}{\alpha}_{s(x)}\circ\big{(}T_{j_{x}^{1}s}(\tau_{\Sigma})_{10}-T_{x}s\circ
T_{j_{x}^{1}s}(\tau_{\Sigma})_{1}\big{)}\wedge\beta\colon$
$\displaystyle\hphantom{I^{\Sigma}_{{\rm
con},2}\big{|}_{j_{x}^{1}s}=\mathcal{L}\big{\\{}}{}{\alpha}_{s(x)}\in
T_{s(x)}^{*}\Sigma,\beta\in\big{(}\Lambda^{m-1}_{1}\big{(}J^{1}\tau_{\Sigma}\big{)}\big{)}_{j_{x}^{1}s}\big{\\}}$

is the corresponding fiber for the contact subbundle on $J^{1}\tau_{\Sigma}$.
The canonical map will be denoted by

$\tau_{\rm EH}\colon\ W_{\rm EH}\to J^{1}\tau_{\Sigma}.$

We will indicate with $\lambda_{\rm EH}$ the pullback of the canonical
$m$-form on $\wedge^{m}\big{(}J^{1}\tau_{\Sigma}\big{)}$ to $W_{\rm EH}$. Then
we have a result analogous to Proposition 4.5 in the context of (first order)
Einstein–Hilbert formulation.

######  Proposition 8.5.

A section $s\colon U\subset M\rightarrow J^{1}\tau_{\Sigma}$ is a critical
holonomic section for the variational problem
$\big{(}J^{1}{\tau}_{\Sigma},{\cal L}_{\rm
EH}^{(1)},\mathcal{I}^{\Sigma}_{{\rm con}}\big{)}$ if and only if there exists
a section $\Gamma\colon U\subset M\rightarrow W_{\rm EH}$ such that

  1. $1)$

$\Gamma$ covers $s$, i.e., $\tau_{\rm EH}\circ\Gamma=s$, and

  2. $2)$

$\Gamma^{*}(X\lrcorner{\rm d}\lambda_{\rm EH})=0$, for all
$X\in\mathfrak{X}^{V((\tau_{\Sigma}\ )_{1}\circ\tau_{\rm EH})}(W_{\rm EH})$.

######  Remark 8.6.

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 [19].

##  9 Contact bundle decomposition for Palatini gravity

We return here to the discussion initiated in Section 5.2, regarding the
splitting of the quotient $W_{\rm PG}/K$; it will be shown that the connection
$\omega_{K}$ is useful also for the identification of elements in $W_{\rm
PG}/K$ that can be seen as elements of an space of forms. Intuitively, it
means that the associated degrees of freedom can be interpreted as
multimomenta.

In order to perform this identification, we will recall some general facts
regarding the decomposition induced for the connection $\omega_{K}$ [6] on the
bundle of forms $W_{\rm PG}$ defined in equation (4.1). The contact structure
on $J^{1}\tau$ gives rise to the contact subbundle on $\mathcal{T}_{0}$ given
by

$\displaystyle I^{m}_{{\rm
con},2}\big{|}_{j_{x}^{1}s}=\mathcal{L}\big{\\{}{\alpha}_{s(x)}\circ(T_{j_{x}^{1}s}\tau_{10}^{\prime}-T_{x}s\circ
T_{j_{x}^{1}s}\tau_{1}^{\prime})\wedge\beta\colon$
$\displaystyle\hphantom{I^{m}_{{\rm
con},2}\big{|}_{j_{x}^{1}s}=\mathcal{L}\big{\\{}}{}{\alpha}_{s(x)}\in
T_{s(x)}^{*}(LM),\beta\in\big{(}\Lambda^{m-1}_{1}(\mathcal{T}_{0})\big{)}_{j_{x}^{1}s}\big{\\}},$
(9.1)

where $\mathcal{L}$ indicates linear closure. There is an splitting of
$I^{m}_{{\rm con},2}$ induced by the choice of a connection on the principal
bundle $p^{LM}_{K}\colon LM\to\Sigma$. Its construction proceeds as follows.
We denote by $\omega_{K}\in\Omega^{1}(LM,\mathfrak{k})$ the chosen connection
and consider the following splitting of the cotangent bundle:

$T^{*}(LM)=\big{(}p^{LM}_{K}\big{)}^{*}(T^{*}\Sigma)\oplus(LM\times\mathfrak{k}^{*}).$

The identification is obtained as follows:

$\displaystyle\big{(}p^{LM}_{K}\big{)}^{*}(T^{*}\Sigma)\oplus(LM\times\mathfrak{k}^{*})$
$\displaystyle\to T^{*}(LM),$ $\displaystyle(e,\widehat{\alpha}_{[u]},\sigma)$
$\displaystyle\mapsto\alpha_{u}=\widehat{\alpha}_{[u]}\circ
T_{u}p_{K}^{LM}+\langle\sigma,\omega_{K}(\cdot)\rangle.$

Accordingly, we have an splitting of contact bundle (9.1)

$\displaystyle I^{m}_{{\rm con},2}=\widetilde{I^{m}_{{\rm con},2}}\oplus
I^{m}_{\mathfrak{k}^{*},2},$

with

$\displaystyle\widetilde{I^{m}_{{\rm
con},2}}\big{|}_{j_{x}^{1}s}=\mathcal{L}\big{\\{}\widehat{\alpha}_{[s(x)]}\circ
T_{s(x)}p_{K}^{LM}\circ\big{(}T_{j_{x}^{1}s}\tau_{10}^{\prime}-T_{x}s\circ
T_{j_{x}^{1}s}\tau_{1}^{\prime}\big{)}\wedge\beta\colon$
$\displaystyle\hphantom{\widetilde{I^{m}_{{\rm
con},2}}\big{|}_{j_{x}^{1}s}=\mathcal{L}\big{\\{}}{}\widehat{\alpha}_{[s(x)]}\in
T_{[s(x)]}^{*}\Sigma,\beta\in\big{(}\Lambda^{m-1}_{1}\mathcal{T}_{0}\big{)}_{j_{x}^{1}s}\big{\\}},$
$\displaystyle
I_{\mathfrak{k}^{*},2}^{m}\big{|}_{j_{x}^{1}s}=\big{\\{}\big{\langle}\sigma\stackrel{{\scriptstyle\wedge}}{{,}}\omega_{K}\circ\big{(}T_{j_{x}^{1}s}\tau_{10}^{\prime}-T_{x}s\circ
T_{j_{x}^{1}s}\tau_{1}^{\prime}\big{)}\big{\rangle}\colon\sigma\in\big{(}\Lambda^{m-1}_{1}\mathcal{T}_{0}\otimes\mathfrak{k}^{*}\big{)}_{j_{x}^{1}s}\big{\\}}.$

The symbol $\langle\cdot\stackrel{{\scriptstyle\wedge}}{{,}}\cdot\rangle$
denotes the natural contraction, defined as follows: For elements of the form
$\alpha_{1}\otimes\nu$, $\alpha_{2}\otimes\eta$ with $\nu,\eta\in\mathfrak{k}$
and $\alpha_{1},\alpha_{2}$ forms, we have
$\langle\alpha_{1}\otimes\nu\stackrel{{\scriptstyle\wedge}}{{,}}\alpha_{2}\otimes\eta\rangle=\langle\nu,\eta\rangle\alpha_{1}\wedge\alpha_{2}$.
For a general element in the linear closure, the definition extends linearly.

We can split our metricity subbundle $I^{m}_{\rm PG}$ using the inclusion

$I^{m}_{\rm PG}\subset I^{m}_{{\rm con},2},$

namely

$I^{m}_{\rm PG}=\big{(}I^{m}_{\rm PG}\cap\widetilde{I^{m}_{{\rm
con},2}}\big{)}\oplus\big{(}I^{m}_{\rm PG}\cap
I^{m}_{\mathfrak{k}^{*},2}\big{)}.$

But we have the following fact.

######  Lemma 9.1.

For every $j_{x}^{1}s\in\mathcal{T}_{0}$

$I^{m}_{\rm PG}\subset\widetilde{I^{m}_{{\rm con},2}}.$

###### Proof.

Let us work in the coordinates considered above; therefore, we have equation
(6.1) for the projector $Tp_{K}^{LM}$ and also

$T_{j_{x}^{1}s}\tau_{10}^{\prime}-T_{x}s\circ
T_{j_{x}^{1}s}\tau_{1}^{\prime}=\frac{\partial}{\partial
e^{\mu}_{k}}\otimes\big{(}{\rm d}e^{\mu}_{k}-e^{\mu}_{k\alpha}{\rm
d}x^{\alpha}\big{)}.$

Then

$\displaystyle
T_{s(x)}p_{K}^{LM}\circ\big{(}T_{j_{x}^{1}s}\tau_{10}^{\prime}-T_{x}s\circ
T_{j_{x}^{1}s}\tau_{1}^{\prime}\big{)}=T_{s(x)}p_{K}^{LM}\left(\frac{\partial}{\partial
e^{\mu}_{k}}\right)\otimes\big{(}{\rm d}e^{\mu}_{k}-e^{\mu}_{k\alpha}{\rm
d}x^{\alpha}\big{)}$ $\displaystyle\qquad{}=\frac{\partial}{\partial
g^{\rho\sigma}}\otimes\big{[}\eta^{kq}\big{(}e_{q}^{\rho}{\rm
d}e_{k}^{\sigma}+e_{q}^{\sigma}{\rm
d}e_{k}^{\rho}\big{)}-\eta^{kq}\big{(}e_{q}^{\rho}e_{k\alpha}^{\sigma}+e_{q}^{\sigma}e_{k\alpha}^{\rho}\big{)}{\rm
d}x^{\alpha}\big{]}$ $\displaystyle\qquad{}=\frac{\partial}{\partial
g^{\rho\sigma}}\otimes\big{[}{\rm
d}g^{\rho\sigma}-\big{(}g^{\rho\beta}e_{\beta}^{k}e^{\sigma}_{k\alpha}+g^{\sigma\beta}e_{\beta}^{k}e^{\rho}_{k\alpha}\big{)}{\rm
d}x^{\alpha}\big{]}$ $\displaystyle\qquad{}=\frac{\partial}{\partial
g^{\rho\sigma}}\otimes\big{[}{\rm
d}g^{\rho\sigma}+\big{(}g^{\rho\beta}\Gamma^{\sigma}_{\beta\alpha}+g^{\sigma\beta}\Gamma^{\rho}_{\beta\alpha}\big{)}{\rm
d}x^{\alpha}\big{]}$

that will define to a set of generators of the bundle $I^{m}_{\rm PG}$ (see
equation (7.4)). ∎

This result is compatible with the fact that the whole subbundle $W_{\rm PG}$
is in the zero level set for the momentum map. We will return to that below.

##  10 First order Einstein–Hilbert Lagrangian as Routhian

As we mentioned in the introductory sections, a crucial role in Routh
reduction is played by the _Routhian_ , which replaces the Lagrangian in
determining the dynamics of the mechanical system. This replacement is
unavoidable, because the dynamics for the Routh reduced problem should happen
on a level set for the momentum map of the theory

$J\colon\ TQ\rightarrow\mathfrak{g}^{*},$

and the constraints imposed by this requirement must be included in the
Lagrangian.

To fix ideas, let us briefly describe how this construction proceeds in the
case of a classical variational problem

$\big{(}{\rm pr}_{1}\colon\mathbb{R}\times TQ\to\mathbb{R},L{\rm
d}t,\langle{\rm d}q-\dot{q}{\rm d}t\rangle\big{)}$

for a mechanical system with configuration space $Q$ and a symmetry described
by a Lie group $G$ acting freely on $Q$. The idea is to incorporate the
constraints imposed by the momentum map into the Lagrangian through a family
of Lagrange multipliers; this prescription tells us that the Routhian becomes

$R_{\mu}:=L-\langle\mu,\omega_{Q}\rangle$

where $\mu\in\mathfrak{g}^{*}$ is the level chosen for the momentum map and
$\omega_{Q}$ is a connection form for the principal bundle $p_{G}^{Q}\colon
Q\to Q/G$. A similar construction can be done when working for classical
variational problems describing a first order field theory [6].

Now, let us try to particularize these considerations for the case of Palatini
gravity. The Routhian form is expected to coincide with the Lagrangian ${\cal
L}_{\rm PG}$ because $J\equiv 0$. Nevertheless, as we have stressed above,
this result would be valid if we were working within the range of [6]; on the
contrary, the variational problem describing Palatini gravity is not reached
by these results, and so this should be properly verified in this particular
case.

First, we write $\overline{p}\colon{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\to\Sigma$ for
the obvious projection. In principle, the bundle ${\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$ would be the
field bundle for the reduced system; nevertheless, we will show in Lemma 10.2
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
$\Sigma$.

In particular, one can consider the map:

$\displaystyle q\colon\ J^{1}\big{(}\tau_{\Sigma}\circ\overline{p}\big{)}$
$\displaystyle\longrightarrow J^{1}\tau_{\Sigma}\times{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}},$
$\displaystyle j_{x}^{1}\sigma$
$\displaystyle\longmapsto\big{(}j_{x}^{1}\big{(}\overline{p}\circ\sigma\big{)},\sigma(x)\big{)}$

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
$\tau_{\Sigma}\circ\overline{p}\colon{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\to M$ as the
basic field bundle. Nevertheless, there are some identifications that will
permit us to simplify this basic bundle further.

In order to proceed, let use the connection $\omega_{K}$ to define the maps
fitting in the following diagram:

${{\mathcal{T}_{0}}}$${{{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}}}$${{J^{1}\big{(}\tau_{\Sigma}\circ\overline{p}\big{)}}}$${{\mathcal{T}_{0}/K}}$${{J^{1}\tau_{\Sigma}\times{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}}}$$\scriptstyle{p_{K}^{J^{1}\tau}\big{|}_{\mathcal{T}_{0}}}$$\scriptstyle{f_{\omega}}$$\scriptstyle{\big{(}\tau_{\Sigma}\circ\overline{p}\big{)}_{10}}$$\scriptstyle{q}$$\scriptstyle{g_{\omega}:=\overline{\Upsilon}_{\omega}|_{\mathcal{T}_{0}/K}}$

The definitions are as follows:

$\displaystyle f_{\omega}\colon\ \mathcal{T}_{0}$
$\displaystyle\longrightarrow{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}},$
$\displaystyle j_{x}^{1}s$ $\displaystyle\longmapsto[s(x),\omega_{K}\circ
T_{x}s]_{K},$ $\displaystyle g_{\omega}\colon\ \mathcal{T}_{0}/{K}$
$\displaystyle\longrightarrow J^{1}\tau_{\Sigma}\times{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}},$
$\displaystyle\big{[}j_{x}^{1}s\big{]}_{K}$
$\displaystyle\longmapsto\big{(}j_{x}^{1}\big{(}p_{K}^{LM}\circ
s\big{)},[s(x),\omega_{K}\circ T_{x}s]_{K}\big{)}.$

The map $g_{\omega}$ is the identification from Corollary 5.6. Since the
Lagrangian form ${\cal L}_{\rm PG}$ is basic for the projection
$p_{\mathcal{T}_{0}}^{K}\colon\mathcal{T}_{0}\to\mathcal{T}_{0}/K$, it defines
a reduced form on $\mathcal{T}_{0}/K$ which can be seen as a form on
$J^{1}\tau_{\Sigma}\times\mathop{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}$. We will
denote it by $\overline{{\cal L}}_{\rm PG}$:

$\big{(}g_{\omega}\circ p_{K}^{J^{1}\tau}\big{)}^{*}\overline{{\cal L}}_{\rm
PG}={\cal L}_{\rm PG},\qquad\overline{{\cal L}}_{\rm
PG}\in\Omega_{2}^{m}\big{(}J^{1}\tau_{\Sigma}\times\mathop{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\big{)}.$

######  Definition 10.1.

The $m$-form $\overline{{\cal L}}_{\rm
PG}\in\Omega^{m}\big{(}J^{1}\tau_{\Sigma}\times\mathop{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\big{)}$ is
the _Routhian_ for the variational problem $\big{(}\mathcal{T}_{0},{\cal
L}_{\rm PG},\mathcal{I}_{\rm PG}^{m}\big{)}$.

Then, we are ready to prove a characteristic property for the Routhian
associated to the reduction of Palatini gravity.

######  Lemma 10.2.

The Routhian $\overline{{\cal L}}_{\rm PG}$ is ${\rm pr}_{1}$-horizontal,
where

${\rm pr}_{1}\colon\ J^{1}\tau_{\Sigma}\times\mathop{\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\longrightarrow
J^{1}\tau_{\Sigma}$

is the projection onto the first factor of the fibred product.

###### Proof.

It follows from equations (7.1) and (7.3) that

${\rm pr}_{1}^{*}{\cal L}_{\rm EH}^{(1)}=\overline{{\cal L}}_{\rm PG},$ (10.1)

as required. ∎

In short, Routhian $\overline{{\cal L}}_{\rm PG}$ does not depend on the fiber
coordinates $A^{\sigma}_{\mu\rho}$ of the bundle

$\overline{p}\colon\ {\rm
Lin}{\big{(}\tau_{\Sigma}^{*}TM,\widetilde{\mathfrak{k}}\big{)}}\to\Sigma;$