# A variational principle for Kaluza–Klein type theories

Frédéric Hélein  IMJ-PRG, UMR CNRS 7586 & Université Denis Diderot, UFR de
Mathématiques, Bâtiment Sophie Germain, 8 place Aurélie Nemours, 75013 Paris,
France<EMAIL_ADDRESS>
(Date: October 25, 2018)

###### Abstract.

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a
definite symmetric bilinear form on $\mathbb{R}^{n}$ and an Ad-invariant
scalar product on the Lie algebra of $\mathfrak{G}$, we construct a
variational problem on fields defined on an arbitrary oriented
$(n+\hbox{dim}\mathfrak{G})$-dimensional manifold $\mathcal{Y}$. We show that,
if $\mathfrak{G}$ is compact and simply connected, any global solution of the
Euler–Lagrange equations leads, through a spontaneous symmetry breaking, to
identify $\mathcal{Y}$ with the total space of a principal bundle over an
$n$-dimensional manifold $\mathcal{X}$. Moreover $\mathcal{X}$ is then endowed
with a (pseudo-)Riemannian metric and a connection which are solutions of the
Einstein–Yang–Mills system of equations with a cosmological constant.

###### 2010 Mathematics Subject Classification:

Primary 49S05, 53C80, 53C05, 83E15

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations
    2. 1.2 The main result
      1. 1.2.1 About Hypothesis (vi)
      2. 1.2.2 Remark
    3. 1.3 Outline of the proof
  2. 2 Notations and description of the obtained equations
    1. 2.1 Intrinsic setting
    2. 2.2 Introducing a basis of $\hat{\mathfrak{g}}$ and using indices
    3. 2.3 Useful relations
    4. 2.4 More precisions on the proof
  3. 3 The Euler–Lagrange equations
    1. 3.1 Variations with respect to coefficients of $\pi$
    2. 3.2 Variations with respect to $\varphi$
    3. 3.3 Variations with respect to $\theta$
  4. 4 The fibration
  5. 5 Trivialization of the bundle
  6. 6 Computation of the connection and the curvature forms
  7. 7 The Einstein–Yang–Mills equations

##  1\. Introduction

###  1.1. Motivations

In 1919 T. Kaluza [10] (after an earlier attempt by G. Nordström [14] in 1914)
discovered that solutions of the Einstein equations of gravity in vacuum on a
5-dimensional manifold could modelize Einstein equations coupled with Maxwell
equations on a 4-dimensional space-time manifold, provided one assumes that
the 5-dimensional manifold is a circle fiber bundle over space-time and that
the metric is constant along these fibers. This was rediscovered more or less
independentely by O. Klein [12] in 1926 (and also by H. Mandel [13]), who
proposed to assume that the size of the extra fifth dimension is sufficientely
tiny in order to explain why this dimension is not directly observed. Since
then this fascinating observation has been an important source of inspiration
and questioning (see e.g. [5]). It has been extended to include non Abelian
gauge theories [4, 11, 2, 3], in order to unify the Einstein equations with
the Yang–Mills equations on a curved space-time and, in particular, it becomes
an important ingredient of the 11-dimensional supergravity and the
superstrings theories. It remains today a subject of questioning (see e.g. [1,
16]).

However some difficulties plag this beautiful idea:

The Kaluza–Klein ansatz is indeed based on the assumption that the metric is
covariantly constant along the fibers. But this raises the question of finding
physical reasons for that. Moreover the initial proposal by Kaluza and Klein
led to inconsistency. This point was raised by P. Jordan [9] and Y. Thiry
[15], who allowed the coefficient of the metric along the fifth dimension to
be an extra scalar field. However this scalar field is a source of
difficulties as to its physical interpretation.

A way to avoid the assumption that the metric is covariantly constant along
the fibers is, as proposed by Klein, to assume that the extra dimension is
tiny. Then by expanding the fields in harmonic modes on each fiber one finds
that, as a consequence of the Heisenberg uncertainty principle, all modes
excepted the zero one should be extremely massive. This would hence explain
why we cannot observe their quantum excitations. This idea is at the origin of
the current hypothesis.

But this does not answer the fundamental question of understanding why these
extra dimensions are fibered and compact (and tiny if we want to support the
above hypothesis or, alternatively, if the smallness assumption is not true,
why the metric is constant along the fiber): could a dynamical mechanism
explain these assumptions ?

In the following we address these questions and we present a variational
principle which satisfies the following properties: provided that the involved
structure Lie group is compact and simply connected, the Euler–Lagrange
equations satisfied by the critical points lead to a mechanism which forces a
spontaneous fibration of the higher dimensional manifold over an emerging
space-time, forces the metric to be covariantly constant along the fibers and
one can build out of these critical points a metric and a connection over the
space-time which are solutions of the Einstein–Yang–Mills system of equations.

Note that our results work partially for e.g. $U(1)$, for which our mechanism
fails to imply the compactness of the fibers without extra _ad hoc_
hypotheses. Hence either there is a need to improve our theory (for example by
taking into account semi-classical or quantum effects), or one may argue that
our results could be sufficient in an Grand Unified Theory, where all
structure gauge groups are supposed to arise from a single compact, simply
connected one, by a symmetry breaking.

###  1.2. The main result

To introduce our model let us first remind the higher dimension generalization
of the so-called _Palatini_ (see [6]) formulation of gravity: Let $N\geq 2$ be
an integer and $E$ be an oriented $N$-dimensional real vector space endowed
with a non degenerate bilinear form h (in most cases the Minkowski scalar
product) and let $so(E,\textsf{h})$ be the Lie algebra of the group of
isometries of $(E,\textsf{h})$. We identify $so(E,\textsf{h})$ with
$\Lambda^{2}E=E\wedge E$ (with a Lie bracket denoted by $[\cdot,\cdot]_{2}$,
see the next section for details). The $N$-dimensional generalization of the
Palatini action on an oriented $N$-dimensional manifold $\mathcal{Y}$ is a
functional defined on pairs $(\theta,\varphi)$ where $\theta$ is a (soldering)
1-form on $\mathcal{Y}$ with coefficient in $E$ and $\varphi$ is a
(connection) 1-form on $\mathcal{Y}$ with coefficient in
$so(E,\textsf{h})\simeq\Lambda^{2}E$. This functional reads

$\mathscr{A}_{P}[\theta,\varphi]=\int_{\mathcal{Y}}\star\theta^{(N-2)}\wedge\left(d\varphi+\frac{1}{2}[\varphi\wedge\varphi]_{2}\right),$

or
$\mathscr{A}_{P}[\theta,\varphi]=\int_{\mathcal{Y}}\star\theta^{(N-2)}\wedge\Phi$
by denoting $\Phi:=d\varphi+\frac{1}{2}[\varphi\wedge\varphi]_{2}$. Here
$\star\theta^{(N-2)}$ is the $(N-2)$-form with coefficient in
$so(E,\textsf{h})^{*}\simeq\Lambda^{2}E^{*}$, with components
$\theta^{(N-2)}_{A_{1}A_{2}}=\frac{1}{(N-2)!}\epsilon_{A_{1}\cdots
A_{N}}\theta^{A_{3}}\wedge\cdots\wedge\theta^{A_{N}}$, where
$\epsilon_{A_{1}\cdots A_{N}}$ is the completely antisymmetric tensor such
that $\epsilon_{1\cdots N}=1$ and, in the product
$\star\theta^{(N-2)}\wedge\Phi$, the duality pairing between
$\Lambda^{2}E^{*}$ and $\Lambda^{2}E$ is implicitely assumed so that
$\star\theta^{(N-2)}\wedge\Phi=\frac{1}{2}\theta^{(N-2)}_{AB}\wedge\Phi^{AB}$
(see the next section for more details).

Then, as it is well-known, the critical points $(\theta,\varphi)$ of
$\mathscr{A}_{P}$ such that the rank of $\theta$ is equal to $N$ everywhere
correspond to solutions of the Einstein equations of gravity in vacuum (with a
metric $\theta^{*}\textsf{h}$ on $\mathcal{Y}$).

Our model can be seen as a deformation of the previous one: we assume that $E$
is itself endowed with a Lie bracket $[\cdot,\cdot]_{1}$ and we denote by
$\hat{\mathfrak{g}}:=(E,[\cdot,\cdot]_{1})$ the resulting Lie algebra. We
assume further that:

  1. (i)

$\hat{\mathfrak{g}}=\mathfrak{s}\oplus\mathfrak{g}$, where $\mathfrak{s}$ is
contained in the center of $\hat{\mathfrak{g}}$ and $\mathfrak{g}$ is a Lie
subalgebra;

  2. (ii)

the Lie bracket
$[\cdot,\cdot]_{1}:\hat{\mathfrak{g}}\times\hat{\mathfrak{g}}\longrightarrow\hat{\mathfrak{g}}$
preserves the metric h;

  3. (iii)

$\mathfrak{s}$ is orthogonal to $\mathfrak{g}$ for the bilinear form h.

We set $n=\hbox{dim}\mathfrak{s}$ and $r=\hbox{dim}\mathfrak{g}$ so that
$N=n+r$. Note that (i) implies that $\mathfrak{s}$ is a _trivial_ Lie
subalgebra and (ii) means that $\forall\xi\in\hat{\mathfrak{g}}$,
$\hbox{ad}_{\xi}\in so(\hat{\mathfrak{g}},\textsf{h})$. We consider the
following space of fields:

$\begin{array}[]{rlr}\mathscr{E}:=\\{\,(\theta,\varphi,\pi);&\theta\in\hat{\mathfrak{g}}\otimes\Omega^{1}(\mathcal{Y}),\;\varphi\in
so(\hat{\mathfrak{g}},\textsf{h})\otimes\Omega^{1}(\mathcal{Y}),&\\\
&\pi\in\hat{\mathfrak{g}}^{*}\otimes\Omega^{N-2}(\mathcal{Y})&\\}\end{array}$

and define on it the action functional $\mathscr{A}$ by:

(1)
$\mathscr{A}[\theta,\varphi,\pi]:=\int_{\mathcal{Y}}\pi\wedge\left(d\theta+\frac{1}{2}[\theta\wedge\theta]_{1}\right)+\star\theta^{(N-2)}\wedge\left(d\varphi+\frac{1}{2}[\varphi\wedge\varphi]_{2}\right)$

where the duality pairing between, respectively, $\hat{\mathfrak{g}}^{*}$ and
$\hat{\mathfrak{g}}$ and $so(\hat{\mathfrak{g}},\textsf{h})^{*}$ and
$so(\hat{\mathfrak{g}},\textsf{h})$ is implicitely used.

We decompose $\theta=\theta^{\mathfrak{s}}+\theta^{\mathfrak{g}}$ according to
the splitting $\hat{\mathfrak{g}}=\mathfrak{s}\oplus\mathfrak{g}$ and we
impose the constraint

(2) $\theta^{\mathfrak{s}}\wedge\theta^{\mathfrak{s}}\wedge\pi=0$

(see the next section for more details) leading hence us to define the
constrained subset:

$\mathscr{C}:=\\{(\theta,\varphi,\pi)\in\mathscr{E};\quad\theta^{\mathfrak{s}}\wedge\theta^{\mathfrak{s}}\wedge\pi=0\\}.$

######  Theorem 1.1.

Assume Hypotheses (i), (ii), (iii). Let $\mathcal{Y}$ be a connected, oriented
$N$-dimensional manifold. Let $(\theta,\varphi,\pi)\in\mathscr{C}$ be a smooth
critical point of the restriction of $\mathscr{A}$ on $\mathscr{C}$. Let
$\textbf{h}:=\theta^{*}\textsf{h}$, a pseudo Riemannian metric on
$\mathcal{Y}$. Assume that:

  1. (iv)

$\mathfrak{g}$ is the Lie algebra of a compact and simply connected Lie group
$\mathfrak{G}$;

  2. (v)

the rank of $\theta$ is equal to $N$ everywhere;

  3. (vi)

$\textbf{h}:=\theta^{*}\textsf{h}$ is _vertically complete_ (see §1.2.1).

Then

  1. (1)

the manifold $\mathcal{Y}$ is the total space of a principal bundle over an
$n$-dimensional manifold $\mathcal{X}$;

  2. (2)

the structure group of this bundle is a group $\mathfrak{G}_{0}$, the
universal cover of which is $\mathfrak{G}$;

  3. (3)

we can construct explicitely out of $\theta$ a pseudo Riemannian metric g and
a $\mathfrak{g}$-valued connection $\nabla$ on $\mathcal{X}$;

  4. (4)

g and $\nabla$ are solution of the Einstein–Yang–Mills system with
cosmological constant equal to $\Lambda=\frac{1}{8}(K,\textsf{h}^{*})$, where
$K$ is the Killing form on $\hat{\mathfrak{g}}$, $\textsf{h}^{*}$ is the
metric on ${\hat{\mathfrak{g}}}^{*}$ and $(\cdot,\cdot)$ is the natural
pairing between both tensors.

####  1.2.1. About Hypothesis (vi)

The pseudo Riemannian metric $\textbf{h}:=\theta^{*}\textsf{h}$ is _vertically
complete_ if, for any continuous map $v$ from $[0,1]$ to
$\mathfrak{g}\subset\hat{\mathfrak{g}}$ and, for any point
$\textsf{y}\in\mathcal{Y}$, there exists an unique $\mathscr{C}^{1}$ map
$\gamma:[0,1]\longrightarrow\mathcal{Y}$, which is a solution of
$(\gamma^{*}\theta)_{t}=v(t)dt$, $\forall t\in[0,1]$, with the initial
condition $\gamma(0)=\textsf{y}$. Such curves $\gamma$ can be interpreted _a
posteriori_ as being _vertical_ curves, i.e. contained in a fiber of the
principal bundle over a point in the space-time. This allows thus singular
space-times with black holes.

####  1.2.2. Remark

Our action may alternatively be written as follows. We endow the direct sum
$\hat{\mathfrak{g}}\oplus so(\hat{\mathfrak{g}},\textsf{h})$ with the product
Lie bracket $[\cdot,\cdot]$ of, respectively,
$(\hat{\mathfrak{g}},[\cdot,\cdot]_{1})$ and
$(so(\hat{\mathfrak{g}},\textsf{h}),[\cdot,\cdot]_{2})$. We consider the space
of fields

$\begin{array}[]{rll}\tilde{\mathscr{E}}:=\\{\,(\theta+\varphi,\pi+\psi);&\theta\in\hat{\mathfrak{g}}\otimes\Omega^{1}(\mathcal{Y}),&\varphi\in
so(\hat{\mathfrak{g}},\textsf{h})\otimes\Omega^{1}(\mathcal{Y}),\\\
&\pi\in\hat{\mathfrak{g}}^{*}\otimes\Omega^{N-2}(\mathcal{Y}),&\psi\in
so(\hat{\mathfrak{g}},\textsf{h})^{*}\otimes\Omega^{N-2}(\mathcal{Y})\\}\end{array}$

and we define

$\tilde{\mathscr{A}}[\theta+\varphi,\pi+\psi]:=\int_{\mathcal{Y}}(\pi+\psi)\wedge\left(d(\theta+\varphi)+\frac{1}{2}[\theta+\varphi\wedge\theta+\varphi]\right).$

We observe that, if the constraint

(3) $\psi=\star\theta^{(N-2)}$

is satisfied, then
$\tilde{\mathscr{A}}[\theta+\varphi,\pi+\psi]=\mathscr{A}[\theta,\varphi,\pi]$.
Hence the study of critical points of $\mathscr{A}$ on $\mathscr{C}$ is
equivalent to the study of critical points of $\tilde{\mathscr{A}}$ on:

$\tilde{\mathscr{C}}:=\\{(\theta+\varphi,\pi+\psi)\in\tilde{\mathscr{E}};\quad\psi=\star\theta^{(N-2)}\hbox{
and }\theta^{\mathfrak{s}}\wedge\theta^{\mathfrak{s}}\wedge\pi=0\\}.$

###  1.3. Outline of the proof

— The action $\mathscr{A}[\theta,\varphi,\pi]$ is the sum of the generalized
Palatini action
$\int_{\mathcal{Y}}\star\theta^{(N-2)}\wedge(d\varphi+\frac{1}{2}[\varphi\wedge\varphi]_{2})$
and of the extra term
$\int_{\mathcal{Y}}\pi\wedge(d\theta+\frac{1}{2}[\theta\wedge\theta]_{1})$.

In the latter term the coefficients of the $(N-2)$-form $\pi$ (constrained by
$\theta^{\mathfrak{s}}\wedge\theta^{\mathfrak{s}}\wedge\pi=0$) play the role
of Lagrange multipliers and, for a critical point, it forces
$d\theta+\frac{1}{2}[\theta\wedge\theta]_{1}$ to be a linear combination of
components of $\theta^{\mathfrak{s}}\wedge\theta^{\mathfrak{s}}$. One can thus
use repeatedly Frobenius theorem: first to the Pfaffian system
$\theta^{\mathfrak{s}}|_{\textsf{f}}=0$, where f is an $r$-dimensional
submanifold of $\mathcal{Y}$, to obtain a local foliation of $\mathcal{Y}$,
the leaves f that we show are actually the fibers of a fibration
$\mathcal{Y}\longrightarrow\mathcal{X}$ thanks to the hypotheses; second by
using the fact that
$d\theta^{\mathfrak{g}}+\frac{1}{2}[\theta^{\mathfrak{g}}\wedge\theta^{\mathfrak{g}}]_{1}$
is a linear combination of components of
$\theta^{\mathfrak{s}}\wedge\theta^{\mathfrak{s}}$ to deduce that the
geometric data associated with $\theta$ are covariantly constant along the
fibers.

On the other hand one uses the fact that the first variation of $\mathscr{A}$
with respect to $\varphi$ vanishes to show that the connection on
$T\mathcal{Y}$ associated to $\varphi$ and $\theta$ is the Levi-Civita
connection for the metric $\textbf{h}:=\theta^{*}\textsf{h}$. Note that this
step is the same as in the standard derivation of the Palatini Euler–Lagrange
equation since $\varphi$ is only present in the integral
$\int_{\mathcal{Y}}\star\theta^{(N-2)}\wedge(d\varphi+\frac{1}{2}[\varphi\wedge\varphi]_{2})$.

Lastly one exploits the fact that the first variation of $\mathscr{A}$ with
respect to $\theta$ vanishes. If the action would only be equal to
$\int_{\mathcal{Y}}\star\theta^{(N-2)}\wedge(d\varphi+\frac{1}{2}[\varphi\wedge\varphi]_{2})$
one would find that the metric h on $\mathcal{Y}$ is a solution of the
Einstein equation in vacuum and consequently the equivariance of the metric
along the fibers derived previously would then give us a solution of an
Einstein–Yang–Mills system of equations on $\mathcal{X}$. However the coupling
of $\theta$ with $\pi$ in the second term
$\int_{\mathcal{Y}}\pi\wedge(d\theta+\frac{1}{2}[\theta\wedge\theta]_{1})$
creates extra source terms in the Einstein–Yang–Mills system which contains an
_a priori_ high degree of arbitrariness and which could hence ruin our
efforts.

_A miraculous cancellation_ : however, apart from a cosmological constant, the
extra sources just cancel! This cancellation is due to the fact that each of
the source terms is covariantly constant along each fiber and hence is equal
to its average value on the fiber, which is compact. But it turns out that
this average value is proportional to the integral of an _exact_ $r$-form on
the fiber and hence vanishes. This phenomenon is similar to the one discovered
in [7] and [8].

##  2\. Notations and description of the obtained equations

###  2.1. Intrinsic setting

Since our action and the resulting Euler–Lagrange equations mix forms with
coefficients in $\hat{\mathfrak{g}}$, $so(\hat{\mathfrak{g}},\textsf{h})$ and
their dual spaces it will be convenient to identify
$so(\hat{\mathfrak{g}},\textsf{h})$ wih
$\hat{\mathfrak{g}}\wedge\hat{\mathfrak{g}}$ as follows.

For any finite dimensional real vector space $E$ and any $k\in\mathbb{N}$ we
let $E^{\otimes k}$ be the $k$-th tensorial power of $E$ and
$\Lambda^{k}E:=E\wedge\cdots\wedge E$ be the subspace of $E^{\otimes k}$ of
skewsymmetric tensors. If $v_{1},\cdots,v_{k}\in E$ we set

$v_{1}\wedge\cdots\wedge
v_{k}:=\sum_{\sigma\in\mathfrak{S}(k)}(-1)^{|\sigma|}v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(k)}\in\Lambda^{k}E$

and, for $p\in\mathbb{N}$ greater than or equal to $k$ and
$\lambda\in\Lambda^{p}E^{*}$, we define the _interior product_ 111Note that,
if we view $v_{1}\wedge\cdots\wedge v_{k}$ and $\lambda$ as elements of,
respectively, $E^{\otimes k}$ and $(E^{*})^{\otimes p}$, then
$v_{1}\wedge\cdots\wedge v_{k}\lrcorner\lambda$ is $1/k!$ times the
contraction of $v_{1}\wedge\cdots\wedge v_{k}$ with $\lambda$.
$v_{1}\wedge\cdots\wedge v_{k}\lrcorner\lambda$ to be the $(p-k)$-form in
$\Lambda^{p-k}E^{*}$ such that:
$\left(v_{1}\wedge\cdots\wedge
v_{k}\lrcorner\lambda\right)(w_{k+1},\cdots,w_{p})=\lambda(v_{1},\cdots,v_{k},w_{k+1},\cdots,w_{p})$,
$\forall,w_{k+1},\cdots,w_{p}\in E$.

To any $\xi\otimes\alpha$ in $E\otimes E^{*}$ we associate the linear map from
$E$ to itself defined by $[\eta\longmapsto\xi\alpha(\eta)]$. By extending
linearly this map, we get a linear isomorphism which allows us to identify
$E\otimes E^{*}$ with $End(E)$. If furthermore $E$ is endowed with a non
degenerate symmetric bilinear form h, it induces a vector space isomorphism
$\zeta\longmapsto\zeta\lrcorner\textsf{h}:=\textsf{h}(\zeta,\cdot)$ from $E$
to $E^{*}$. We hence get an unique linear map $\mathscr{L}:E\otimes
E\longrightarrow E\otimes E^{*}\simeq End(E)$ such that, for any
$\xi,\zeta,\eta\in E$,

$\mathscr{L}(\xi\otimes\zeta)=\xi\otimes(\zeta\lrcorner\textsf{h})\simeq[E\ni\eta\longmapsto\xi\textsf{h}(\zeta,\eta)\in
E].$

Then $\mathscr{L}$ is an isomorphism. We endow $E\otimes E$ with the unique
product law $\ast$ such that
$\mathscr{L}(\alpha\ast\beta)=\mathscr{L}(\alpha)\circ\mathscr{L}(\beta)$,
$\forall\alpha,\beta\in E\otimes E$. We also get a Lie algebra bracket
$[\cdot,\cdot]_{2}$ on $E\otimes E$ defined by
$[\alpha,\beta]_{2}=\alpha\ast\beta-\beta\ast\alpha$.

The subspace $\Lambda^{2}E=E\wedge E\subset E\otimes E$ is then a Lie
subalgebra of $(E\otimes E,[\cdot,\cdot]_{2})$ which coincides with the
inverse image by $\mathscr{L}$ of the Lie subalgebra $so(E,\textsf{h})$. This
allows us to identify $so(E,\textsf{h})$ with $\Lambda^{2}E$ endowed with the
bracket $[\cdot,\cdot]_{2}$.

###  2.2. Introducing a basis of $\hat{\mathfrak{g}}$ and using indices

We let $(\textbf{t}_{A})_{1\leq A\leq N}$ be a basis of $\hat{\mathfrak{g}}$
such that $(\textbf{t}_{a})_{1\leq a\leq n}$ is a basis of $\mathfrak{s}$ and
$(\textbf{t}_{\alpha})_{n+1\leq\alpha\leq N}$ is a basis of $\mathfrak{g}$. We
will systematically use the following conventions for the indices: $1\leq
A,B,C,\ldots\leq N$ and $1\leq a,b,c,\ldots\leq
n<\alpha,\beta,\gamma,\ldots\leq N$.

We denote by b the restriction of h to $\mathfrak{s}$ and k the restriction of
h to $\mathfrak{k}$ and we set
$\textsf{h}_{AB}:=\textsf{h}(\textbf{t}_{A},\textbf{t}_{B})$,
$\textsf{b}_{ab}:=\textsf{b}(\textbf{t}_{a},\textbf{t}_{b})$ and
$\textsf{k}_{\alpha\beta}:=\textsf{k}(\textbf{t}_{\alpha},\textbf{t}_{\beta})$,
so that Hypothesis (iii) translates as

$\left(\textsf{h}_{AB}\right)=\left(\begin{array}[]{cc}\textsf{h}_{ab}&\textsf{h}_{a\beta}\\\
\textsf{h}_{\alpha
b}&\textsf{h}_{\alpha\beta}\end{array}\right)=\left(\begin{array}[]{cc}\textsf{b}_{ab}&0\\\
0&\textsf{k}_{\alpha\beta}\end{array}\right).$

We denote by $c^{A}_{BC}$ the structure constants of $\hat{\mathfrak{g}}$ in
the basis $(\textbf{t}_{A})_{1\leq A\leq N}$, defined by
$[\textbf{t}_{B},\textbf{t}_{C}]_{1}=\textbf{t}_{A}c^{A}_{BC}$. We observe
that, due to Hypothesis (i),

$\left(c^{A}_{BC}\right)=\left(\begin{array}[]{ccc}c^{a}_{bc}&c^{a}_{b\gamma}&c^{a}_{\beta\gamma}\\\
c^{\alpha}_{bc}&c^{\alpha}_{b\gamma}&c^{\alpha}_{\beta\gamma}\end{array}\right)=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&c^{\alpha}_{\beta\gamma}\end{array}\right).$

For any $A,B=1,\cdots,N$, we let
$\textbf{t}_{AB}:=\textbf{t}_{A}\wedge\textbf{t}_{B}$. Then
$\left(\textbf{t}_{AB}\right)_{1\leq A<B\leq N}$ is a basis of
$\Lambda^{2}\hat{\mathfrak{g}}=\hat{\mathfrak{g}}\wedge\hat{\mathfrak{g}}$.
Hence using the isomorphism $\mathscr{L}$ defined previously to identify
$so(\hat{\mathfrak{g}},\textsf{h})$ with $\Lambda^{2}\hat{\mathfrak{g}}$ we
can view $\left(\textbf{t}_{AB}\right)_{1\leq A<B\leq N}$ as a basis of
$so(\hat{\mathfrak{g}},\textsf{h})$ as well. Through this identification we
have
$\textbf{t}_{AB}(\textbf{t}_{C})=\textbf{t}_{A}\textsf{h}_{BC}-\textbf{t}_{B}\textsf{h}_{AC}$.
Moreover

$[\textbf{t}_{A_{1}B_{1}},\textbf{t}_{A_{2}B_{2}}]_{2}=\textbf{t}_{A_{1}B_{2}}\textsf{h}_{B_{1}A_{2}}-\textbf{t}_{A_{1}A_{2}}\textsf{h}_{B_{1}B_{2}}-\textbf{t}_{B_{1}B_{2}}\textsf{h}_{A_{1}A_{2}}+\textbf{t}_{B_{1}A_{2}}\textsf{h}_{A_{1}B_{2}}.$

We denote by $(\textbf{t}^{A})_{1\leq A\leq N}$ the basis of
$\hat{\mathfrak{g}}^{*}$ which is dual to $(\textbf{t}_{A})_{1\leq A\leq N}$
and by $\left(\textbf{t}^{AB}\right)_{1\leq A<B\leq N}$ the basis of
$\Lambda^{2}\hat{\mathfrak{g}}^{*}\simeq(\Lambda^{2}\hat{\mathfrak{g}})^{*}$
which is dual to $\left(\textbf{t}_{AB}\right)_{1\leq A<B\leq N}$.

If $\Phi$ is a form with coefficients in $so(\hat{\mathfrak{g}},\textsf{h})$
with coordinates $\left(\Phi^{AB}\right)_{1\leq A<B\leq N}$ we set
$\Phi^{BA}:=-\Phi^{AB}$, for $A\geq B$, so that

$\Phi=\sum_{1\leq A<B\leq
N}\textbf{t}_{AB}\Phi^{AB}=\frac{1}{2}\sum_{A,B=1}^{N}\textbf{t}_{AB}\Phi^{AB}=\frac{1}{2}\textbf{t}_{AB}\Phi^{AB}$

and we will systematically use the last writing
$\Phi=\frac{1}{2}\textbf{t}_{AB}\Phi^{AB}$, where the summation over $1\leq
A,B\leq N$ is implicitely assumed. Similarly if $\psi$ is a
$\Lambda^{2}\hat{\mathfrak{g}}^{*}$-valued form, we will use the same
convention $\psi=\frac{1}{2}\psi_{AB}\textbf{t}^{AB}$ for its decomposition in
the basis $\left(\textbf{t}^{AB}\right)_{1\leq A<B\leq N}$. The duality
pairing between a $\Lambda^{2}\hat{\mathfrak{g}}$-valued form $\Phi$ and a
$\Lambda^{2}\hat{\mathfrak{g}}^{*}$-valued form $\psi$ then reads
$\psi\wedge\Phi=\frac{1}{2}\psi_{AB}\wedge\Phi^{AB}$.

Lastly we use $\textsf{h}_{AB}$ and $\textsf{h}^{AB}$ to rise and lower the
indices: $\varphi{{}^{A}}_{B}=\textsf{h}_{BB^{\prime}}\varphi^{AB^{\prime}}$,
$\varphi^{AB}=\varphi{{}^{A}}_{B^{\prime}}\textsf{h}^{B^{\prime}B}$, etc.

With these conventions, if
$\theta\in\hat{\mathfrak{g}}\otimes\Omega^{1}(\mathcal{Y})$ we write
$\theta=\textbf{t}_{A}\theta^{A}$ and
$[\theta\wedge\theta]_{1}=\textbf{t}_{A}[\theta\wedge\theta]^{A}_{1}$ with
$[\theta\wedge\theta]^{A}_{1}:=c^{A}_{BC}\theta^{B}\wedge\theta^{C}$ and hence

$d\theta^{A}+\frac{1}{2}[\theta\wedge\theta]^{A}_{1}=d\theta^{A}+\frac{1}{2}c^{A}_{BC}\theta^{B}\wedge\theta^{C}$

If $\varphi\in
so(\hat{\mathfrak{g}},\textsf{h})\otimes\Omega^{1}(\mathcal{Y})\simeq\Lambda^{2}\hat{\mathfrak{g}}\otimes\Omega^{1}(\mathcal{Y})$
we write $\varphi=\frac{1}{2}\textbf{t}_{AB}\varphi^{AB}$ and
$[\varphi\wedge\varphi]_{2}=\frac{1}{2}\textbf{t}_{AB}[\varphi\wedge\varphi]_{2}^{AB}$
with
$[\varphi\wedge\varphi]_{2}^{AB}:=2\textsf{h}_{A^{\prime}B^{\prime}}\varphi^{AA^{\prime}}\wedge\varphi^{B^{\prime}B}=2\varphi{{}^{A}}_{A^{\prime}}\wedge\varphi^{A^{\prime}B}$
and hence

$d\varphi^{AB}+\frac{1}{2}[\varphi\wedge\varphi]_{2}^{AB}=d\varphi^{AB}+\varphi{{}^{A}}_{A^{\prime}}\wedge\varphi^{A^{\prime}B}.$

Constraint (2) then reads $\theta^{a}\wedge\theta^{b}\wedge\pi=0$, $\forall
a,b=1,\cdots,n$.

###  2.3. Useful relations

Assume that the rank of
$\theta\in\hat{\mathfrak{g}}\otimes\Omega^{1}(\mathcal{Y})$ is equal to $N$
everywhere and decompose $\theta=\textbf{t}_{A}\theta^{A}$. Then
$(\theta^{1},\cdots,\theta^{N})$ is a coframe on $\mathcal{Y}$. We denote by
$(\frac{\partial}{\partial\theta^{1}},\cdots,\frac{\partial}{\partial\theta^{N}})$
its dual frame. We define recursively

(4)
$\theta^{(N-1)}_{A}:=\frac{\partial}{\partial\theta^{A}}\lrcorner\theta^{(N)},\quad\theta^{(N-2)}_{AB}:=\frac{\partial}{\partial\theta^{B}}\lrcorner\theta^{(N-1)}_{A},\quad\theta^{(N-3)}_{ABC}:=\frac{\partial}{\partial\theta^{C}}\lrcorner\theta^{(N-2)}_{AB}.$

Using the fact that $\theta^{(N)}=\frac{1}{N!}\epsilon_{A_{1}\cdots
A_{N}}\theta^{A_{1}}\wedge\cdots\wedge\theta^{A_{N}}$ one may prove that

(5) $\theta^{(N-1)}_{A}=\frac{1}{(N-1)!}\epsilon_{AA_{2}\cdots
A_{N}}\theta^{A_{2}}\wedge\cdots\wedge\theta^{A_{N}},$ (6)
$\theta^{(N-2)}_{AB}=\frac{1}{(N-2)!}\epsilon_{ABA_{3}\cdots
A_{N}}\theta^{A_{3}}\wedge\cdots\wedge\theta^{A_{N}},$ (7)
$\theta^{(N-3)}_{ABC}=\frac{1}{(N-3)!}\epsilon_{ABCA_{4}\cdots
A_{N}}\theta^{A_{4}}\wedge\cdots\wedge\theta^{A_{N}},\hbox{ etc.}$

Moreover we have the following

(8)
$\theta^{A}\wedge\theta^{(N-1)}_{A^{\prime}}=\delta^{A}_{A^{\prime}}\theta^{(N)},$
(9)
$\theta^{A}\wedge\theta^{(N-2)}_{A^{\prime}B^{\prime}}=\delta^{A}_{B^{\prime}}\theta^{(N-1)}_{A^{\prime}}-\delta^{A}_{A^{\prime}}\theta^{(N-1)}_{B^{\prime}}$

and

(10)
$\theta^{A}\wedge\theta^{(N-3)}_{A^{\prime}B^{\prime}C^{\prime}}=\delta^{A}_{C^{\prime}}\theta^{(N-2)}_{A^{\prime}B^{\prime}}+\delta^{A}_{B^{\prime}}\theta^{(N-2)}_{C^{\prime}A^{\prime}}+\delta^{A}_{A^{\prime}}\theta^{(N-2)}_{B^{\prime}C^{\prime}}.$

Indeed (8) can be proved by developping the relation
$0=\frac{\partial}{\partial\theta^{A^{\prime}}}\lrcorner
0=\frac{\partial}{\partial\theta^{A^{\prime}}}\lrcorner(\theta^{A}\wedge\theta^{(N)})$.
Computing the interior product by
$\frac{\partial}{\partial\theta^{B^{\prime}}}$ to both sides of (8) leads to
(9) and computing the interior product by
$\frac{\partial}{\partial\theta^{C^{\prime}}}$ to both sides of (9) leads to
(10).

Lastly we have the following formulas

(11) $d\theta^{(N-1)}_{A}=d\theta^{B}\wedge\theta^{(N-2)}_{AB},$ (12)
$d\theta^{(N-2)}_{AB}=d\theta^{C}\wedge\theta^{(N-3)}_{ABC},$

which can be proved, e.g., by using (5), (6) and (7).

As an application, assuming that the rank of
$\theta\in\hat{\mathfrak{g}}\otimes\Omega^{1}(\mathcal{Y})$ is equal to $N$,
we have

$\star\theta^{(N-2)}=\frac{1}{2}\theta_{AB}^{(N-2)}\textbf{t}^{AB}.$

(Thus Condition (3) reads $\psi=\frac{1}{2}\theta^{(N-2)}_{AB}\textbf{t}^{AB}$
or, equivalentely,
$\theta^{A}\wedge\theta^{B}\wedge\psi=\textbf{t}^{AB}\theta^{(N)}$, $\forall
A,B=1,\cdots,N$.)

###  2.4. More precisions on the proof

In the proof of the Theorem, once we prove the existence of a fibration of
$\mathcal{Y}$ over $\mathcal{X}$ and once a local trivialization of this
bundle has been chosen (characterized by a projection map from $\mathcal{Y}$
to $\mathcal{X}$ and a map $g$ from an open subset of $\mathcal{Y}$ to
$\mathfrak{G}_{0}$), one can write that $\theta^{a}=e^{a}$ and
$\theta^{\alpha}=(gAg^{-1}+g^{-1}dg)^{\alpha}$, where $e^{a}$ and
$A=\textbf{t}_{\alpha}A^{\alpha}$ are pull-back forms of 1-forms on
$\mathcal{X}$. Then a metric g on $\mathcal{X}$ is defined by
$\textbf{g}=(\theta^{\mathfrak{s}})^{*}\textsf{b}=\textsf{b}_{ab}e^{a}e^{b}$
and $A$ is the expression of the connection $\nabla$ in the trivialization. We
then set $F:=dA+\frac{1}{2}[A\wedge A]$, the curvature 2-form of $A$. The
variation with respect to $\theta$ leads to the equation

$\frac{1}{2}\theta_{ABC}^{(-3)}\wedge\left(d\varphi^{AB}+\varphi{{}^{A}}_{D}\wedge\varphi^{DB}\right)=-d\pi_{C}-c^{B}_{CA}\theta^{A}\wedge\pi_{B}\hbox{
mod}[\theta_{\gamma}^{(N-1)}].$

One can recognize on the left hand side the Einstein tensor of $\mathbf{h}$ on
$\mathcal{Y}$. After a gauge transformation
$e^{\alpha}=S^{\alpha}_{\beta}\theta^{\beta}$ and
$\omega{{}^{\alpha}}_{\beta}=S^{\alpha}_{\alpha^{\prime}}\varphi{{}^{\alpha^{\prime}}}_{\beta^{\prime}}(S^{-1})^{\beta^{\prime}}_{\beta}-dS^{\alpha}_{\beta^{\prime}}(S^{-1})^{\beta^{\prime}}_{\beta}$,
where $S=\hbox{Ad}_{g}$ (see Section 5), the previous equation translates as

$\frac{1}{2}e_{ABC}^{(-3)}\wedge\left(d\omega^{AB}+\omega{{}^{A}}_{D}\wedge\omega^{DB}\right)=-dp_{C}\hbox{
mod}[e_{\gamma}^{(-1)}]$

The key observations are that the left hand side is constant on any fiber,
whereas the restriction of the right hand side to any fiber is an exact form.
Both observations lead to the conclusion that
$\frac{1}{2}e_{ABC}^{(-3)}\wedge\left(d\omega^{AB}+\omega{{}^{A}}_{C}\wedge\omega^{CB}\right)=0\hbox{
mod}[e_{\gamma}^{(-1)}]$, i.e. the two blocks
$\hbox{Ein}(\textbf{h}){{}^{a}}_{c}$ and
$\hbox{Ein}(\textbf{h}){{}^{a}}_{\gamma}$ of the Einstein tensor of $\omega$
vanish.

The final equations, after a long computation (see Section 6) then read

(13)
$\left\\{\begin{array}[]{ccl}\hbox{Ein}(\textbf{g}){{}^{a}}_{d}+\frac{1}{2}\left(F{{}_{\gamma}}^{ac}F{\gamma}_{dc}-\frac{1}{4}F{{}_{\gamma}}^{bc}F{\gamma}_{bc})\delta^{a}_{d}\right)+\Lambda\delta^{a}_{d}&=&0\\\
\nabla_{c}F{{}_{\gamma}}^{ca}-c^{\beta}_{\alpha\gamma}A^{\alpha}_{c}F{{}_{\beta}}^{ca}&=&0\end{array}\right.$

where
$\hbox{Ein}(\textbf{g}){{}^{a}}_{d}:=\hbox{Ric}(\textbf{g}){{}^{a}}_{d}-\frac{1}{2}\hbox{R}(\textbf{g})\delta^{a}_{d}$
is the Einstein tensor of g, $F:=dA+\frac{1}{2}[A\wedge A]$ and
$\Lambda:=-\frac{1}{8}c^{\alpha}_{\beta\gamma}c^{\beta}_{\alpha\epsilon}\textsf{h}^{\gamma\epsilon}=-\frac{1}{8}(K,\textsf{h}^{*})$,
where $K$ is the Killing form on $\mathfrak{g}$.

##  3\. The Euler–Lagrange equations

In the following we assume that $(\theta,\varphi,\pi)\in\mathscr{C}$ is a
critical point of $\mathscr{A}$ such that $\hbox{rank}\theta=N$ (Hypothesis
(v)). We denote by
$\textbf{h}=\textsf{b}_{ab}\theta^{a}\theta^{b}+\textsf{k}_{\alpha\beta}\theta^{\alpha}\theta^{\beta}$
the induced metric on $\mathcal{Y}$ and we assume that h is vertically
complete (Hypothesis (vi)).

###  3.1. Variations with respect to coefficients of $\pi$

Since $\hbox{rank}\theta=N$, the family $(\theta^{1},\cdots,\theta^{N})$ is a
coframe on $\mathcal{Y}$, there exists unique coefficients $H^{A}_{BC}$ such
that
$d\theta^{A}+\frac{1}{2}c^{A}_{BC}\theta^{B}\wedge\theta^{C}=\frac{1}{2}H^{A}_{BC}\theta^{B}\wedge\theta^{C}$
and $H^{A}_{BC}+H^{A}_{CB}=0$. We decompose $\pi=\pi_{A}\textbf{t}^{A}$ and
each $\pi_{A}$ as $\pi_{A}=\frac{1}{2}\pi_{A}^{BC}\theta^{(N-2)}_{BC}$, where
$\pi_{A}^{BC}+\pi_{A}^{CB}=0$. The constraint (2) then reads $\pi_{A}^{ab}=0$
or

(14)
$\pi_{A}=\pi_{A}^{b\gamma}\theta^{(N-2)}_{b\gamma}+\frac{1}{2}\pi_{A}^{\beta\gamma}\theta^{(N-2)}_{\beta\gamma}$

A first order variation of $(\theta,\varphi,\pi)$ keeping $\theta$ and
$\varphi$ constant and respecting (14) thus induces a variation of $\pi$ of
the form
$\delta\pi_{A}=\chi_{A}^{b\gamma}\theta^{(N-2)}_{b\gamma}+\frac{1}{2}\chi_{A}^{\beta\gamma}\theta^{(N-2)}_{\beta\gamma}$.
The fact that the action $\mathscr{A}$ is stationary with respect to such
variations of $\pi$ thus reads

$\int_{\mathcal{Y}}\delta\pi_{A}\wedge\left(d\theta^{A}+\frac{1}{2}[\theta\wedge\theta]^{A}\right)=\int_{\mathcal{Y}}\left(\chi_{A}^{b\gamma}H^{A}_{b\gamma}+\frac{1}{2}\chi_{A}^{\beta\gamma}H^{A}_{\beta\gamma}\right)\theta^{(N)}=0,\quad\forall\chi_{A}^{b\gamma},\chi_{A}^{\beta\gamma}$

and lead to the Euler–Lagrange equations
$H^{A}_{b\gamma}=H^{A}_{\beta\gamma}=0$, $\forall A,b,\beta,\gamma$. We thus
deduce that

(15)
$\Theta^{A}:=d\theta^{A}+\frac{1}{2}c^{A}_{BC}\theta^{B}\wedge\theta^{C}=\frac{1}{2}H^{A}_{bc}\theta^{b}\wedge\theta^{c}$

or equivalentely

(16)
$\left\\{\begin{array}[]{ccl}d\theta^{a}&=&\frac{1}{2}H^{a}_{bc}\theta^{b}\wedge\theta^{c}\\\
d\theta^{\alpha}+\frac{1}{2}c^{\alpha}_{\beta\gamma}\theta^{\beta}\wedge\theta^{\gamma}&=&\frac{1}{2}H^{\alpha}_{bc}\theta^{b}\wedge\theta^{c}\end{array}\right.$

###  3.2. Variations with respect to $\varphi$

Keeping $\theta$ and $\pi$ fixed we look at first order variations
$\delta\varphi=\lambda$ of $\varphi$. This induces the condition that, for all
$\lambda$,

$\frac{1}{2}\int_{\mathcal{Y}}d\left(\lambda^{AB}\wedge\theta^{(N-2)}_{AB}\right)+\lambda^{AB}\wedge\left(d\theta^{(N-2)}_{AB}-\varphi{{}^{A^{\prime}}}_{A}\wedge\theta^{(N-2)}_{A^{\prime}B}-\varphi{{}^{B^{\prime}}}_{B}\wedge\theta^{(N-2)}_{AB^{\prime}}\right)=0$

Assuming that $\lambda$ has compact support and using (10) and (12) we deduce
the relation

$\left(d\theta^{C}+\varphi{{}^{C}}_{C^{\prime}}\wedge\theta^{C^{\prime}}\right)\wedge\theta^{(N-3)}_{ABC}=d\theta^{(N-2)}_{AB}-\varphi{{}^{A^{\prime}}}_{A}\wedge\theta^{(N-2)}_{A^{\prime}B}-\varphi{{}^{B^{\prime}}}_{B}\wedge\theta^{(N-2)}_{AB^{\prime}}=0$

which implies that the torsion 2-form
$d\theta^{A}+\varphi{{}^{A}}_{A^{\prime}}\wedge\theta^{A^{\prime}}$ vanishes.
Hence the connection on $T\mathcal{Y}$ associated to $\varphi$ coincides with
the Levi-Civita connection of $(\mathcal{Y},\textbf{h})$, where
$\textbf{h}=\textsf{h}_{AB}\theta^{A}\theta^{B}$.

###  3.3. Variations with respect to $\theta$

We first observe that, through a variation $\delta\theta=\tau$ of $\theta$
keeping $\varphi$ and the coefficients $\pi_{A}^{b\gamma}$ and
$\pi_{A}^{\beta\gamma}$ fixed, we have

$\delta\left(\theta_{AB}^{(N-2)}\right)=\tau^{C}\wedge\theta^{(N-3)}_{ABC},$

plus the relation
$\delta\Theta^{A}=d\tau^{A}+c^{A}_{BC}\tau^{B}\wedge\theta^{C}$ which implies

$\pi_{A}\wedge\delta\Theta^{A}=d\left(\tau^{A}\wedge\pi_{A}\right)+\tau^{A}\wedge\left(d\pi_{A}+c^{C}_{AB}\theta^{B}\wedge\pi_{C}\right)$

and lastly
$\delta\pi_{A}=\pi_{A}^{b\gamma}\left(\tau^{d}\wedge\theta^{(N-3)}_{b\gamma
d}+\tau^{\delta}\wedge\theta^{(N-3)}_{b\gamma\delta}\right)+\frac{1}{2}\pi_{A}^{\beta\gamma}\left(\tau^{d}\wedge\theta^{(N-3)}_{\beta\gamma
d}+\tau^{\delta}\wedge\theta^{(N-3)}_{\beta\gamma\delta}\right)$ which, thanks
to
$\Theta^{A}\wedge\theta^{(N-3)}_{b\gamma\delta}=\Theta^{A}\wedge\theta^{(N-3)}_{\beta\gamma
d}=\Theta^{A}\wedge\theta^{(N-3)}_{\beta\gamma\delta}=0$ by (15), leads to

$\left(\delta\pi_{A}\right)\wedge\Theta^{A}=-\pi_{A}^{b\gamma}H^{A}_{bd}\tau^{d}\wedge\theta^{(N-1)}_{\gamma}.$

In conclusion, by assuming that $\tau$ has compact support, we obtain

$\int_{\mathcal{Y}}\tau^{C}\wedge\left(\frac{1}{2}\theta^{(N-3)}_{ABC}\wedge\Phi^{AB}-\pi_{A}^{b\gamma}H^{A}_{bC}\theta^{(N-1)}_{\gamma}+d\pi_{C}-c^{B}_{AC}\theta^{A}\wedge\pi_{B}\right)=0$

where we set $\Phi:=d\varphi+\frac{1}{2}[\varphi\wedge\varphi]$. Hence we
deduce the Euler–Lagrange equation

(17)
$\frac{1}{2}\theta^{(N-3)}_{ABC}\wedge\Phi^{AB}+d\pi_{C}-c^{B}_{AC}\theta^{A}\wedge\pi_{B}=0\hbox{
mod}[\theta^{(N-1)}_{\mathfrak{g}}]$

where, for any 3-form $\lambda$, $\lambda=0\hbox{
mod}[\theta^{(N-1)}_{\mathfrak{g}}]$ means that there exists coefficients
$\lambda^{\alpha}$ such that
$\lambda=\lambda^{\alpha}\theta^{(N-1)}_{\alpha}$.

##  4\. The fibration

From the first equation in (16) we deduce that $d\theta^{a}=0\hbox{
mod}[\theta^{b}]$, $\forall a=1,\cdots,n$. Since the rank of
$(\theta^{1},\cdots,\theta^{n})$ is equal to $n$ everywhere, Frobenius’
theorem implies that, for any point $\textsf{y}\in\mathcal{Y}$, there exists a
neighbourhood of y in which there exists a unique submanifold f of dimension
$r$ crossing y such that $\theta^{a}|_{\textsf{f}}=0$, $\forall a=1,\cdots,n$.
Hence $\mathcal{Y}$ is foliated by integral leaves of dimension $r$.

Consider on the product manifold $\mathcal{Y}\times\mathfrak{G}$ the
$\mathfrak{g}$-valued 1-form $\tau:=h^{-1}dh-\theta^{\mathfrak{g}}$, where
$(\textsf{y},h)$ denotes a point in $\mathcal{Y}\times\mathfrak{G}$ and where
$\theta^{\mathfrak{g}}:=\textbf{t}_{\alpha}\theta^{\alpha}$. It satisfies the
identity
$d\tau+d\theta^{\mathfrak{g}}+\frac{1}{2}[\theta^{\mathfrak{g}}\wedge\theta^{\mathfrak{g}}]+[\theta^{\mathfrak{g}}\wedge\tau]+\frac{1}{2}[\tau\wedge\tau]=0$.
However the second equation in (16) implies that, for any integral leaf f,
$d\theta^{\mathfrak{g}}+\frac{1}{2}[\theta^{\mathfrak{g}}\wedge\theta^{\mathfrak{g}}]|_{\textsf{f}}=0$
and thus $d(\tau|_{\textsf{f}\times\mathfrak{G}})=0\hbox{ mod}[\tau]$. Hence,
again by Frobenius’ theorem, for any
$(\textsf{y}_{0},g_{0})\in\textsf{f}\times\mathfrak{G}$, there exist a unique
$r$-dimensional submanifold $\Gamma\subset\textsf{f}\times\mathfrak{G}$ which
is a solution of $\tau|_{\Gamma}=0$ and which contains
$(\textsf{y}_{0},g_{0})$. This implies the existence of a unique map $g$ (the
graph of which is $\Gamma$) from a neighbourhood of $\textsf{y}_{0}$ in f to
$\mathfrak{G}$ such that $g(\textsf{y}_{0})=g_{0}$ and
$dg-g\theta^{\mathfrak{g}}|_{\textsf{f}}=0$. Moreover $g$ is clearly
invertible.

Consider any smooth path $\gamma:[0,1]\longrightarrow\mathfrak{G}$ such that
$\gamma(0)=1_{\mathfrak{G}}$ and a point $\textsf{y}_{0}\in\mathcal{Y}$. By
Hypothesis (vi) we can associate to it a unique path
$u:[0,1]\longrightarrow\textsf{f}$ such that $u(0)=\textsf{y}_{0}$ and
$(u,\gamma)^{*}\tau=0$. The image of $(u,\gamma)$ is contained in some
integral submanifold $\Gamma$ which coincides locally with the graph of an
invertible map $g$ as previously. Thus to any path homotopic to $\gamma$ in
$\mathfrak{G}$ with fixed extremities it corresponds a path homotopic to $u$
in f with fixed extremities. Since $\mathfrak{G}$ is simply connected we can
thus define a unique map $T:\mathfrak{G}\longrightarrow\textsf{f}$ such that
$T(1_{\mathfrak{G}})=\textsf{y}_{0}$ and $(T\times Id)^{*}\tau=0$. Hence
$\mathfrak{G}$ is a universal cover of f and, in particular, since
$\mathfrak{G}$ is compact f is _compact_.

To any _fixed_ $x=(x^{1},\cdots,x^{n})\in\mathbb{R}^{n}$ we associated the
vector field $X$ on $\mathcal{Y}$ defined by
$X=x^{a}\frac{\partial}{\partial\theta^{a}}$. Let $\textsf{f}_{0}$ be some
integral leaf. Let us assume that $x$ is in the unit ball $B^{n}$ of
$\mathbb{R}^{n}$. Since $\textsf{f}_{0}$ is compact there exists a
neighbourhood $\mathcal{T}$ of $\textsf{f}_{0}$ in $\mathcal{Y}$ and some
$\varepsilon>0$ such that the flow map $(t,\textsf{y})\longmapsto
e^{tX}(\textsf{y})$ is defined on
$[-\varepsilon,\varepsilon]\times\mathcal{T}$. We observe that, due to (16),
$L_{X}\theta^{a}=H^{a}_{bc}x^{b}\theta^{c}$, $\forall a$. Hence there exists
functions $M^{a}_{bc}$ on $\mathcal{Y}$ (depending on $x$) such that
$\left(e^{tX}\right)^{*}\theta^{a}=M^{a}_{c}\theta^{c}$, $\forall a$. For any
leaf $\textsf{f}\subset\mathcal{T}$, let
$\iota:\textsf{f}\longrightarrow\mathcal{Y}$ its embedding map and
$\iota_{t}:=e^{tX}\circ\iota$. Note that the image of $\iota_{t}$ is
$e^{tX}(\textsf{f})$. We have then

$\iota_{t}^{*}\theta^{a}=\left(e^{tX}\circ\iota\right)^{*}\theta^{a}=\iota^{*}\left(e^{tX}\right)^{*}\theta^{a}=\iota^{*}\left(M^{a}_{c}\theta^{c}\right),\quad\forall
a.$

Thus the 1-form $\textbf{t}_{a}\theta^{a}$ vanishes on $e^{tX}(\textsf{f})$
iff it vanishes on f, i.e. f is an integral leaf iff $e^{tX}(\textsf{f})$ is
also an integral leaf. As a consequence the map
$B^{n}\times\textsf{f}_{0}\ni(x,\textsf{y})\longmapsto e^{\varepsilon
x^{a}\frac{\partial}{\partial\theta^{a}}}(\textsf{y})$ is a local
diffeomorphism onto a neighbourhood of $\textsf{f}_{0}$, which provides us
with a local trivialization of the set of leaves. Hence the set $\mathcal{X}$
of integral leaves has the structure of an $n$-dimensional manifold and the
quotient map $P:\mathcal{Y}\longrightarrow\mathcal{X}$ is a bundle fibration.

Set $e^{a}:=\theta^{a}$, for $1\leq a\leq n$. From
$\frac{\partial}{\partial\theta^{\beta}}\lrcorner
e^{a}=\frac{\partial}{\partial\theta^{\beta}}\lrcorner de^{a}=0$ we deduce
that there exists a coframe $\left(\underline{e}^{a}\right)_{1\leq a\leq n}$
on $\mathcal{X}$ such that $e^{a}=P^{*}\underline{e}^{a}$, $\forall a$. Thus
we can equipp $\mathcal{X}$ with the pseudo Riemannian metric
$\underline{\textbf{g}}:=\textsf{b}_{ab}\underline{e}^{a}\underline{e}^{b}$.

In the following we choose an $n$-dimensional submanifold
$\Sigma\subset\mathcal{Y}$ transverse to the fibration. Without loss of
generality (replacing $\mathcal{Y}$ by an open subset of $\mathcal{Y}$ if
necessary) we can assume that $\Sigma$ intersects all fibers of $P$ and we
define the map $g:\mathcal{Y}\longrightarrow\mathfrak{G}$ which is constant
equal to $1_{\mathfrak{G}}$ on $\Sigma$ and such that
$dg-g\theta^{\mathfrak{g}}|_{\textsf{f}}=0$ for any integral leaf f. We then
define $A:=g\theta^{\mathfrak{g}}g^{-1}-dg\cdot g^{-1}$. The relation
$dg-g\theta^{\mathfrak{g}}|_{\textsf{f}}=0$ then translates as
$A|_{\textsf{f}}=0$ and hence we have the decomposition $A=A_{a}\theta^{a}$.
Moreover since

(18) $\theta^{\mathfrak{g}}=g^{-1}Ag+g^{-1}dg,$

we have
$d\theta^{\mathfrak{g}}+\frac{1}{2}[\theta\wedge\theta]^{\mathfrak{g}}=g^{-1}(dA+\frac{1}{2}[A\wedge
A])g=g^{-1}Fg$, where $F:=dA+\frac{1}{2}[A\wedge A]$. By using (16) we deduce
that $\frac{\partial}{\partial\theta^{\alpha}}\lrcorner dA=0$,
$\forall\alpha=n+1,\cdots,N$, i.e. the coefficients $A_{a}$ are constants on
the fibers f. Moreover we have

(19) $F^{\alpha}=\frac{1}{2}F^{\alpha}_{bc}e^{b}\wedge e^{c},$

where the coefficients $F_{bc}=gH_{bc}^{\mathfrak{g}}g^{-1}$ are constant on
the fibers.

##  5\. Trivialization of the bundle

Using the map $g:\mathcal{Y}\longrightarrow\mathfrak{G}$ defined previously we
define the map $S$ from $\mathcal{Y}$ to $\hbox{End}(\hat{\mathfrak{g}})$
which, to any $\textsf{y}\in\mathcal{Y}$, associates
$\hbox{Ad}_{g(\textsf{y})}$. In other words,
$\forall(v,\xi)\in\mathfrak{s}\times\mathfrak{g}$,
$S(v+\xi):=g(v+\xi)g^{-1}=v+g\xi g^{-1}$. We remark that $S$ takes values in
$SO(\hat{\mathfrak{g}},\textsf{h})$ because of Hypothesis (ii). Let
$\left(S^{A}_{B}\right)_{1\leq A,B\leq N}$ be the matrix of $S$ in the basis
$\left(\textbf{t}_{A}\right)_{1\leq A\leq N}$, i.e. such that
$S(\textbf{t}_{A})=\textbf{t}_{B}S^{B}_{A}$. We define a new coframe
$\left(e^{A}\right)_{1\leq A\leq N}$ by $e^{A}=S^{A}_{B}\theta^{B}$.
Equivalentely

$\left\\{\begin{array}[]{cccl}e^{a}&:=&\theta^{a}&\forall a=1,\cdots,n\\\
e^{\alpha}&:=&S^{\alpha}_{\beta}\theta^{\beta}&\forall\alpha=n+1,\cdots,N\end{array}\right.$

Then $e^{\alpha}=(g\theta^{\mathfrak{g}}g^{-1})^{\alpha}$ and (18) imply

(20) $e^{\alpha}=A^{\alpha}+(dg\
g^{-1})^{\alpha},\quad\forall\alpha=n+1,\cdots,N.$

We deduce that

$\begin{array}[]{ccl}de^{\alpha}-\frac{1}{2}[e\wedge e]^{\alpha}+[A\wedge
e]^{\alpha}&=&de^{\alpha}-\frac{1}{2}[e\wedge
e]^{\alpha}+[(e-dg\,g^{-1})\wedge e]^{\alpha}\\\
&=&de^{\alpha}+\frac{1}{2}[e\wedge e]^{\alpha}-[dg\,g^{-1}\wedge
e]^{\alpha}\\\ &=&\left(gd\theta g^{-1}+[dgg^{-1}\wedge
e]\right)^{\alpha}+\frac{1}{2}[e\wedge e]^{\alpha}-[dgg^{-1}\wedge
e]^{\alpha}\\\
&=&\left(g\left(d\theta+\frac{1}{2}[\theta\wedge\theta]\right)g^{-1}\right)^{\alpha}\end{array}$

from which we get the useful identity

(21) $de^{\alpha}-\frac{1}{2}[e\wedge e]^{\alpha}+[A\wedge
e]^{\alpha}=F^{\alpha}:=\frac{1}{2}F_{bc}^{\alpha}e^{b}\wedge e^{c}.$

Let us translate the left hand side of (17) in the new coframe. First we
define $e^{(N)}:=e^{1}\wedge\cdots\wedge e^{N}$ and note that
$e^{(N)}=\theta^{(N)}$. Moreover defining
$e^{(N-1)}_{A}:=\frac{\partial}{\partial e^{A}}\lrcorner e^{(N)}$,
$e^{(N-2)}_{AB}:=\frac{\partial}{\partial e^{A}}\wedge\frac{\partial}{\partial
e^{B}}\lrcorner e^{(N)}$, we observe that, since
$\frac{\partial}{\partial\theta^{A}}=\frac{\partial}{\partial
e^{B}}S^{B}_{A}$, we have
$\theta^{(N-1)}_{A}=e^{(N-1)}_{A^{\prime}}S^{A^{\prime}}_{A}$,
$\theta^{(N-2)}_{AB}=e^{(N-2)}_{A^{\prime}B^{\prime}}S^{A^{\prime}}_{A}S^{B^{\prime}}_{B}$
and
$\theta^{(N-3)}_{ABC}=e^{(N-3)}_{A^{\prime}B^{\prime}C^{\prime}}S^{A^{\prime}}_{A}S^{B^{\prime}}_{B}S^{C^{\prime}}_{C}$.

Second let $\omega$ be the $so(\hat{\mathfrak{g}},\textsf{h})$-valued
connection 1-form in the coframe $(e^{A})_{1\leq A\leq N}$, which is equal to
$\omega:=S\varphi S^{-1}-dS\,S^{-1}$. Let
$\Omega:=d\omega+\frac{1}{2}[\omega\wedge\omega]=S\Phi S^{-1}$, where
$\Phi=d\varphi+\frac{1}{2}[\varphi\wedge\varphi]$. Then
$\Phi^{AB}=(S^{-1})^{A}_{A^{\prime}}(S^{-1})^{B}_{B^{\prime}}\Omega^{A^{\prime}B^{\prime}}$.

We deduce that
$\theta^{(N-3)}_{ABC}\wedge\Phi^{AB}=e^{(N-3)}_{ABC^{\prime}}\wedge\Omega^{AB}S^{C^{\prime}}_{C}$.
Hence (17) is equivalent to

(22)
$\frac{1}{2}e^{(N-3)}_{ABC}\wedge\Omega^{AB}+\left(d\pi_{C^{\prime}}-c^{B}_{AC^{\prime}}\theta^{A}\wedge\pi_{B}\right)(S^{-1})^{C^{\prime}}_{C}=0\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}],$

where, for any $(N-1)$-form $\lambda$, we write:

$\lambda=0\hbox{ mod}[e^{(N-1)}_{\mathfrak{g}}]$

iff there exists forms $\lambda^{\alpha}$ such that
$\lambda=\lambda^{\alpha}e^{(N-1)}_{\alpha}$.

In the following we use extensively Relations (8), (9), (10), (11) and (12).

######  Lemma 5.1.

We have

(23)
$\left(d\pi_{C^{\prime}}-c^{B}_{AC^{\prime}}\theta^{A}\wedge\pi_{B}\right)(S^{-1})^{C^{\prime}}_{C}=d\left(\pi_{C^{\prime}}(S^{-1})^{C^{\prime}}_{C}\right)\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}].$

_Proof_ — From the definition of $S$ we deduce that,
$\forall\xi\in\hat{\mathfrak{g}}$,

$d(S^{-1}(\xi))=[g^{-1}\xi
g,g^{-1}dg]=g^{-1}[\xi,\textbf{t}_{\alpha}(e^{\alpha}-A^{\alpha})]g=-S^{-1}([e-A,\xi]),$

where, in the last equality we used the fact that
$[\xi,\textbf{t}_{a}e^{a}]=0$ because of Hypothesis (i). Thus we can write
$d(S^{-1})^{C^{\prime}}_{C}=-(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}(e^{B}-A^{B})$.
Hence

$\begin{array}[]{ccl}d\left((S^{-1})^{C^{\prime}}_{C}\pi_{C^{\prime}}\right)&=&-(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}(e^{B}-A^{B})\wedge\pi_{C^{\prime}}+(S^{-1})^{C^{\prime}}_{C}\left(d\pi_{C^{\prime}}\right)\\\
&=&-(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}S^{B}_{B^{\prime}}\theta^{B^{\prime}}\wedge\pi_{C^{\prime}}+(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}A^{B}\wedge\pi_{C^{\prime}}+(S^{-1})^{C^{\prime}}_{C}\left(d\pi_{C^{\prime}}\right)\end{array}$

But because of
$[\hbox{Ad}_{g}(\xi),\hbox{Ad}_{g}(\eta)]=\hbox{Ad}_{g}([\xi,\eta])$,
$\forall\xi,\eta\in\mathfrak{g}$, which is equivalent to
$c^{A}_{B^{\prime}C^{\prime}}S^{B^{\prime}}_{B}S^{C^{\prime}}_{C}=S^{A}_{A^{\prime}}c^{A^{\prime}}_{BC}$,
we have
$(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}S^{B}_{B^{\prime}}=c^{C^{\prime}}_{B^{\prime}C^{\prime\prime}}(S^{-1})^{C^{\prime\prime}}_{C}$.
Thus for the first term on the r.h.s.,

$(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}S^{B}_{B^{\prime}}\theta^{B^{\prime}}\wedge\pi_{C^{\prime}}=c^{C^{\prime}}_{B^{\prime}C^{\prime\prime}}(S^{-1})^{C^{\prime\prime}}_{C}\theta^{B^{\prime}}\wedge\pi_{C^{\prime}}=c^{B}_{AC^{\prime}}\theta^{A}\wedge\pi_{B}(S^{-1})^{C^{\prime}}_{C}$

and hence

$d\left((S^{-1})^{C^{\prime}}_{C}\pi_{C^{\prime}}\right)=\left(d\pi_{C^{\prime}}-c^{B}_{AC^{\prime}}\theta^{A}\wedge\pi_{B}\right)(S^{-1})^{C^{\prime}}_{C}+(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}A^{B}\wedge\pi_{C^{\prime}}$

However it follows from (14) that
$\pi_{C^{\prime}}=S^{\gamma}_{\gamma^{\prime}}\pi^{b\gamma^{\prime}}_{C^{\prime}}e^{(N-2)}_{b\gamma}+\frac{1}{2}S^{\beta}_{\beta^{\prime}}S^{\gamma}_{\gamma^{\prime}}\pi^{\beta^{\prime}\gamma^{\prime}}_{C^{\prime}}e^{(N-2)}_{\beta\gamma}$
and, since $A^{B}=A^{B}_{c}e^{c}$, we get
$(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}A^{B}\wedge\pi_{C^{\prime}}=-(S^{-1})^{C^{\prime}}_{A}c^{A}_{BC}A^{B}_{b}S^{\gamma}_{\gamma^{\prime}}\pi_{C^{\prime}}^{b\gamma^{\prime}}e^{(N-1)}_{\gamma}=0\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}]$. Hence (23) follows. $\square$

Thus if we define $p_{C}:=\pi_{C^{\prime}}(S^{-1})^{C^{\prime}}_{C}$ we deduce
from (23) that (22) is equivalent to

(24) $\frac{1}{2}e^{(N-3)}_{ABC}\wedge\Omega^{AB}+dp_{C}=0\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}].$

We need to compute $dp_{C}$. For that purpose we use the _a priori_
decomposition
$p_{C}=p^{b\gamma}_{C}e^{(N-2)}_{b\gamma}+\frac{1}{2}p^{\beta\gamma}_{C}e^{(N-2)}_{\beta\gamma}$.
We first compute using (9), (10), (16) and (21)

$\begin{array}[]{ccl}de^{(N-2)}_{b\gamma}&=&de^{a}\wedge e_{b\gamma
a}^{(N-3)}+de^{\alpha}\wedge e_{b\gamma\alpha}^{(N-3)}\\\
&=&H^{a}_{ab}e_{\gamma}^{(N-1)}+c^{\alpha}_{\gamma\alpha}e_{b}^{(N-1)}-c^{\alpha}_{\beta\gamma}(A^{\beta})_{b}e_{\gamma}^{(N-1)}\\\
&=&\left(H^{a}_{ab}-c^{\alpha}_{\beta\gamma}(A^{\beta})_{b}\right)e_{\gamma}^{(N-1)}=0\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}],\end{array}$

where we have used the fact that, since $\mathfrak{G}$ is compact, its Lie
algebra $\mathfrak{g}$ is _unimodular_ , which reads
$c^{\alpha}_{\gamma\alpha}=0$. Similarly

$\begin{array}[]{ccl}de^{(N-2)}_{\beta\gamma}&=&de^{a}\wedge e_{\beta\gamma
a}^{(N-3)}+de^{\alpha}\wedge e_{\beta\gamma\alpha}^{(N-3)}\\\
&=&0+c^{\alpha}_{\gamma\alpha}e_{\beta}^{(N-1)}+c^{\alpha}_{\alpha\beta}e_{\gamma}^{(N-1)}+c^{\alpha}_{\beta\gamma}e_{\alpha}^{(N-1)}\\\
&=&0\hbox{ mod}[e^{(N-1)}_{\mathfrak{g}}],\end{array}$

Thus by writing
$dp^{b\gamma}_{C}=p^{b\gamma}_{C,c}e^{c}+p^{b\gamma}_{C,\gamma}e^{\gamma}$ and
$dp^{\beta\gamma}_{C}=p^{\beta\gamma}_{C,c}e^{c}+p^{\beta\gamma}_{C,\delta}e^{\delta}$,
we get

$dp_{C}=p^{b\gamma}_{C,\gamma}e^{(N-1)}_{b}-p^{b\gamma}_{C,b}e^{(N-1)}_{\gamma}+p^{\beta\gamma}_{C,\gamma}e^{(N-1)}_{\beta}\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}]=p^{b\gamma}_{C,\gamma}e^{(N-1)}_{b}\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}]$

Lastly by decomposing $\Omega^{AB}=\frac{1}{2}\Omega{{}^{AB}}_{CD}e^{C}\wedge
e^{D}$, we find that

$\frac{1}{2}e^{(N-3)}_{ABC}\wedge\Omega^{AB}=-\hbox{Ein}(\omega){{}^{A}}_{C}e^{(N-1)}_{A},$

where
$\hbox{Ein}(\omega){{}^{A}}_{C}:=\hbox{Ric}(\omega){{}^{A}}_{C}-\frac{1}{2}\hbox{R}(\omega)\delta{{}^{A}}_{C}$,
$\hbox{Ric}(\omega){{}^{A}}_{C}:=\Omega{{}^{AB}}_{CB}$ and
$\hbox{R}(\omega):=\hbox{Ric}(\omega){{}^{A}}_{A}$. Obviously
$\hbox{Ric}(\omega){{}^{A}}_{C}$ is the Ricci curvature, $\hbox{R}(\omega)$
the scalar curvature and $\hbox{Ein}(\omega){{}^{A}}_{C}$ the Einstein tensor
of h in the coframe $\left(e^{A}\right)_{1\leq A\leq N}$. Hence we find that
(24) is equivalent to
$\left(\hbox{Ein}(\omega){{}^{a}}_{C}-p^{a\gamma}_{C,\gamma}\right)e^{(N-1)}_{a}=0\hbox{
mod}[e^{(N-1)}_{\mathfrak{g}}]$, or

(25) $\hbox{Ein}(\omega){{}^{a}}_{C}=p^{a\gamma}_{C,\gamma},\quad\forall
a=1,\cdots,n,\forall C=1,\cdots,N.$

We will come back to this equation later on.

##  6\. Computation of the connection and the curvature forms

We need to compute the connection 1-form $\omega$ and its curvature 2-form. As
a preliminary we first set $\underline{\gamma}{{}^{a}}_{c}$ to be the
connection 1-form on $(\mathcal{X},\underline{\textbf{g}})$ in the coframe
$\underline{e}^{a}$, i.e. which satisfies
$\underline{\gamma}^{ac}+\underline{\gamma}^{ca}=0$ and
$d\underline{e}^{a}+\underline{\gamma}{{}^{a}}_{c}\wedge\underline{e}^{b}=0$.
Then we set $\gamma{{}^{a}}_{c}:=P^{*}\underline{\gamma}{{}^{a}}_{c}$ which
satisfies similar relations, which, together with (21), leads to

(26) $\left\\{\begin{array}[]{ccc}de^{a}+\gamma{{}^{a}}_{c}\wedge e^{c}&=&0\\\
de^{\alpha}-\frac{1}{2}F^{\alpha}_{bc}e^{b}\wedge
e^{c}-\frac{1}{2}c^{\alpha}_{\beta\gamma}(e^{\beta}-2A^{\beta})\wedge
e^{\gamma}&=&0\end{array}\right.$

Now the connexion 1-form $\omega$ is uniquely characterized by the condition
$\omega^{AB}+\omega^{BA}=0$ (preservation of the metric) and
$de^{A}+\omega{{}^{A}}_{C}\wedge e^{C}=0$ (the torsion vanishes), which can be
written

(27) $\left\\{\begin{array}[]{ccl}de^{a}+\omega{{}^{a}}_{c}\wedge
e^{c}+\omega{{}^{a}}_{\gamma}\wedge e^{\gamma}&=&0\\\
de^{\alpha}+\omega{{}^{\alpha}}_{c}\wedge
e^{c}+\omega{{}^{\alpha}}_{\gamma}\wedge e^{\gamma}&=&0\end{array}\right.$

Comparing with (26) we are tempted to assume that
$\omega{{}^{\alpha}}_{\gamma}=-\frac{1}{2}c^{\alpha}_{\beta\gamma}(e^{\beta}-2A^{\beta})$,
which fulfills the condition $\omega^{\alpha\beta}+\omega^{\beta\alpha}=0$,
since
$c^{\alpha}_{\beta\gamma^{\prime}}\textsf{k}^{\gamma^{\prime}\gamma}+c^{\gamma}_{\beta\alpha^{\prime}}\textsf{k}^{\alpha^{\prime}\alpha}=0$
because the metric k is preserved by the adjoint action of $\mathfrak{g}$. We
also guess that $\omega{{}^{\alpha}}_{c}=-\frac{1}{2}F^{\alpha}_{bc}e^{b}$,
which forces automatically
$\omega{{}^{a}}_{\gamma}=\frac{1}{2}\textsf{k}_{\gamma\gamma^{\prime}}F^{\gamma^{\prime}}_{bc^{\prime}}\textsf{g}^{c^{\prime}c}e^{b}$,
in order to satisfy $\omega^{\alpha b}+\omega^{b\alpha}=0$. Then in order to
fulfill the first relation of (27), one needs to assume that
$\omega{{}^{a}}_{c}=\gamma{{}^{a}}_{c}-\frac{1}{2}\textsf{k}_{\gamma\gamma^{\prime}}F^{\gamma^{\prime}}_{a^{\prime}c}\textsf{b}^{a^{\prime}a}e^{\gamma}$.
We then check that
$\omega^{ac}=\gamma^{ac}-\frac{1}{2}\textsf{k}_{\gamma\gamma^{\prime}}F^{\gamma^{\prime}}_{a^{\prime}c^{\prime}}\textsf{b}^{a^{\prime}a}\textsf{b}^{c^{\prime}c}e^{\gamma}$
is skew symmetric in $(a,c)$. Thus we see that the forms $\omega^{AC}$ defined
by:

$\left(\begin{array}[]{cc}\omega{{}^{a}}_{c}&\omega{{}^{a}}_{\gamma}\\\
\omega{{}^{\alpha}}_{c}&\omega{{}^{\alpha}}_{\gamma}\end{array}\right)=\left(\begin{array}[]{cc}\gamma{{}^{a}}_{c}-\frac{1}{2}\textsf{k}_{\gamma\gamma^{\prime}}F^{\gamma^{\prime}}_{a^{\prime}c}\textsf{b}^{a^{\prime}a}e^{\gamma}&\frac{1}{2}\textsf{k}_{\gamma\gamma^{\prime}}F^{\gamma^{\prime}}_{ba^{\prime}}\textsf{b}^{a^{\prime}a}e^{b}\\\
-\frac{1}{2}F^{\alpha}_{bc}e^{b}&-\frac{1}{2}c^{\alpha}_{\beta\gamma}(e^{\beta}-2A^{\beta})\end{array}\right)$

satisfy (27) and $\omega^{AC}+\omega^{CA}=0$. Hence this is the Levi-Civita
connection 1-form of $(\mathcal{Y},\textbf{h})$. In the following it will
convenient to set $F{{}^{\gamma}}_{bc}:=F^{\gamma}_{bc}$,
$F{{}_{\gamma}}{{}^{a}}_{c}:=\textsf{k}_{\gamma\gamma^{\prime}}F{{}^{\gamma^{\prime}}}_{a^{\prime}c}\textsf{b}^{a^{\prime}a}$
and $F{{}_{\gamma
b}}^{c}:=\textsf{k}_{\gamma\gamma^{\prime}}F{{}^{\gamma^{\prime}}}_{bc^{\prime}}\textsf{b}^{c^{\prime}c}$.
Then

$\left(\begin{array}[]{cc}\omega{{}^{a}}_{c}&\omega{{}^{a}}_{\gamma}\\\
\omega{{}^{\alpha}}_{c}&\omega{{}^{\alpha}}_{\gamma}\end{array}\right)=\left(\begin{array}[]{cc}\gamma{{}^{a}}_{c}-\frac{1}{2}F{{}_{\gamma}}{{}^{a}}_{c}e^{\gamma}&\frac{1}{2}F{{}_{\gamma
b}}^{a}e^{b}\\\
-\frac{1}{2}F{{}^{\alpha}}_{bc}e^{b}&-\frac{1}{2}c^{\alpha}_{\beta\gamma}(e^{\beta}-2A^{\beta})\end{array}\right)$

We can thus compute the curvature 2-form
$\Omega{{}^{A}}_{C}=d\omega{{}^{A}}_{C}+\omega{{}^{A}}_{B}\wedge\omega{{}^{B}}_{C}$.

$\Omega{{}^{a}}_{c}=d\left(\gamma{{}^{a}}_{c}-\frac{1}{2}F{{}_{\gamma}}{{}^{a}}_{c}e^{\gamma}\right)+\left(\gamma{{}^{a}}_{b}-\frac{1}{2}F{{}_{\beta^{\prime}}}{{}^{a}}_{b}e^{\beta^{\prime}}\right)\wedge\left(\gamma{{}^{b}}_{c}-\frac{1}{2}F{{}_{\gamma^{\prime}}}{{}^{b}}_{c}e^{\gamma^{\prime}}\right)-\frac{1}{4}F{{}_{\beta
b^{\prime}}}^{a}F{{}^{\beta}}_{c^{\prime}c}e^{b^{\prime}}\wedge
e^{c^{\prime}}$ $\Omega{{}^{a}}_{\gamma}=d\left(F{{}_{\gamma
b}}^{a}e^{b}\right)+\frac{1}{2}\left(\gamma{{}^{a}}_{b}-\frac{1}{2}F{{}_{\beta^{\prime}}}{{}^{a}}_{b}e^{\beta^{\prime}}\right)\wedge\left(F{{}_{\gamma
b^{\prime}}}^{a}e^{b^{\prime}}\right)+\frac{1}{4}F{{}_{\beta
b^{\prime}}}^{a}c^{\beta}_{\beta^{\prime}\gamma}e^{b^{\prime}}\wedge(2A^{\beta^{\prime}}-e^{\beta^{\prime}})$
$\Omega{{}^{\alpha}}_{c}=-d\left(F{{}^{\alpha}}_{bc}e^{b}\right)-\frac{1}{2}\left(F{{}^{\alpha}}_{b^{\prime}b}e^{b^{\prime}}\right)\wedge\left(\gamma{{}^{b}}_{c}-\frac{1}{2}F{{}_{\gamma^{\prime}}}{{}^{b}}_{c}e^{\gamma^{\prime}}\right)-\frac{1}{4}F{{}^{\beta}}_{bc}c^{\alpha}_{\beta^{\prime}\beta}(2A^{\beta^{\prime}}-e^{\beta^{\prime}})\wedge
e^{b}$
$\Omega{{}^{\alpha}}_{\gamma}=\frac{1}{2}d\left(c^{\alpha}_{\beta\gamma}(2A^{\beta}-e^{\beta})\right)-\frac{1}{4}\left(F{{}^{\alpha}}_{b^{\prime}b}e^{b^{\prime}}\right)\wedge\left(F{{}_{\gamma
c^{\prime}}}^{b}e^{c^{\prime}}\right)+\frac{1}{4}c^{\alpha}_{\beta^{\prime}\beta}c^{\beta}_{\gamma^{\prime}\gamma}(2A^{\beta^{\prime}}-e^{\beta^{\prime}})\wedge(2A^{\gamma^{\prime}}-e^{\gamma^{\prime}}).$

Lastly we obtain the components of the Ricci tensor $\hbox{Ric}(\omega)$
through a lengthy computation.

(28)
$\hbox{Ric}(\omega){{}^{a}}_{d}=\hbox{Ric}(\gamma){{}^{a}}_{d}-\frac{1}{2}F{{}_{\beta}}^{ac}F{{}^{\beta}}_{dc}$

where
$\hbox{Ric}(\gamma){{}^{a}}_{d}:=\left(d\gamma{{}^{a}}_{c}+\gamma{{}^{a}}_{b}\wedge\gamma{{}^{b}}_{c}\right)_{de}\textsf{b}^{ce}$
is the Ricci curvature of $\gamma$, and using the decompositions
$dF_{\delta}{{}^{ac}}=F_{\delta}{{}^{ac}}_{,c}e^{c}$ and
$\gamma{{}^{a}}_{b}=(\gamma{{}^{a}}_{b})_{c}e^{c}$,

(29)
$\hbox{Ric}(\omega){{}^{a}}_{\delta}=\frac{1}{2}\left(F_{\delta}{{}^{ac}}_{,c}+(\gamma{{}^{a}}_{b})_{c}F{{}_{\delta}}^{bc}+(\gamma{{}^{c}}_{b})_{c}F{{}_{\delta}}^{ab}-c^{\gamma}_{\alpha\delta}A^{\alpha}_{c}F{{}_{\gamma}}^{ac}\right)$
(30)
$\hbox{Ric}(\omega){{}^{\alpha}}_{\delta}=\frac{1}{4}F{{}_{\delta}}{{}^{bc}}F{{}^{\alpha}}{{}_{bc}}-\frac{1}{4}c^{\alpha}_{\beta\gamma}c^{\beta}_{\delta\epsilon}\textsf{k}^{\gamma\epsilon}$

We deduce the scalar curvature $\hbox{R}(\omega)$ of $\omega$ in function of
the scalar curvature $\hbox{R}(\gamma):=\hbox{Ric}(\gamma){{}^{a}}_{a}$:

(31)
$\hbox{R}(\omega)=\hbox{R}(\gamma)-\frac{1}{4}F{{}_{\alpha}}^{ab}F{{}^{\alpha}}_{ab}-\frac{1}{4}c^{\alpha}_{\beta\gamma}c^{\beta}_{\alpha\delta}\textsf{k}^{\gamma\delta}$

Hence the Einstein tensor of $\omega$ is

(32)
$\hbox{Ein}(\omega){{}^{a}}_{d}=\hbox{Ein}(\gamma){{}^{a}}_{d}-\frac{1}{2}\left(F{{}_{\beta}}^{ac}F{{}^{\beta}}_{dc}-\frac{1}{4}F{{}_{\alpha}}^{bc}F{{}^{\alpha}}_{bc}\delta{{}^{a}}_{d}\right)+\frac{1}{8}c^{\alpha}_{\beta\gamma}c^{\beta}_{\alpha\delta}\textsf{k}^{\gamma\delta}\delta{{}^{a}}_{d}$

and $\hbox{Ein}(\omega){{}^{a}}_{\delta}=\hbox{Ric}(\omega){{}^{a}}_{\delta}$
is given by (29).

An important observation is that _the components of_
$\hbox{Ein}(\omega){{}^{a}}_{d}$ _and_ $\hbox{Ein}(\omega){{}^{a}}_{\delta}$
_are constant on the fibers f_.

##  7\. The Einstein–Yang–Mills equations

We conclude by exploiting the fact that the fibers f are compact without
boundary. Let $\mu^{(r)}:=e^{n+1}\wedge\cdots e^{N}$ and set
$\mu^{(r-1)}_{\alpha}:=\frac{\partial}{\partial
e^{\alpha}}\lrcorner\mu^{(r)}$, $\forall\alpha$. By integrating both sides of
(25) on a fiber f we obtain

$\int_{\textsf{f}}\hbox{Ein}(\omega){{}^{a}}_{C}\mu^{(r)}=\int_{\textsf{f}}p^{a\gamma}_{C,\gamma}\mu^{(r)}=\int_{\textsf{f}}d\left(p^{a\gamma}_{C}\mu^{(r-1)}_{\gamma}\right)=0.$

But on the one hand, the components of $\hbox{Ein}(\omega){{}^{a}}_{C}$ are
constant on the fiber f, as seen in the previous section. Hence

(33)
$\hbox{Ein}(\omega){{}^{a}}_{C}=\frac{\int_{\textsf{f}}\hbox{Ein}(\omega){{}^{a}}_{C}\mu^{(r)}}{\int_{\textsf{f}}\mu^{(r)}}=0.$

Thus, using (29) and (32) we deduce that $\gamma$ and $A$ are solutions of the
Einstein–Yang–Mills system

$\left\\{\begin{array}[]{ccl}\hbox{Ein}(\gamma){{}^{a}}_{d}-\frac{1}{2}\left(F{{}_{\beta}}^{ac}F{{}^{\beta}}_{dc}-\frac{1}{4}F{{}_{\alpha}}^{bc}F{{}^{\alpha}}_{bc}\delta{{}^{a}}_{d}\right)+\frac{1}{8}c^{\alpha}_{\beta\gamma}c^{\beta}_{\alpha\delta}\textsf{k}^{\gamma\delta}\delta{{}^{a}}_{d}&=&0\\\
F_{\delta}{{}^{ac}}_{,c}+(\gamma{{}^{a}}_{b})_{c}F{{}_{\delta}}^{bc}+(\gamma{{}^{c}}_{b})_{c}F{{}_{\delta}}^{ab}-c^{\gamma}_{\alpha\delta}A^{\alpha}_{c}F{{}_{\gamma}}^{ac}&=&0\end{array}\right.$

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