# Para-Sasaki-like Manifolds with Generalized Symmetric Metric Connection

###### Abstract

The present paper deals with the generalized symmetric metric connection
defined on para-Sasaki-like manifolds. We derive a relation between the Levi-
Civita connection and the generalized symmetric metric conneciton on the
considered manifold. We investigate the curvature tensor, the Ricci tensor and
scalar curvature tensor with respect to the generalized symmetric metric
connection. We study para-Sasaki-like solitons on para-Sasaki-like manifolds
with the generalized symmetric metric connection. Finally, we construct two
examples of para-Sasaki-like manifolds admitting generalized symmetric metric
connection and verify our some results.

Şenay BULUT, Pınar İNSELÖZ

Eskişehir Technical University, Department of Mathematics, Eskişehir, TURKEY

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Key Words Almost paracontact almost paracomplex Riemannian manifold, Para-
Sasaki-like manifold, para-Ricci-like soliton, para-Einstein-like manifold,
Generalized symmetric metric connection.

2000 MR Subject Classification 53C15, 53C25, 53C50.

##  1 Introduction

The study of a semisymmetric linear connection on a differentiable manifold
was initiated by Friedmann and Schouten [22] in 1924. Later, in [16] the idea
of semisymmetric linear connection with a torsion different from zero was
introduced and studied. This connection has been studied on manifolds equipped
with different structures such as almost contact B-metric manifolds, almost
contact manifolds, Kenmotsu manifolds [14, 15, 19]. A quarter-symmetric linear
connection on a differentiable manifold was introduced by Golab [1]. Rastogi
[20, 21] started the systematic study of the quarter-symmetric metric
connection. Some properties of semisymmetric metric connections have been
investigated on manifolds having special structures [2, 18, 23, 24]. A
generalized symmetric metric connection which is the generalized form of the
semisymmetric metric connection and the quarter-symmetric metric connection
for Kenmotsu manifolds, $L_{p}$-Sasakian manifolds and Golden Lorentzian
manifolds has been defined and studied [11, 12, 13].

The study of Riemannian Ricci solitons was carried out in [7]. The
investigations into the Ricci solitons have been generalized in contact
geometry, paracontact geometry and pseudo-Riemannian geometry [8, 9, 10]. A
detailed study on para-Ricci-like soliton has been performed in [4].

The geometry of almost paracontact almost paracomplex Riemannian manifolds
(briefly apapR manifolds) has been examined [6]. After that, para-Sasaki-like
manifolds, a special class of apapR manifolds, have been presented in [4].
Para-Ricci-like solitons with Reeb vector field potential, arbitrary potential
and vertical potential have been investigated and proved some additional
geometric properties in [3, 4, 5].

Motivated by these circumstances we initiate the study of para-Sasaki-like
manifolds with generalized symmetric metric connection which is the
generalization of semi-symmetric and quarter-symmetric metric connection. In
Section 2, we present some basic definitions and results about para-Sasaki-
like manifolds. In Section 3, we define the generalized symmetric metric
connection on apapR manifolds and calculate curvature tensor and the Ricci
tensor with respect to this connection for para-Sasaki-like Riemannian
manifolds. Moreover, we investigate some operators with this connection. In
Section 4, we prove that the cases of para-Einstein-like of the manifold and
admitting para-Ricci-like soliton are preserved in this connection. Finally,
we give two examples to illustrate the obtained results.

##  2 Para-Sasaki-like Riemannian Manifolds

Let $M$ be a differentiable manifold of dimension $(2n+1)$ endowed with a
tensor field $\phi$ of type $(1,1)$, a vector field $\xi$, a $1-$form $\eta$
and a Riemannian metric $g$, which satisfies

$\begin{array}[]{l}\eta(\xi)=1,\ \ \ \phi^{2}x=x-\eta(x)\xi,\ \ \ \phi\xi=0\\\
g(\phi x,\phi y)=g(x,y)-\eta(x)\eta(y),\ \ \ tr\phi=0\end{array}$ (1)

for any $x,y\in\chi(M)$. Such manifold $(M,\phi,\xi,\eta,g)$ is called an
almost paracontact almost paracomplex Riemannian manifold (shortly, apapR
manifold)[6]. Moreover, by using the latter equalities we get the following:

$\begin{array}[]{lll}g(x,\xi)=\eta(x),&&g(x,\phi y)=g(\phi x,y),\\\
g(\xi,\xi)=1,&&\eta(\nabla_{x}\xi)=0,\end{array}$ (2)

where $\nabla$ denotes the Levi-Civita connection of $g$.

The associated metric $\widetilde{g}$ of $g$ on $(M,\phi,\xi,\eta,g)$
determined by the equality $\widetilde{g}(x,y)=g(x,\phi y)+\eta(x)\eta(y)$ is
a pseudo-Riemannian metric of signature $(n+1,n)$.

An apapR manifold $M$ is said to be a para-Sasaki-like manifold if the
following is provided:

$\begin{array}[]{lll}(\nabla_{x}\phi)y&=&-g(x,y)\xi-\eta(y)x+2\eta(x)\eta(y)\xi,\\\
&=&-g(\phi x,\phi y)\xi-\eta(y)\phi^{2}x.\end{array}$ (3)

In [17], it is proven that the following identities hold for any para-Sasaki-
like manifold $(M,g,\phi,\xi,\eta)$:

$\begin{array}[]{lll}\nabla_{x}\xi=\phi x,&&(\nabla_{x}\eta)y=g(x,\phi y),\\\
R(x,y)\xi=-\eta(y)x+\eta(x)y,&&\text{Ric}(x,\xi)=-2n\ \eta(x),\\\
R(\xi,y)\xi=\phi^{2}y,&&\text{Ric}(x,\xi)(\xi,\xi)=-2n,\end{array}$ (4)

where $R$ and Ric denote the curvature tensor and the Ricci tensor,
respectively. If $M$ is para-Sasaki-like Riemannian manifold, then we have the
following identities:

$\begin{array}[]{lll}\phi\nabla_{x}\xi=\nabla_{\phi x}\xi=x-\eta(x)\xi,\\\
\end{array}$ (5)

##  3  Para-Sasaki-like Riemannian Manifolds with Generalized Symmetric Metric
Connection

In this section we consider a generalized symmetric metric connection on an
apapR manifold.

Let $(M,\phi,\xi,\eta,g)$ be an apapR manifold. We define a generalized
symmetric metric connection $\overline{\nabla}$ on $M$ by

$\overline{\nabla}_{x}y=\nabla_{x}y+\alpha\\{g(x,y)\xi-\eta(y)x\\}+\beta\\{g(\varphi
x,y)\xi-\eta(y)\varphi x\\},$ (6)

where $\nabla$ is a Levi-Civita connection of $g$. By the latter equality the
torsion tensor $T$ of $M$ corresponding to the connection $\overline{\nabla}$
is given by

$\begin{array}[]{lll}T(x,y)&=&\overline{\nabla}_{x}y-\overline{\nabla}_{y}x-[x,y]\\\
&=&\alpha\\{\eta(x)y-\eta(y)x\\}+\beta\\{\eta(x)\varphi y-\eta(y)\varphi
x\\},\end{array}$ (7)

for any vector field $x,y$ on $M$, where $\alpha$ and $\beta$ are smooth
functions. The connection $\overline{\nabla}$ is said to be generalized
symmetric connection. If $(\alpha,\beta)=(1,0)$ or $(\alpha,\beta)=(0,1)$,
then the generalized symmetric connection is reduced to a semi-symmetric
connection and quarter-symmetric connection, respectively. If there is a
Riemannian metric $g$ on $M$ such that the connection $\overline{\nabla}$
satisfies the condition

$(\overline{\nabla}_{x}g)(y,z)=0,$ (8)

for all $x,y,z\in\chi(M)$, then $\overline{\nabla}$ is called a generalized
symmetric metric connection, otherwise it is called a generalized symmetric
nonmetric connection.

Now we will show the existence of the generalized symmetric metric connection
$\overline{\nabla}$ on an apapR manifold $(M,\phi,\xi,\eta,g)$.

###### Theorem 1

There exists a unique linear connection $\overline{\nabla}$ satisfying (7) and
(8) on an apapR manifold $(M,\phi,\xi,\eta,g)$.

Proof Suppose that $\overline{\nabla}$ is a linear connection determined by

$\overline{\nabla}_{x}y=\nabla_{x}y+H(x,y),$ (9)

for any vector field $x$ and $y$, where $H$ is a tensor of type $(1,2)$. Now,
we will specify the tensor field $H$ such that $\overline{\nabla}$ satisfies
the relations (7) and (8). By using the definition of torsion tensor and (9),
we obtain

$T(x,y)=H(x,y)-H(y,x),$ (10)

for all $x,y\in\chi(M)$. The latter equality leads to

$g(T(x,y),z)=g(H(x,y),z)-g(H(y,x),z).$ (11)

The equation (8) yields to

$0=xg(y,z)-g(\overline{\nabla}_{x}y,z)-g(\overline{\nabla}_{x}z,y)=g(H(x,y),z)+g(H(x,z),y).$

Then, the latter equality leads to

$g(H(x,y),z)=-g(H(x,z),y).$ (12)

By virtue of (11) and (12) we can easily calculate the following:

$g(T(x,y),z)+g(T(z,x),y)+g(T(z,y),x)=2g(H(x,y),z).$ (13)

By using (7) and (13) we get

$\begin{array}[]{lll}2g(H(x,y),z)&=&g(T(x,y),z)+g(T(z,x),y)+g(T(z,y),x)\\\
&=&2g(\alpha(g(x,y)\xi-\eta(y)x)+\beta(g(\varphi x,y)\xi-\eta(y)\varphi
x),z).\end{array}$ (14)

Hence, (14) implies

$H(x,y)=\alpha\\{g(x,y)\xi-\eta(y)x\\}+\beta\\{g(\varphi
x,y)\xi-\eta(y)\varphi x\\}.$ (15)

The latter equality leads to Equation (6). $\square$

By means of (3), (4) and (5), we obtain the following proposition.

###### Proposition 2

Let $M$ be a para-Sasaki-like Riemannian manifold with the generalized
symmetric metric connection. We have the following relations:

$\begin{array}[]{lll}\overline{\nabla}_{x}\varphi&=&(\beta-1)\\{g(\varphi
x,\varphi y)\xi+\eta(y)\varphi^{2}(x)\\}+\alpha\\{g(x,\varphi
y)\xi+\eta(y)\varphi x\\},\\\ \overline{\nabla}_{x}\xi&=&(1-\beta)\varphi
x-\alpha\varphi^{2}x,\\\ (\overline{\nabla}_{x}\eta)y&=&(1-\beta)g(\varphi
x,y)-\alpha g(x,y)+\eta(x)\eta(y),\\\ \end{array}$ (16)

for any $x,y\in\chi(M)$.

###### Proposition 3

Let $(M,g,\phi,\xi,\eta)$ be a $(2n+1)-$dimensional para-Sasaki-like
Riemannian manifold with the generalized symmetric metric connection. The
relation between the curvature tensors of the generalized symmetric metric
connection $\overline{\nabla}$ and the Levi-Civita connection $\nabla$ is
given by the following formulas:

$\begin{array}[]{lll}\overline{R}(x,y)z&=&R(x,y)z+x(\alpha)g(y,z)\xi+(\alpha-\alpha\beta)g(y,z)\varphi
x\\\ &&-x(\alpha)\eta(z)y+x(\beta)g(\varphi
y,z)\xi+(2\beta-\beta^{2})g(\varphi y,z)\varphi x\\\ &&-x(\beta)\eta(z)\varphi
y+(\alpha^{2}+\beta)g(y,z)\eta(x)\xi+\alpha\beta g(\varphi y,z)\eta(x)\xi\\\
&&-\alpha^{2}g(y,z)x+(\alpha^{2}+\beta)\eta(y)\eta(z)x+(\alpha-\alpha\beta)g(\varphi
y,z)x\\\ &&+\alpha\beta\eta(y)\eta(z)\varphi
x-y(\alpha)g(x,z)\xi+y(\alpha)\eta(z)x\\\ &&-y(\beta)g(\varphi
x,z)\xi+(\beta^{2}-2\beta)g(\varphi x,z)\varphi y+y(\beta)\eta(z)\varphi x\\\
&&-(\alpha^{2}+\beta)\eta(y)g(x,z)\xi-\alpha\beta g(\varphi
x,z)\eta(y)\xi+\alpha^{2}g(x,z)y\\\
&&-(\alpha^{2}+\beta)\eta(x)\eta(z)y+(\alpha\beta-\alpha)g(\varphi x,z)y\\\
&&-\alpha\beta\eta(x)\eta(z)\varphi y+(\alpha\beta-\alpha)g(x,z)\varphi y.\\\
\end{array}$ (17)

Proof The curvature tensor $\overline{R}$ of the generalized symmetric metric
connection $\overline{\nabla}$ on $M$ is defined by

$\overline{R}(x,y)z=\overline{\nabla}_{x}\overline{\nabla}_{y}z-\overline{\nabla}_{y}\overline{\nabla}_{x}z-\overline{\nabla}_{[x,y]}z.$
(18)

Substituting (6) in the last equation and using Proposition 1, through (1),
(2), (3), we complete the proof by direct computation. $\square$

If $\alpha,\beta$ are constant functions, then the curvature tensor
$\overline{R}$ is given by

$\begin{array}[]{lll}\overline{R}(x,y)z&=&R(x,y)z+(\alpha-\alpha\beta)g(y,z)\varphi
x+(2\beta-\beta^{2})g(\varphi y,z)\varphi x\\\
&&+(\alpha^{2}+\beta)g(y,z)\eta(x)\xi+\alpha\beta g(\varphi y,z)\eta(x)\xi\\\
&&-\alpha^{2}g(y,z)x+(\alpha^{2}+\beta)\eta(y)\eta(z)x+(\alpha-\alpha\beta)g(\varphi
y,z)x\\\ &&+\alpha\beta\eta(y)\eta(z)\varphi x+(\beta^{2}-2\beta)g(\varphi
x,z)\varphi y\\\ &&-(\alpha^{2}+\beta)\eta(y)g(x,z)\xi-\alpha\beta g(\varphi
x,z)\eta(y)\xi+\alpha^{2}g(x,z)y\\\
&&-(\alpha^{2}+\beta)\eta(x)\eta(z)y+(\alpha\beta-\alpha)g(\varphi x,z)y\\\
&&+(\alpha\beta-\alpha)g(x,z)\varphi y-\alpha\beta\eta(x)\eta(z)\varphi y.\\\
\end{array}$ (19)

Setting $z=\xi$ in the above equality and using (1), (2) and (4), we get

$\begin{array}[]{lll}\overline{R}(x,y)\xi&=&\alpha\eta(y)\varphi
x+(\beta-1)\eta(y)x-(\beta-1)\eta(x)y-\alpha\eta(x)\varphi y.\\\ \end{array}$
(20)

Plugging $x=\xi$ in (20) and using (1) we obtain the following equality:

$\begin{array}[]{lll}\overline{R}(\xi,y)\xi&=&(1-\beta)\varphi^{2}y-\alpha\varphi(y).\\\
\end{array}$ (21)

Let $\\{\xi,e_{1},\ldots,e_{2n}\\}$ be an orthonormal basis of the tangent
space at each point of the manifold $M$. If
$\overline{R}(x,y,z,w)=g(\overline{R}(x,y)z,w)$, then the Ricci tensor
$\overline{\text{Ric}}$ and the scalar curvature $\overline{\text{scal}}$ with
respect to generalized symmetric metric connection $\overline{\nabla}$ are
presented by

$\overline{\mathrm{Ric}}(x,y)=\sum_{i=0}^{2n}\overline{R}(e_{i},x,y,e_{i})$

and

$\overline{\mathrm{scal}}=\sum_{i=0}^{2n}\overline{\mathrm{Ric}}(e_{i},e_{i}),$
(22)

respectively.

###### Theorem 4

Let $(M,g,\phi,\xi,\eta)$ be a $(2n+1)-$dimensional para-Sasaki-like
Riemannian manifold with the generalized symmetric metric connection. Then,
there exists the following relationship between the Ricci curvatures
$\mathrm{Ric}$ and $\overline{\mathrm{Ric}}$, respectively:

$\begin{array}[]{lll}\overline{\mathrm{Ric}}(x,y)&=&\mathrm{Ric}(x,y)+[\beta^{2}-\beta+(1-2n)\alpha^{2}]g(x,y)\\\
&&+[(2n+1)\beta-\beta^{2}+(2n-1)\alpha^{2}]\eta(x)\eta(y)\\\
&&+[(\alpha\beta-\alpha)(2-2n)+\alpha]g(x,\varphi y)\end{array}$ (23)

Proof Taking into account (3) and (19), we immediately obtain the Ricci tensor
$\overline{\mathrm{Ric}}$. $\square$

Using (22) and (23), we obtain the following theorem.

###### Theorem 5

Let $(M,g,\phi,\xi,\eta)$ be a $(2n+1)-$dimensional para-Sasaki-like
Riemannian manifold with the generalized symmetric metric connection. The
scalar curvatures $\mathrm{scal}$ and $\overline{\mathrm{scal}}$ satisfy the
following relation:

$\begin{array}[]{lll}\overline{\mathrm{scal}}&=&\mathrm{scal}+2n[\beta^{2}+(1-2n)\alpha^{2}].\\\
\end{array}$ (24)

###### Theorem 6

The Hessian, the divergence and the Laplace operators with respect to the
generalized symmetric metric connection $\overline{\nabla}$ satisfy the
following relations:

$\displaystyle\overline{\mathrm{div}}\ (x)$ $\displaystyle=$
$\displaystyle\mathrm{div}\ x-2n\alpha\ \eta(x)$ (25)
$\displaystyle\overline{\mathrm{Hess}}f(x,y)$ $\displaystyle=$
$\displaystyle\mathrm{Hess}f(x,y)+\alpha(xf)\eta(y)-\alpha(\xi f)g(x,y)$ (27)
$\displaystyle+\beta(\varphi x)f\eta(y)-\beta(\xi f)g(\varphi x,y)$
$\displaystyle\overline{\Delta}f$ $\displaystyle=$ $\displaystyle\Delta
f-2n\alpha\ \xi f$ (28)

Proof Let $\\{\xi,e_{1},\ldots,e_{2n}\\}$ be an orthonormal basis of $T_{p}M$,
$p\in M$. By taking the trace of $\overline{\nabla}x$, we get

$\overline{\mathrm{div}}\
(x)=\displaystyle\sum_{i=0}^{2n}g(\overline{\nabla}_{e_{i}}x,e_{i}).$

The latter equality together with (6) completes the first part of proof.

By definition we have

$\overline{\mathrm{Hess}}f(x,y)=g(\overline{\nabla}_{x}grad\ f,y).$

We know that $g(grad\ f,x)=x(f)$. By virtue of (6), we prove the second part.

Substituting $x=grad\ f$ in (25) and using (6), we obtain the Laplace operator
$\overline{\Delta}f$.

$\square$

##  4 Para-Ricci-like solitons on Para-Sasaki-like Riemannian Manifolds With
Generalized Symmetric Metric Connection

It is known from [3] that the apapR manifold $(M,\varphi,\xi,\eta,g)$ is
called para-Einstein-like with constants $(a,b,c)$ if its Ricci tensor
$\mathrm{Ric}$ satisfies the following formula:

$\mathrm{Ric}=a\ g+b\ \widetilde{g}+c\ \eta\otimes\eta$ (29)

where $a,b,c$ are constants. In particular, if $b=0$ and $b=c=0$, then the
manifold is called an $\eta-$Einstein manifold and an Einstein manifold,
respectively. If $a,b,c$ are functions on $M$, the manifold $M$ is called
almost para-Einstein-like, almost $\eta-$Einstein-like or an almost Einstein
manifold, respectively.

###### Theorem 7

If the para-Sasaki-like manifold $(M,\varphi,\xi,\eta,g)$ is a para-Einstein-
like manifold with constants $(a,b,c)$, then the apapR manifold
$(M,\varphi,\xi,\eta,g)$ with the generalized symmetric metric connection is
also a para-Einstein-like manifold with constants $(\lambda,\mu,\nu)$
determined by

$\begin{array}[]{lll}\lambda&=&\beta^{2}-\beta+(1-2n)\alpha^{2}+a,\\\
\mu&=&(2-2n)(\alpha\beta-\alpha)+\alpha+b\\\
\nu&=&(2n-1)\alpha^{2}+(2n-2)\alpha\beta+(1-2n)\alpha+(2n+1)\beta-\beta^{2}+c\end{array}$
(30)

Proof Since the para-Sasaki-like manifold $(M,\varphi,\xi,\eta,g)$ is a para-
Einstein-like manifold with constants $(a,b,c)$, (29) is valid. Substituting
(23) in (29), the theorem is proved by straightforward calculations.

$\square$

The concept of the para-Ricci-like soliton with potential $\xi$ is introduced
in [3]. An apapR manifold $(M,\varphi,\xi,\eta,g)$ admits a para-Ricci-like
soliton with potential $\xi$ and constants $(\lambda,\mu,\nu)$ if the
following identity is valid:

$\frac{1}{2}\mathcal{L}_{\xi}g+\mathrm{Ric}+\lambda
g+\mu\widetilde{g}+\nu\eta\otimes\eta=0,$ (31)

where $\mathcal{L}$ and $\mathrm{Ric}$ denote the Lie derivative and the Ricci
tensor, respectively. The Lie derivative of $g$ along $v$ is defined by

$(\mathcal{L}_{v}g)(x,y)=g(\nabla_{x}v,y)+g(x,\nabla_{y}v).$ (32)

If $\mu=0$ or $\mu=\nu=0$, then $M$ admits an $\eta-$Ricci soliton or a Ricci
soliton, respectively. If $\lambda,\mu,\nu$ are functions on $M$, then the
soliton is said to be almost para-Ricci-like soliton, almost $\eta-$Ricci
soliton or almost Ricci soliton. Moreover, this notion can be generalized for
any potential. We say that $(M,\varphi,\xi,\eta,g)$ admits a para-Ricci-like
soliton with any potential vector field $v$ if the following condition is
valid for a triplet of constants $(\lambda,\mu,\nu)$:

$\frac{1}{2}\mathcal{L}_{v}g+\mathrm{Ric}+\lambda
g+\mu\widetilde{g}+\nu\eta\otimes\eta=0.$ (33)

For a para-Sasaki-like manifold $(M,\varphi,\xi,\eta,g)$, we obtain

$\mathcal{L}_{\xi}g(x,y)=g(\nabla_{x}\xi,y)+g(x,\nabla_{y}\xi)=2g(x,\phi y).$
(34)

By using (32) and (34) we get

$\overline{\mathcal{L}}_{\xi}g(x,y)=\mathcal{L}_{\xi}g(x,y)-2\alpha g(\varphi
x,\varphi y)-2\beta g(\varphi x,y).$ (35)

###### Theorem 8

If the para-Sasaki-like manifold $(M,\varphi,\xi,\eta,g)$ admits a para-Ricci-
like soliton with potential $\xi$ and constants $(a,b,c)$, then
$(M,\varphi,\xi,\eta,g)$ with the generalized symmetric metric conneciton
$\overline{\nabla}$ also admits a para-Ricci-like soliton with constants
$(\lambda,\mu,\nu)$ determined by

$\begin{array}[]{lll}\lambda&=&(2n-1)\alpha^{2}-\beta^{2}+\alpha+\beta+a\\\
\mu&=&2(n-1)\alpha\beta+(1-2n)\alpha+\beta+b\\\
\nu&=&\beta^{2}+(1-2n)\alpha^{2}+(\alpha-\alpha\beta)(2n-2)-(2n+2)\beta+c\end{array}$
(36)

Proof Let the para-Sasaki-like manifold $(M,\varphi,\xi,\eta,g)$ admit a para-
Ricci-like soliton with potential $\xi$ and constants $(a,b,c)$. Then, the
relation (31) is valid. If (35) and (23) are substituted in (31), we infer the
assertion.

$\square$

###### Theorem 9

Let $(M,\varphi,\xi,\eta,g)$ be a para-Sasaki-like Riemannian manifold of
dimension $(2n+1)$ and let it admit a para-Ricci-like soliton with constants
$(a,b,c)$ whose potential vector field $v$ satisfies the condition $v=k\xi$,
i.e., it is pointwise collinear with the Reeb vector field $\xi$, where $k$ is
a differentiable function on $M$. Then, $M$ also admits a para-Ricci-like
soliton with respect to the generalized symmetric metric conneciton
$\overline{\nabla}$ with constants $(\lambda,\mu,\nu)$ determined by

$\begin{array}[]{l}\lambda=a+\alpha b-\beta^{2}+\beta+(2n-1)\alpha^{2}\\\
\mu=b+\beta b+(\alpha-\alpha\beta)(2n-2)\\\ \nu=\alpha
b-(2n+1)\beta+\beta^{2}+(1-2n)\alpha^{2}+c-\beta
b+(\alpha\beta-\alpha)(2n-2)\end{array}$ (37)

Proof Bearing in mind (6) and (32), we compute Lie derivative of $g$ along
$v=k\xi$ corresponding the generalized symmetric metric conneciton
$\overline{\nabla}$ as follows

$\overline{\mathcal{L}}_{v}g(x,y)=\mathcal{L}_{v}g(x,y)+2\alpha
k\eta(x)\eta(y)-2\alpha kg(x,y)-2\beta g(x,\varphi y).$ (38)

Withdrawing $\mathcal{L}_{v}g(x,y)$ in the latter equality and using (23) and
(33), we complete the proof. $\square$

Note that if $(M,\varphi,\xi,\eta,g)$ admits a para-Ricci-like soliton with
any potential $v$, it may not admit a para-Ricci-like soliton with respect to
a generalized symmetric metric connection.

##  5 Examples

In this section, we construct two examples of the para-Sasaki-like manifold
admitting a generalized symmetric metric connection and after that validate
our results.

###  5.1 Example 1

In [6], a $3-$dimensional Lie group $L$ is considered with a global basis
$\\{\xi=e_{0},e_{1},e_{2}\\}$ of the left invariant vector fields on $L$ such
that the commutators of its associated Lie algebra are determined as follows

$[e_{0},e_{1}]=-e_{2},\hskip 14.22636pt[e_{0},e_{2}]=-e_{1},\hskip
14.22636pt[e_{1},e_{2}]=0.$ (39)

$L$ is endowed with an apapR structure $(\varphi,\xi,\eta,g)$ by

$\begin{array}[]{ll}\varphi(e_{0})=0,\hskip
14.22636pt\varphi(e_{1})=e_{2},\hskip 14.22636pt\varphi(e_{2})=e_{1},\hskip
14.22636pt\eta(e_{0})=1,\\\ g(e_{0},e_{0})=g(e_{1},e_{1})=g(e_{2},e_{2})=1,\\\
g(e_{i},e_{j})=0\text{ for }i,j\in\\{0,1,2\\},\ \ \ i\neq j.\end{array}$ (40)

In the same paper, it is obtained that the basic components of the Levi-Civita
connection $\nabla$, $R_{ijkl}=R(e_{i},e_{j},e_{k},e_{l})$ and
$\text{Ric}_{ij}=\text{Ric}(e_{i},e_{j})$ are calculated by

$\begin{array}[]{l}\nabla_{e_{1}}e_{0}=e_{2},\hskip
14.22636pt\nabla_{e_{2}}e_{0}=e_{1},\hskip
14.22636pt\nabla_{e_{1}}e_{2}=\nabla_{e_{2}}e_{1}=-e_{0};\\\
R_{1221}=-R_{1001}=-R_{2002}=1,\hskip 14.22636pt\text{
Ric}_{00}=-2.\end{array}$ (41)

It is proved that the constructed manifold $L$ is a para-Einstein-like para-
Sasaki-like Riemannian manifold with constants $(a,b,c)=(0,0,-2)$. Moreover,
$L$ is a para-Ricci-like soliton with constants $(0,-1,3)$. The only non-zero
component of $(\mathcal{L}_{\xi}g)(e_{i},e_{j})=(\mathcal{L}_{\xi}g)_{ij}$ is
given by $(\mathcal{L}_{\xi}g)_{12}=2$.

Now, let us consider the generalized symmetric metric connection
$\overline{\nabla}$ on $L$. Taking into account (6) and (41), we compute the
non-zero components of $\overline{\nabla}$ as follows

$\begin{array}[]{l}\overline{\nabla}_{e_{1}}e_{0}=(1-\beta)e_{2}-\alpha
e_{1},\hskip 14.22636pt\overline{\nabla}_{e_{2}}e_{0}=(1-\beta)e_{1}-\alpha
e_{2},\\\
\overline{\nabla}_{e_{1}}e_{2}=\overline{\nabla}_{e_{2}}e_{1}=(\beta-1)e_{0},\hskip
14.22636pt\overline{\nabla}_{e_{1}}e_{1}=\overline{\nabla}_{e_{2}}e_{2}=\alpha
e_{0}.\end{array}$ (42)

By virtue of (19) and (42), the components
$\overline{R}_{ijkl}=\overline{R}(e_{i},e_{j},e_{k},e_{l})=g(\overline{R}(e_{i},e_{j})e_{k},e_{l})$
excluding the well-known their symmetries and antisymmetries are calculated as
follows:

$\begin{array}[]{l}\overline{R}_{0110}=-\overline{R}_{0202}=(\beta-1),\\\
\overline{R}_{0102}=-\alpha\\\
\overline{R}_{1212}=\alpha^{2}-(\beta-1)^{2}\end{array}$ (43)

Using above relations, we compute the nonzero components of the Ricci tensor
as follows:

$\begin{array}[]{lll}\overline{\mathrm{Ric}}_{00}=2(\beta-1),&\overline{\mathrm{Ric}}_{11}=\overline{\mathrm{Ric}}_{22}=\beta(\beta-1)-\alpha^{2},&\overline{\mathrm{Ric}}_{12}=\alpha.\\\
\end{array}$ (44)

Taking into account (44), we obtain
$\overline{\text{scal}}=2(\beta^{2}-\alpha^{2}-1)$.

Therefore, by virtue of (44), it can be easily seen that the manifold
$(L,\phi,\xi,\eta,g)$ admits a para-Einstein-like manifold with regard to the
generalized symmetric metric connection $\overline{\nabla}$ with constants
$(\lambda,\mu,\nu)$ determined by

$\begin{array}[]{lll}\lambda=\beta(\beta-1)-\alpha^{2},&\mu=\alpha,&\nu=\alpha(\alpha-1)-(\beta-1)(\beta-2).\\\
\end{array}$ (45)

The non-zero components of $\mathcal{L}_{\xi}g$ are determined by the
following equalities:

$\begin{array}[]{lll}(\overline{\mathcal{L}}_{\xi}g)_{11}=(\overline{\mathcal{L}}_{\xi}g)_{22}=-2\alpha,&(\overline{\mathcal{L}}_{\xi}g)_{12}=2(1-\beta).\\\
\end{array}$ (46)

Therefore, using (44) and (46), we establish that the constructed manifold
admits a paraRicci-like soliton with regard to the generalized symmetric
metric connection $\overline{\nabla}$ with the following constants:

$\begin{array}[]{lll}\lambda=\alpha^{2}+\alpha+\beta-\beta^{2},&\mu=\beta-\alpha-1,&\nu=(\beta-1)(\beta+3)-\alpha^{2}.\\\
\end{array}$ (47)

In conclusion, the constructed $3-$dimensional example of a para-Sasaki-like
Riemannian manifold $(L,\phi,\xi,\eta,g)$ with the generalized symmetric
metric connection with the results in (45) and (47) support the proven
assertions in Theorem 7 and 8.

###  5.2 Example 2

Let us consider as an explicit example a $5-$dimensional Para-Sasaki-like
manifold $G$ given in [17]. Then, the Lie group $G$ has a basis of left-
invariant vector fields $\\{e_{0},\ldots,e_{4}\\}$ with corresponding Lie
algebra determined as follows

$\begin{array}[]{l}[e_{0},e_{1}]=pe_{2}-e_{3}+qe_{4},\ \ \
[e_{0},e_{2}]=-pe_{1}-qe_{3}-e_{4},\\\ \end{array}$
$\begin{array}[]{l}[e_{0},e_{3}]=-e_{1}+qe_{2}+pe_{4},\ \ \
[e_{0},e_{4}]=-qe_{1}-e_{2}-pe_{3},\\\ \end{array}$

where $p,q\in\mathbb{R}$. The Lie group $G$ is endowed with an almost
paracontact almost paracomplex structure $(\varphi,\xi,\eta,g)$ as follows:

$\begin{array}[]{l}\xi=e_{0},\ \ \ \varphi e_{1}=e_{3},\ \ \ \varphi
e_{2}=e_{4},\ \ \ \varphi e_{3}=e_{1},\ \ \ \varphi_{4}=e_{2},\\\
g(e_{i},e_{i})=1,\ \ \ g(e_{i},e_{j})=0,\ \ \ i,j\in\\{0,1,\ldots,4\\},\ i\neq
j.\end{array}$ (48)

The non-zero components of the Levi-Civita connection are given in the
following way:

$\begin{array}[]{l}\nabla_{e_{0}}e_{1}=pe_{2}+qe_{4},\ \ \
\nabla_{e_{1}}e_{0}=e_{3},\ \ \ \nabla_{e_{0}}e_{2}=-pe_{1}-qe_{3},\ \ \
\nabla_{e_{2}}e_{0}=e_{4},\\\ \nabla_{e_{0}}e_{3}=qe_{2}+pe_{4},\ \ \
\nabla_{e_{3}}e_{0}=e_{1},\ \ \ \nabla_{e_{0}}e_{4}=-qe_{1}-pe_{3},\ \ \
\nabla_{e_{4}}e_{0}=e_{2},\\\
\nabla_{e_{1}}e_{3}=\nabla_{e_{2}}e_{4}=\nabla_{e_{3}}e_{1}=\nabla_{e_{4}}e_{2}=-e_{0}.\\\
\end{array}$ (49)

In [4], it is proved that $(G,\varphi,\xi,\eta,g)$ is a para-Sasaki-like
manifold and
$\eta-$Einstein with constants $(a,b,c)=(0,0,-4)$. Moreover, it admits a para-
Ricci-like soliton with potential $\xi$ constants $(0,-1,5)$. The only non-
zero component of the Ricci tensor $\mathrm{Ric}$ is $\mathrm{Ric}_{00}=-4$.
The non-zero of the components
$R_{ijkl}=R(e_{i},e_{j},e_{k},e_{l})=g(R(e_{i},e_{j})e_{k},e_{l})$ are
determined by the following equalities with the well-known their symmetries
and antisymmetries:

$\begin{array}[]{l}R_{0110}=R_{0220}=R_{0330}=R_{0440}=-1,\\\
R_{1234}=R_{1432}=R_{1331}=R_{2442}=1.\end{array}$ (50)

The non-zero components of $\mathcal{L}_{\xi}g$ are determined by the
following equalities:

$\begin{array}[]{l}(\mathcal{L}_{\xi}g)_{13}=(\mathcal{L}_{\xi}g)_{24}=(\mathcal{L}_{\xi}g)_{31}=(\mathcal{L}_{\xi}g)_{42}=2.\\\
\end{array}$

The non-zero components of the generalized symmetric metric connection
$\overline{\nabla}$ are given in the following way:

$\begin{array}[]{l}\overline{\nabla}_{e_{0}}e_{1}=\nabla_{e_{0}}e_{1}=pe_{2}+qe_{4},\
\ \ \ \ \ \overline{\nabla}_{e_{1}}e_{0}=(1-\beta)e_{3}-\alpha e_{1},\\\
\overline{\nabla}_{e_{0}}e_{2}=\nabla_{e_{0}}e_{2}=-pe_{1}-qe_{3},\ \ \ \
\overline{\nabla}_{e_{2}}e_{0}=(1-\beta)e_{4}-\alpha e_{2},\\\
\overline{\nabla}_{e_{0}}e_{3}=\nabla_{e_{0}}e_{3}=qe_{2}+pe_{4},\ \ \ \ \ \ \
\overline{\nabla}_{e_{3}}e_{0}=(1-\beta)e_{1}-\alpha e_{3},\\\
\overline{\nabla}_{e_{0}}e_{4}=\nabla_{e_{0}}e_{4}=-qe_{1}-pe_{3},\ \ \ \ \
\overline{\nabla}_{e_{4}}e_{0}=(1-\beta)e_{2}-\alpha e_{4},\\\
\overline{\nabla}_{e_{1}}e_{3}=\overline{\nabla}_{e_{3}}e_{1}=\overline{\nabla}_{e_{2}}e_{4}=\overline{\nabla}_{e_{4}}e_{2}=(\beta-1)e_{0},\\\
\overline{\nabla}_{e_{i}}e_{i}=\alpha e_{0},\ \ i\neq 0.\end{array}$ (51)

Taking into account (19) and (50), we compute the components
$\overline{R}_{ijkl}=\overline{R}(e_{i},e_{j},e_{k},e_{l})=g(\overline{R}(e_{i},e_{j})e_{k},e_{l})$
excluding the well-known their symmetries and antisymmetries as follows:

$\begin{array}[]{l}\overline{R}_{0101}=-\overline{R}_{0440}=\overline{R}_{0202}=-\overline{R}_{0330}=1-\beta,\\\
\overline{R}_{0103}=\overline{R}_{0402}=-\alpha,\\\
\overline{R}_{1412}=-\overline{R}_{3423}=-\overline{R}_{3441}=-\overline{R}_{2312}=\alpha(\beta-1),\\\
\overline{R}_{3434}=\overline{R}_{2323}=\overline{R}_{1414}=\overline{R}_{1212}=\alpha^{2},\\\
\overline{R}_{1313}=\overline{R}_{2424}=\alpha^{2}-(\beta-1)^{2},\\\
\overline{R}_{1234}=-\overline{R}_{2314}=(\beta-1)^{2}.\end{array}$ (52)

The nonzero components of the Ricci tensor $\overline{Ric}$ are calculated as
follows:

$\begin{array}[]{l}\overline{Ric}_{00}=4(\beta-1),\\\
\overline{Ric}_{11}=\overline{Ric}_{22}=\overline{Ric}_{33}=\overline{Ric}_{44}=\beta^{2}-\beta-3\alpha^{2},\\\
\overline{Ric}_{13}=\overline{Ric}_{24}=3\alpha-2\alpha\beta.\end{array}$ (53)

Hence, by using above relations the scalar curvature is obtained by

$\overline{scal}=4(\beta^{2}-3\alpha^{2}-1).$

Moreover, the non-zero components of the Lie derivative of $g$ along $\xi$
with connection $\overline{\nabla}$ are given by

$\begin{array}[]{l}(\overline{\mathcal{L}}_{\xi}g)_{11}=(\overline{\mathcal{L}}_{\xi}g)_{22}=(\overline{\mathcal{L}}_{\xi}g)_{33}=(\overline{\mathcal{L}}_{\xi}g)_{44}=-2\alpha,\\\
(\overline{\mathcal{L}}_{\xi}g)_{13}=(\overline{\mathcal{L}}_{\xi}g)_{24}=2(1-\beta)\end{array}$
(54)

By virtue of (53) and (54), the considered manifold $(G,\varphi,\xi,\eta,g)$
is para-Einstein-like manifold with constants $(\lambda,\mu,\nu)$ given by

$\begin{array}[]{l}\lambda=\beta^{2}-\beta-3\alpha^{2},\\\
\mu=-2\alpha\beta+3\alpha,\\\
\nu=3\alpha^{2}+2\alpha\beta-3\alpha+5\beta-\beta^{2}-4.\end{array}$ (55)

and it admits a para-Ricci-like soliton with potential $\xi$ with constants
$(\lambda,\mu,\nu)$ determined by

$\begin{array}[]{l}\lambda=3\alpha^{2}+\alpha-\beta^{2}+\beta,\\\
\mu=2\alpha\beta+\beta-3\alpha-1,\\\
\nu=\beta^{2}-3\alpha^{2}-2\alpha\beta-6\beta+2\alpha+5.\end{array}$ (56)

Consequently, the obtained manifold $(G,\phi,\xi,\eta,g)$ with the results in
(55) and (56) support the proven assertions in Theorem 7 and 8.

###### Acknowledgement 10

The first author was supported by Project 22ADP356 of Scientific Research
Projects Commission, Eskişehir Technical University.

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# Quadratic DHOST theories revisited

David Langlois  Université de Paris, CNRS, Astroparticule et Cosmologie,
F-75006 Paris, France  Karim Noui  Institut Denis Poisson, Université de
Tours, Université d’Orléans, CNRS, UMR 7013, 37200 Tours, France  Université
de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France  Hugo
Roussille  Université de Paris, CNRS, Astroparticule et Cosmologie, F-75006
Paris, France  Institut Denis Poisson, Université de Tours, Université
d’Orléans, CNRS, UMR 7013, 37200 Tours, France

###### Abstract

We present a novel and remarkably simple formulation of degenerate higher-
order scalar-tensor (DHOST) theories whose Lagrangian is quadratic in second
derivatives of some scalar field. Using disformal transformations of the
metric, we identify a special “frame” (or metric) for which the Lagrangian of
quadratic DHOST theories reduces to the usual Einstein-Hilbert term plus a few
terms that depend on simple geometric quantities characterizing the uniform
scalar field hypersurfaces. In particular, for quadratic DHOST theories in the
physically interesting class Ia, the Lagrangian simply consists of the
Einstein-Hilbert term plus a term proportional to the three-dimensional scalar
curvature of the uniform scalar field hypersurfaces. The classification of all
quadratic DHOST theories becomes particularly transparent in this geometric
reformulation, which also applies to scalar-tensor theories that are
degenerate only in the unitary gauge.

##  I Introduction

Scalar-tensor theories have attracted considerable attention as alternatives
to General Relativity. As ever more sophisticated models have been considered,
special attention was recently devoted to scalar-tensor Lagrangians that
contain second order derivatives of the scalar field. A crucial requirement
for such theories to be physically relevant is the absence of the so-called
Ostrogradski ghost Ostrogradsky:1850fid , in order to avoid disastrous
instabilities (see Woodard:2015zca ; Ganz:2020skf  and references therein). It
was believed for some time that the absence of an Ostrogradski ghost requires
the Euler-Lagrange equations to be at most second-order, which explains why
the literature was limited, until a few years ago, to the study of Horndeski’s
theories Horndeski:1974wa , rediscovered in the guise of the so-called
galileons Nicolis:2008in ; Deffayet:2011gz ; Kobayashi:2011nu . But the
discovery of viable theories “beyond Horndeski” Zumalacarregui:2013pma ;
Gleyzes:2014dya ; Gleyzes:2014qga  possessing Euler-Lagrange equations of
order higher than two, challenged this preconception.

It was then realised, in Langlois:2015cwa ; Langlois:2015skt , that the
absence of the Ostrogradski ghost is automatically guaranteed in Lagrangians
whose degeneracy entails constraints that eliminate the potentially dangerous
extra scalar degree of freedom, even if the associated Euler-Lagrange
equations are higher-order. This led to the systematic classification of
Degenerate Higher-Order Scalar-Tensor (DHOST) theories Langlois:2015cwa ;
Langlois:2015skt ; Crisostomi:2016tcp ; Crisostomi:2016czh ; Achour:2016rkg ;
BenAchour:2016fzp  (see Langlois:2018dxi  for a recent review).

Within the family of DHOST theories, one can use the correspondence between
different actions that transform into each other via a disformal
transformation of the metric,

$g_{\mu\nu}\longrightarrow\tilde{g}_{\mu\nu}\equiv
A(\phi,X)g_{\mu\nu}+B(\phi,X)\phi_{\mu}\phi_{\nu}\,,$ (1)

where $A$ and $B$ are functions of $\phi$, $\phi_{\mu}\equiv\nabla_{\mu}\phi$
and $X\equiv\nabla_{\mu}\phi\nabla^{\mu}\phi$. Provided this transformation is
invertible, the actions $\tilde{S}[\phi,\tilde{g}_{\mu\nu}]$ and
$S[\phi,g_{\mu\nu}]\equiv\tilde{S}[\phi,A(\phi,X)g_{\mu\nu}+B(\phi,X)\phi_{\mu}\phi_{\nu}]$
are physically equivalent111Note that this is only true in absence of matter.
Indeed, when matter is included (in the form of a field $\psi_{m}$), the total
actions
$S_{tot}[g_{\mu\nu},\phi,\psi_{m}]=S[g_{\mu\nu},\phi]+S_{m}[g_{\mu\nu},\psi_{m},]$
and
$\tilde{S}_{tot}[\tilde{g}_{\mu\nu},\phi,\psi_{m}]=\tilde{S}[\tilde{g}_{\mu\nu},\phi]+S_{m}[\tilde{g}_{\mu\nu},\psi_{m}]$
correspond to different physical theories, since matter is minimally coupled
to the metric $g_{\mu\nu}$ in the former case and to $\tilde{g}_{\mu\nu}$ in
the latter case.. One is thus allowed to pick up one of these disformally
related metrics $\tilde{g}_{\mu\nu}$, or “frame”, to write the corresponding
theory in a convenient way.

In this article, we identify a “frame” where the action of quadratic DHOST
theories takes a remarkably simple form, with a natural geometric
interpretation based on the three-dimensional hypersurface $\Sigma_{\phi}$
where the scalar field $\phi$ is constant. Remarkably, we find that all
quadratic DHOST theories in class Ia (see Achour:2016rkg  for the
classification), the most interesting for physical applications, can always be
mapped, via a disformal transformation, to the very simple form222Notice that
the action (3) is a special case of the so-called spatially covariant gravity
actions introduced in Gao:2014soa  and studied further in Gao:2014fra .

$\displaystyle S[g_{\mu\nu},\phi]=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm
P}^{2}\,{}^{(4)}\\!{R}+\lambda_{1}\,{}^{(3)}\\!{R}\,\right)\,,$ (2)

where $M_{\rm P}$ is the 4-dimensional Planck mass, ${}^{(4)}\\!{R}$ the four-
dimensional Ricci scalar, ${}^{(3)}\\!{R}$ the three-dimensional Ricci scalar
of the hypersurfaces $\Sigma_{\phi}$ and $\lambda_{1}$ an arbitrary function
of $X$ and $\phi$.

More generally, all quadratic DHOST theories can be written in the form

$\displaystyle S[g_{\mu\nu},\phi]=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm
P}^{2}\,{}^{(4)}\\!{R}+\lambda_{1}\,{}^{(3)}\\!{R}+\lambda_{2}\,\varepsilon\,K^{2}+\lambda_{3}\,a^{2}\,\right),$
(3)

where $K$ is the trace of the extrinsic curvature of $\Sigma_{\phi}$ and
$a^{2}\equiv a_{\mu}a^{\mu}$ the square of the “acceleration” vector of the
unit vector normal to $\Sigma_{\phi}$, $\varepsilon$ is the sign333The sign
$\varepsilon$ can be absorbed into a redefinition of $\lambda_{2}$ but we
leave it here for later convenience. of $X$, while $\lambda_{A}$ are arbitrary
functions of $\phi$ and $X$. Interestingly, quadratic U-degenerate theories
introduced in DeFelice:2018mkq , which are degenerate only in the unitary
gauge, where the scalar field is a function of time only, can also be
reformulated in the form (3).

The formulations (2) and (3) can be seen as an analog of the “Einstein frame”
for traditional scalar-tensor theories, where the dynamical structure of the
gravitational theory is governed by the usual Einstein-Hilbert action with a
constant Planck mass. The possibility to reformulate degenerate theories, via
disformal transformations, in this “geometric” frame is the main result of
this paper. We also show how the classification of quadratic DHOST theories
becomes transparent in this new formulation, as well as the reason behind the
instability of several classes of DHOST theories.

Note that three-dimensional quantities based on the uniform scalar field
hypersurfaces have already been used in several earlier works, for instance in
the context of the EFT of inflation Cheung:2007st , of Horava’s gravity and
its extensions Blas:2010hb , in the study of the cosmology of Horndeski and
Beyond Horndeski theories Gleyzes:2013ooa ; Gleyzes:2014rba . The novelty here
is that we combine this three-dimensional formalism with the systematic
exploitation of disformal transformations, in order to simplify the
description of DHOST theories.

The paper is organized as follows. In the next section, we recall the
definition of quadratic DHOST, define the notion of “weakly degenerate”
theories which are degenerate in the unitary gauge (when $\varepsilon=-1$), as
well as the parametrisations that have been introduced in previous works to
describe them. In Section III, we present the new formulation of quadratic
DHOST and weakly degenerate actions. In section IV, we revisit the
classification of DHOST theories in this new perspective. We also show how the
action (3) is related to the very familiar Horndeski action by computing
explicitly the disformal transformation which links them. Finally, we present
a very simple argument based on this new formulation which proves that DHOST
theories in the classes II and III are plagued by instabilities (or do not
propagate gravitational waves). We conclude in section V with some
perspectives. And we also provide some details and additional results in an
Appendix.

##  II Higher-order scalar-tensor theories and degeneracy

We consider scalar-tensor theories with a quadratic dependence on the second
derivatives of the scalar field. Quite generally, their Lagrangian can be
written in the form

$\displaystyle
L=f(\phi,X)\,{}^{(4)}\\!{R}+\sum_{A=1}^{5}\alpha_{A}(\phi,X)\,L_{A}\,,$ (4)

where the elementary quadratic Lagrangians $L_{A}$ are defined by

$\displaystyle L_{1}=\phi_{\mu\nu}\phi^{\mu\nu},\quad L_{2}=\Box\phi^{2},\quad
L_{3}=\phi^{\mu}\phi_{\mu\nu}\phi^{\nu}\Box\phi,\quad
L_{4}=(\phi_{\mu\nu}\phi^{\nu})^{2}\,,\quad
L_{5}=(\phi^{\mu}\phi_{\mu\nu}\phi^{\nu})^{2}\,,$ (5)

where $\phi_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}\phi$. Note that one can add
in the Lagrangian terms of lower order in $\phi_{\mu\nu}$, which can be
written in the form $P(\phi,X)$ (order 0) and $Q(\phi,X)\Box\phi$ (order 1),
but we will ignore them here as they do not affect the degeneracy of the
action.

If the six functions $f$ and $\alpha_{A}$ are arbitrary, one finds generically
an extra scalar degree of freedom which leads in general to an Ostrogradski
instability. To avoid the presence of any extra degree of freedom, one needs
to impose degeneracy conditions, which define the DHOST theories. For
quadratic theories, the degeneracy imposes three conditions, derived in
Langlois:2015cwa ,

$D_{0}(X)=0\,,\qquad D_{1}(X)=0\,,\qquad D_{2}(X)=0\,,$ (6)

where the functions $D_{0}$, $D_{1}$ and $D_{2}$ are given in Appendix C.

Instead of working directly with DHOST theories, it will be convenient at this
stage to consider a larger family of theories, which are degenerate in a
weaker sense as we now explain. Assuming that $\varepsilon={\rm sgn}(X)=-1$,
one can introduce the so-called unitary gauge, where the scalar field is
spatially uniform and thus depends only on time. We will name weakly
degenerate all the theories that are degenerate in the unitary gauge. This
includes obviously the DHOST theories, which are degenerate in any gauge and
thus a fortiori in the unitary gauge, but also theories that are degenerate
only in the unitary gauge, dubbed U-degenerate theories in DeFelice:2018mkq .
Since U-degenerate theories are not DHOST theories, they contain an extra
degree of freedom, but this scalar mode is not propagating in the unitary
gauge, as it satisfies an elliptic partial differential equation.

Weakly degenerate theories have been classified in DeFelice:2018mkq . For
quadratic theories, they satisfy a single degeneracy condition (to be
contrasted with the three conditions (6) obeyed by the quadratic DHOST
theories), which reads Langlois:2015cwa

$D_{0}(X)-XD_{1}(X)+X^{2}D_{2}(X)=0\,.$ (7)

Hence only five out of the six functions in (4) are independent. In
DeFelice:2018mkq , the unitary degeneracy condition (7) was solved by
expressing the five functions $\alpha_{A}$ in terms of $f$ and four
independent functions $\kappa_{1}$, $\kappa_{2}$, $\alpha$ and $\sigma$ as
follows

$\displaystyle\alpha_{1}=\kappa_{1}+\frac{f}{X}\,,\quad\alpha_{2}=\kappa_{2}-\frac{f}{X}\,,\quad\alpha_{3}=\frac{2f-4Xf_{X}}{X^{2}}+2\sigma\kappa_{1}+2\left(3\sigma-\frac{1}{X}\right)\kappa_{2}\,,$
(8)
$\displaystyle\alpha_{4}=\alpha+2\frac{X(f_{X}-\kappa_{1})-f}{X^{2}}\,,\quad\alpha_{5}=\frac{2f_{X}-X\alpha}{X^{2}}+\kappa_{1}\left(\frac{1}{X^{2}}+3\sigma^{2}-\frac{2\sigma}{X}\right)+\kappa_{2}\left(3\sigma-\frac{1}{X}\right)^{2}.$

Quadratic DHOST theories can be inferred from weakly degenerate theories by
imposing two more additional conditions (say $D_{0}=0$ and $D_{1}=0$) and
therefore can be parametrised in terms of three independent functions only.
Quadratic DHOST theories contain several subclasses and the explicit form of
their parametrisation depends on the specific subclass considered
Langlois:2015cwa .

##  III Geometric formulation

###  III.1 Geometric Lagrangian

As emphasised in the introduction, the scalar field $\phi$ naturally induces a
preferred slicing of spacetime, which we are going to exploit. Let us first
introduce various geometric tensors associated with this slicing. The
intrinsic geometry of any constant $\phi$ hypersurface $\Sigma_{\phi}$ is
characterized by the three-dimensional induced metric

$\displaystyle h_{\mu\nu}\equiv
g_{\mu\nu}-\varepsilon\,n_{\mu}n_{\nu}\,,\qquad
n_{\mu}\equiv\frac{\phi_{\mu}}{\sqrt{|X|}}\,,$ (9)

where $n_{\mu}$ is the unit vector orthogonal to $\Sigma_{\phi}$. One can also
introduce the Riemann tensor ${}^{(3)}\\!{R}_{\mu\nu\rho\sigma}$ associated
with $h_{\mu\nu}$, the extrinsic curvature tensor and the “acceleration”
vector, the components of the latter being respectively given by

$\displaystyle K_{\mu\nu}\equiv
h_{\mu}^{\alpha}\,h_{\nu}^{\beta}\,\nabla_{\alpha}n_{\beta}\,,\qquad
a_{\mu}\equiv n^{\nu}\nabla_{\nu}n_{\mu}\,.$ (10)

By expressing these quantities explicitly in terms of the scalar field and of
its derivatives, one notes that both $K_{\mu\nu}$ and $a^{\mu}$ are linear in
$\phi_{\mu\nu}$ as shown in (34). This suggests to rewrite the quadratic
Lagrangian in terms of the square of these quantities. In the following, we
will therefore examine theories whose action reads

$\displaystyle S[g_{\mu\nu},\phi]=\int d^{4}x\sqrt{-g}\left(\frac{M_{\rm
P}^{2}}{2}\,{}^{(4)}\\!{R}+L_{\phi}\,\right)\,,$ (11)

with a Lagrangian term $L_{\phi}$ of the form

$L_{\phi}=\lambda_{1}\,{}^{(3)}\\!{R}+\lambda_{2}\,\varepsilon\,K^{2}+\lambda_{3}\,a^{2}+\lambda_{4}\,\varepsilon\,K_{\mu\nu}K^{\mu\nu}\,,$
(12)

where $\lambda_{A}$ are arbitrary functions of $\phi$ and $X$. We have also
included a dependence on the three-dimensional scalar curvature of
$\Sigma_{\phi}$.

It turns out that using only the first three terms, i.e. choosing
$\lambda_{4}=0$, will be sufficient for our purpose as we will explain later.
The total action is thus

$\displaystyle S[g_{\mu\nu},\phi]=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm
P}^{2}\,{}^{(4)}\\!{R}+\lambda_{1}\,{}^{(3)}\\!{R}+\lambda_{2}\,\varepsilon\,K^{2}+\lambda_{3}\,a^{2}\,\right)\,,$
(13)

which, using the relations in Appendix A, yields a quadratic action of the
form (4), plus lower order terms also given in Appendix A which we do not
discuss here, with the coefficients

$\displaystyle f=\frac{M_{\rm
P}^{2}}{2}+\lambda_{1}\,,\quad\alpha_{1}=\frac{\lambda_{1}}{X}\,,\quad\alpha_{2}=\frac{\lambda_{2}-\lambda_{1}}{X}\,,$
$\displaystyle\alpha_{3}=\frac{2\lambda_{1}-4X\lambda_{1X}-2\lambda_{2}}{X^{2}}\,,\quad\alpha_{4}=\frac{4X\lambda_{1X}-2\lambda_{1}+\lambda_{3}}{X^{2}}\,,\quad\alpha_{5}=\frac{\lambda_{2}-\lambda_{3}}{X^{3}}\,.$
(14)

It is immediate to check that this quadratic Lagangian is indeed weakly
degenerate, as it is of the form (8) with

$f=\frac{M_{\rm P}^{2}}{2}+\lambda_{1}\,,\quad\kappa_{1}=-\frac{M_{\rm
P}^{2}}{2X}\,,\quad\kappa_{2}=\frac{M_{\rm
P}^{2}+2\lambda_{2}}{2X}\,,\quad\sigma=0\,,\quad\alpha=\frac{\lambda_{3}+2X{\lambda_{1X}}}{X^{2}}\,.$
(15)

###  III.2 Disformal transformations

We now perform a disformal transformation (1) of the action (13), using the
formulas derived in Achour:2016rkg  and recalled in Appendix D. The quadratic
part of the new action is still weakly degenerate, i.e. of the form (8) with
the coefficients

$\displaystyle\sigma$
$\displaystyle=\frac{A_{X}}{A},\quad\kappa_{1}=-\frac{M_{\rm
P}^{2}}{2X}\frac{A^{3/2}}{\sqrt{A+BX}},\quad\kappa_{2}=\frac{M_{\rm
P}^{2}+2\lambda_{2}}{2X}\frac{A^{3/2}}{\sqrt{A+BX}},$ (16) $\displaystyle\quad
f$ $\displaystyle=\frac{M_{\rm
P}^{2}+2\lambda_{1}}{2}\sqrt{A(A+BX)},\quad\alpha={\cal
A}(\lambda_{1},\lambda_{3},A,B)\,,$

where we do not write explicity $\cal A$ as it is rather cumbersome (but it
can be deduced from the expression (17) given below444Indeed, the expression
of $\cal A$ can be obtained directly from (17) which gives
$\alpha=\frac{M^{2}A^{2}(3\kappa_{1}(X\sigma-1)-2X\kappa_{1X})^{2}}{2X^{3}\kappa_{1}^{3}}\lambda_{3}+\frac{2f\sigma(2+X\sigma)-2f_{X}(1+4X\sigma)}{X}\,.$
Then, one substitutes $f$, $\kappa_{1}$ and $\sigma$ by their expressions (16)
in terms of $A$, $B$ and $\lambda_{1}$ to get $\cal A$ explicitly.).
Interestingly, the new theory (16) is parametrised by five independent
functions, as many functions as required to span the whole family of weakly
degenerate theories.

Conversely, given a weakly degenerate theory defined by the set of functions
$(f,\kappa_{1},\kappa_{2},\sigma,\alpha)$, one can successively invert the
relations (16) and determine $A$ from $\sigma$, $B$ from $\kappa_{1}$,
$\lambda_{2}$ from $\kappa_{2}$ and $\lambda_{1}$ from $f$. The last relation
yields $\lambda_{3}$, which reads

$\displaystyle\lambda_{3}=\frac{2(X\alpha-2f\sigma(2+X\sigma)+2f_{X}(1+4X\sigma))X^{2}\kappa_{1}^{3}}{M^{2}A^{2}(3\kappa_{1}(X\sigma-1)-2X\kappa_{1X})^{2}}\,.$
(17)

This proves that any weakly degenerate action can be obtained from a disformal
transformation of the action (13). As a conclusion, (13) provides us with a
complete parametrization, up to disformal transformations, of quadratic weakly
degenerate theories.

Since the family of weakly degenerate theories is parametrised by five free
functions and that disformal transformations depend on two free functions,
only three functions are needed to parametrise the geometric frame555This is
the reason why we chose $\lambda_{4}=0$ in (12). Note that one could have made
different choices to reduce the number of free functions, e.g. one could
impose $\lambda_{2}=0$, $\lambda_{1}=0$ or another linear relation between the
four functions $\lambda_{A}$. However, one would lose the remarkable
simplicity of the geometric frame, specially in class I (18)..

Finally, let us note that, although our intuitive reasoning was based on the
unitary gauge, which implicitly assumes that the hypersurfaces $\Sigma_{\phi}$
are spacelike, i.e. $\varepsilon=-1$, the relation between the actions (4)
with coefficients (8) and (13) can be obviously extended to the case
$\varepsilon=+1$ even if they can no longer be interpreted as weakly
degenerate actions.

##  IV Geometric formulation of DHOST theories

All the subclasses of DHOST theories are stable under disformal
transformations, as shown in Achour:2016rkg . This implies that once the DHOST
theories in the geometric frame have been classified, the classification can
immediately be extended to the whole family of DHOST theories, as they are all
generated by disformal transformations from the geometric frame actions.

###  IV.1 DHOST theories in class I

Class I is characterized by the relation $\alpha_{1}=-\alpha_{2}$, which is
equivalent to $\lambda_{2}=0$ according to (14). Under this assumption, the
first degeneracy condition $D_{0}=0$ is automatically satisfied and the two
other conditions, remarkably, both reduce to $\lambda_{3}=0$ (see Appendix C).
DHOST theories in class I, when expressed in the geometric frame, are thus of
the very simple form

$\displaystyle S_{\rm I}=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm
P}^{2}\,{{}^{(4)}\\!{R}}+\lambda_{1}\,{}^{(3)}\\!{R}\right)\qquad{\rm(class\
I)}\,.$ (18)

Within the class I, one finds the subclass Ib characterized by
$\alpha_{1}=f/X$. It is then clear from the first relation in (14) that this
subclass corresponds to the action (18) with $M_{\rm P}=0$, i.e. a pure three-
dimensional curvature term, which does not contain tensor modes Achour:2016rkg
.

Let us discuss further the subclass Ia, which is the most interesting from a
physical point of view. The simplest theory in the geometric frame is
obviously General Relativity, with $\lambda_{1}=0$. Via disformal
transformations, it generates a family of DHOST theories parametrised by two
functions, $A$ and $B$. Let us stress that the speed $c_{g}$ of gravitational
waves is modified via a disformal transformation as the causal structure is
modified. From an initial value $c_{g}^{2}=1+2\lambda_{1}/M_{\rm P}^{2}$ in
the geometric frame666The quantity $c_{g}^{2}$ is easily obtained from the
coefficients of the time derivatives and spatial gradients of the tensor modes
in the action. In practice, using the Gauss-Codazzi identity (36), we express
${}^{(4)}\\!{R}$ in terms of ${}^{(3)}\\!{R}$ and $K_{\mu\nu}$ so that the
action (18) becomes  $\displaystyle S_{\rm I}=\frac{1}{2}M_{\rm P}^{2}\int
d^{4}x\sqrt{-g}\left[\left(1+2\lambda_{1}/M_{\rm
P}^{2}\right)\,{{}^{(3)}\\!{R}}+\varepsilon\left(K^{2}-K_{\mu\nu}K^{\mu\nu}\right)\right]\,,$
(19) and then we obtain $c_{g}^{2}$ as the ratio of the coefficient of
${}^{(3)}\\!{R}$ with the coefficient of $-\varepsilon K_{\mu\nu}K^{\mu\nu}$.,
one gets after disformal transformation

$c_{g}^{2}=\left(1+\frac{BX}{A}\right)\left(1+\frac{2\lambda_{1}}{M_{\rm
P}^{2}}\right)\,.$ (20)

As expected, only conformal transformations, i.e. with $B=0$, leave $c_{g}$
invariant. Consequently, theories Ia such that $c_{g}=c$ (for any solution)
can be obtained from a theory with arbitrary $\lambda_{1}$ in the geometric
frame (for which $c_{g}\neq c$) via a disformal transformation that
compensates the initial detuning of $c_{g}$ from $c$ so that the final DHOST
theory verifies $c_{g}=c$ (which is equivalent to the condition $\alpha_{1}=0$
in the original DHOST formulation Langlois:2017dyl ; Crisostomi:2017lbg ;
deRham:2016wji ).

Note that the DHOST theories that can play the role of dark energy while
satisfying both $c_{g}=c$ and the GW decay constraint Creminelli:2018xsv , as
suggested by the GW170817 observation TheLIGOScientific:2017qsa , correspond
to the theories generated via conformal transformations ($B=0$) from General
Relativity in the geometric frame, i.e. with $\lambda_{1}=0$. The cosmology of
such theories has been explored in Frusciante:2018tvu . However, since the
LIGO-Virgo measurements probe wavelengths many orders of magnitude smaller
than cosmological scales, these constraints do not necessarily apply on
cosmological scales (see e.g. deRham:2018red ), leaving the other DHOST
theories still relevant for cosmology Boumaza:2020klg .

In addition to the geometric frame introduced in the present work, another
convenient frame for the theories Ia, is the “Horndeski frame” where the
corresponding action is of the Horndeski form (up to lower order terms), i.e.

$S_{H}[g_{\mu\nu},\phi]=\int
d^{4}x\sqrt{-g}\left(F\,{}^{(4)}\\!{R}+2F_{X}(L_{1}-L_{2})\right)\,.$ (21)

It is thus interesting to derive explicity the disformal transformation that
relates these two frames. Given any (quadratic) Horndeski action,
characterized by a function $F(\phi,X)$, one can define

$\lambda_{1}=-\frac{M_{\rm P}^{2}}{2}+\frac{2}{M_{\rm P}^{2}}F(F-2XF_{X})\,,$
(22)

and verify that the disformal transformation of (18) with
$A=\text{sg}(F-2XF_{X})\in\\{+1,-1\\}$ and

$\displaystyle AB=\frac{M_{\rm P}^{4}}{4X(2XF_{X}-F)^{2}}-\frac{1}{X}\,,$ (23)

gives exactly (21). Note that we have necessarily $2XF_{X}-F\neq 0$ since we
are not in the class Ib.

###  IV.2 DHOST theories in the classes II and III

####  IV.2.1 Classification

Let us first discuss DHOST theories in the class III, characterized by $f=0$,
i.e.

$\lambda_{1}=-\frac{M_{\rm P}^{2}}{2}\qquad{\rm(class\ III)}\,.$ (24)

If $M_{\rm P}\neq 0$, the degeneracy conditions $D_{1}=0$ and $D_{2}=0$ then
imply

$\lambda_{3}=0\quad{\rm(subclass\ IIIa)}\,\qquad{\rm
or}\qquad\lambda_{2}=-\frac{M_{\rm P}^{2}}{3}\quad{\rm(subclass\ IIIb)}\,,$
(25)

corresponding to the subclasses IIIa and IIIb respectively. Finally, the case
$M_{\rm P}=0$ and thus $\lambda_{1}=0$, with $\lambda_{3}$ and $\lambda_{2}$
free, corresponds to the subclass IIIc, which does not contain tensor modes.

Let us now turn to the class II, which contains the theories that are neither
in class I or in class III. From the degeneracy conditions (68), one finds
that the class IIa is characterized by

$\displaystyle\lambda_{1}+\frac{M_{\rm
P}^{2}}{2}=f(X)=\xi\sqrt{|X|}\,,\qquad\lambda_{3}=0\,,\qquad{\rm(subclass\
IIa)}\,.$ (26)

where $\xi$ is a constant. The class IIb is characterized by two free
functions, $\lambda_{1}$ and $\lambda_{2}$, and the conditions

$M_{\rm
P}=0\,,\qquad\lambda_{3}=2\frac{(\lambda_{1}-2X{\lambda_{1}}_{X})^{2}}{\lambda_{1}}\qquad{\rm(subclass\
IIb)}\,.$ (27)

which solve the degeneracy conditions.

As a consequence, any DHOST theory in the class II can be described by the
action

$\displaystyle S_{II}[g_{\mu\nu},\phi]$ $\displaystyle=$ $\displaystyle\xi\int
d^{4}x\sqrt{-g|X|}\,{}^{(3)}\\!{R}-\frac{M_{\rm P}^{2}}{2}\varepsilon\int
d^{4}x\sqrt{-g}\left(K_{\mu\nu}K^{\mu\nu}-\left(1-{\mu}_{2}\right)K^{2}\right)$
(28) $\displaystyle=$ $\displaystyle\xi\int
d^{4}x\sqrt{|h|}\,{}^{(3)}\\!{R}-\frac{M_{\rm P}^{2}}{2}\varepsilon\int
d^{4}x\sqrt{-g}\left(K_{\mu\nu}K^{\mu\nu}-\left(1-\mu_{2}\right)K^{2}\right)\,,$
(29)

where we used the relation (41) between the determinants $g$ and $h$ and we
introduced the dimensionless parameter $\mu_{2}\equiv{2\lambda_{2}}/{M_{\rm
P}^{2}}$. Interestingly the three-dimensional Ricci term in this action
reduces exactly to the three-dimensional Einstein-Hilbert Lagrangian for the
metric $h_{\mu\nu}$ but integrated over the four-dimensional space-time.

####  IV.2.2 Instabilities

The geometric frame reformulation (29) of DHOST theories in class II is
particularly convenient to see that these theories are plagued by
instabilities, as originally shown in Langlois:2017mxy . Indeed, if one
considers solutions with $\varepsilon=-1$ in the unitary gauge, spatial
gradients in the equations of motion of the (scalar and tensor) fields can
only originate from the ${}^{(3)}\\!{R}$ term in the action.

Let us consider a homogeneous and isotropic background, with scale factor
$a(t)$ and scalar field $\phi(t)$. In the unitary gauge, the perturbations
about such a background are fully encoded in the scalar perturbation $\zeta$
and the tensor perturbation $\gamma_{ij}$ of the three-dimensional metric
$h_{ij}$,

$\displaystyle h_{ij}=a^{2}(t)e^{2\zeta}(\delta_{ij}+\gamma_{ij})\,,$ (30)

where latin letters $(i,j,\cdots)$ hold for spatial indices. When we
substitute this expression into the the ${}^{(3)}\\!{R}$ term which appears in
(29), we obtain at quadratic order in the perturbations

$\displaystyle\int d^{4}x\sqrt{h}\,{}^{(3)}\\!{R}=\int
d^{4}x\,a\,\left(-\frac{1}{4}\partial_{k}\gamma_{ij}\,\partial^{k}\gamma^{ij}+\partial_{i}\zeta\,\partial^{i}\zeta+{o}(\gamma^{2},\zeta^{2},\gamma\zeta)\right)\,.$
(31)

Hence, we immediately see that the gradient term of the scalar mode $\zeta$
has an opposite sign compared to the tensor modes and therefore there will be
necessarily a gradient instability either in the tensor sector or in the
scalar sector777In the case where $\varepsilon=+1$, one cannot take the
unitary gauge anymore, but instead one could fix $\phi$ to be one of the three
spatial coordinates. Hence, a similar analysis would lead to a quadratic
action of the form (31) with $(i,j,k)$ three-dimensional space-time indices
while the terms with $K_{\mu\nu}$ in (29) would involve now (space-like)
gradients of the fields only. Therefore, the only kinetic terms of the scalar
field and of the tensor modes would be contained in (31) and we clearly see
that they have opposite signs, which means that there would be a ghost
instability.. Thus, we recover very easily the result that DHOST theories in
class II are unstable Langlois:2017mxy . We see that this instability is
closely related to the form (26) of $\lambda_{1}$ which is necessary to select
a DHOST theory among weakly degenerate theories.

##  V Conclusion

In this work, we have obtained a strikingly simple reformulation of quadratic
DHOST theories (and of weakly degenerate theories), based on a Lagrangian
involving a few geometrical terms associated with the three-dimensional
constant $\phi$ hypersurfaces, in addition to the standard Einstein-Hilbert
term. This geometric frame action describes only a subset of DHOST theories
but the rest of the family can be “generated” from this subset via disformal
transformations.

Moreover, since the various subclasses of DHOST theories are stable under
disformal transformations, it is sufficient to classify the subset of theories
in the geometric frame to automatically generalise this classification to the
whole family. A compelling illustration is given by the subclass Ia, the most
interesting class from a phenomenological perspective, which includes
Horndeski’s theories. In the original classification of DHOST theories, this
subclass is parametrised by the three functions $f$, $\alpha_{1}$ and
$\alpha_{3}$, while the other functions are expressed in terms of these, with
rather ugly expressions for $\alpha_{4}$ and $\alpha_{5}$. By contrast, in
this new geometric perspective, the subclass Ia arises from an geometric frame
action that depends on a single function $\lambda_{1}$ multiplying the three-
dimensional curvature, all the other theories being obtained via disformal
transformations. The subclass Ia is thus parametrised by the three functions
$\lambda_{1}$, $A$ and $B$. One can proceed similarly for the other DHOST
subclasses, as well as for all the quadratic weakly degenerate theories which
are also of the form (13).

As we have argued in this article, the geometric frame perspective is
appealing to understand and analyse the underlying dynamical structure of
DHOST theories, similarly to the Einstein frame in traditional scalar-tensor
theories. However we should stress that, in the presence of matter, it is much
more convenient in general, from a practical point of view, to stick to the
physical frame where matter is minimally coupled to the metric rather than to
move to the geometric frame. Indeed, matter would be disformally coupled to
geometric frame metric, with the functions $A$ and $B$ usually defined only
implicitly. This would make the calculations in this frame very cumbersome.
For concrete applications, it is thus more appropriate to work directly with
the original formulation and to add matter minimally coupled to the metric in
(4). Scalar-tensor theories that are disformally related thus correspond to
physically distinct theories since matter is minimally coupled to both
theories.

By contrast, the geometric frame approach should be useful to get a better
intuitive understanding of DHOST theories as well as for their classification,
or to study their generic properties invariant under disformal
transformations, such as the instabilities in classes II and III. In this
respect, it would be interesting to extend the geometric frame description to
include the cubic DHOST theories, whose classification in the standard
formulation is even more involved than for the quadratic case. We plan to
explore this question in a future work.

###### Acknowledgements.

We would like to thank Christos Charmousis for very instructive discussions.
KN acknowledges support from the CNRS grant 80PRIME and thanks ENS for their
hospitality.

##  Appendix A Uniform scalar field hypersurfaces $\Sigma_{\phi}$ and the
geometric frame

This section is devoted to recall useful geometrical properties of the uniform
scalar field hypersurfaces $\Sigma_{\phi}$ and its consequences of the
reformulation of DHOST and weakly degenerate theories.

###  A.1 Geometry of the hypersurfaces $\Sigma_{\phi}$

The geometry of $\Sigma_{\phi}$ is fully characterized by the induced metric
$h_{\mu\nu}$ and its four-dimensional normal $n_{\mu}$ (9),

$\displaystyle h_{\mu\nu}\equiv g_{\mu\nu}-\varepsilon n_{\mu}n_{\nu}\,,\qquad
n_{\mu}\equiv\frac{\phi_{\mu}}{\sqrt{|X|}}\,,\qquad\varepsilon\equiv\text{sg}(X)\,.$
(32)

From these two tensors one can construct the extrinsic curvature tensor and
the “acceleration” vector, their components being respectively given by

$\displaystyle K_{\mu\nu}\equiv
h_{\mu}^{\alpha}\,h_{\nu}^{\beta}\,\nabla_{\alpha}n_{\beta}\,,\qquad
a_{\mu}\equiv n^{\nu}\nabla_{\nu}n_{\mu}\,.$ (33)

Their explicit expressions in terms of $\phi$, its first derivatives
$\phi_{\mu}$ and its second derivatives $\phi_{\mu\nu}$ can be easily
computed:

$\displaystyle a_{\mu}=-\frac{1}{2|X|}h_{\mu\nu}X^{\nu}\,,\quad
K_{\mu\nu}=\frac{1}{\sqrt{|X|}}\left[\phi_{\mu\nu}+\frac{\phi^{\alpha}X_{\alpha}}{2X^{2}}\phi_{\mu}\phi_{\nu}-\frac{1}{2X}\left(\phi_{\mu}X_{\nu}+\phi_{\nu}X_{\mu}\right)\right],$
(34)

where $X_{\mu}\equiv\partial_{\mu}X=2\phi_{\mu\nu}\phi^{\nu}$.

The induced three-dimensional Riemann tensor is also of great importance. It
is given by

$\displaystyle{}^{(3)}\\!{R}_{\mu\nu\rho\sigma}=h^{\alpha}_{\mu}h^{\beta}_{\nu}h^{\gamma}_{\rho}h^{\delta}_{\sigma}\,{}^{(4)}\\!{R}_{\alpha\beta\gamma\delta}+\varepsilon(K_{\mu\rho}K_{\nu\sigma}-K_{\nu\rho}K_{\mu\sigma})\,.$
(35)

This relation enables us to compute the three-dimensional Ricci tensor and the
Ricci scalar which is given by the Gauss-Codazzi relation,

$\displaystyle{}^{(3)}\\!{R}={}^{(4)}\\!{R}-\varepsilon\left[K^{2}-K_{\mu\nu}K^{\mu\nu}-2\nabla_{\mu}(a^{\mu}-Kn^{\mu})\right]\,.$
(36)

To obtain a more explicit form, we can use the following equations,

$\displaystyle\varepsilon\,K^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{X}L_{2}-\frac{2}{X^{2}}L_{3}+\frac{1}{X^{3}}L_{5}\,,$
(37) $\displaystyle\varepsilon\,K_{\mu\nu}K^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{X}L_{1}-\frac{2}{X^{2}}L_{4}+\frac{1}{X^{3}}L_{5}\,,$
(38) $\displaystyle a^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{X^{2}}L_{4}-\frac{1}{X^{3}}L_{5}\,,$ (39)

where $L_{A}$ are the elementary quadratic Lagrangians (5), together with the
relation

$\displaystyle\varepsilon\int
d^{4}x\,\sqrt{-g}\,f\,\nabla_{\mu}(a^{\mu}-Kn^{\mu})=\int
d^{4}x\,\sqrt{-g}\left[-\frac{2f_{X}}{X}L_{3}+\frac{2f_{X}}{X}L_{4}+\frac{f_{\phi}}{X}(X\Box\phi-\phi_{\mu\nu}\phi^{\mu}\phi^{\nu})\right]\,,$
(40)

where $f$ is an arbitrary function of $\phi$ and $X$, $f_{X}$ its derivative
with respect to $X$ and $f_{\phi}$ its derivative with respect to $\phi$.

Finally, one can express the determinant $g$ of the metric $g_{\mu\nu}$ in
terms of the determinant $h$ of $h_{\mu\nu}$ and $X$ as follows

$\displaystyle g$ $\displaystyle=$
$\displaystyle\frac{1}{24}\epsilon^{\mu_{1}\nu_{1}\rho_{1}\sigma_{1}}\epsilon^{\mu_{2}\nu_{2}\rho_{2}\sigma_{2}}g_{\mu_{1}\mu_{2}}g_{\nu_{1}\nu_{2}}g_{\rho_{1}\rho_{2}}g_{\sigma_{1}\sigma_{2}}$
(41) $\displaystyle=$
$\displaystyle\frac{1}{6X}\epsilon^{\mu_{1}\nu_{1}\rho_{1}\sigma_{1}}\epsilon^{\mu_{2}\nu_{2}\rho_{2}\sigma_{2}}h_{\mu_{1}\mu_{2}}h_{\nu_{2}\nu_{2}}h_{\rho_{1}\rho_{2}}\phi_{\sigma_{1}}\phi_{\sigma_{2}}=\frac{h}{X}\,,$

which follows from the very definition of the determinant and
$\epsilon_{\mu\nu\rho\sigma}$ is the fully antisymmetric four-dimensional
tensor.

###  A.2 DHOST theories in the geometric frame

As a consequence, the simplified action (3)

$\displaystyle S[g_{\mu\nu},\phi]=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm
P}^{2}\,{}^{(4)}\\!{R}+\lambda_{1}\,{}^{(3)}\\!{R}+\lambda_{2}\,\varepsilon\,K^{2}+\lambda_{3}\,a^{2}\,\right),$
(42)

can be reformulated as a sum of a quadratic action in the more usual form (4)
with the coefficients,

$\displaystyle f=\frac{M_{\rm
P}^{2}}{2}+\lambda_{1}\,,\quad\alpha_{1}=\frac{\lambda_{1}}{X}\,,\quad\alpha_{2}=\frac{\lambda_{2}-\lambda_{1}}{X}\,,$
$\displaystyle\alpha_{3}=\frac{2\lambda_{1}-4X\lambda_{1X}-2\lambda_{2}}{X^{2}}\,,\quad\alpha_{4}=\frac{4X\lambda_{1X}-2\lambda_{1}+\lambda_{3}}{X^{2}}\,,\quad\alpha_{5}=\frac{\lambda_{2}-\lambda_{3}}{X^{3}}\,,$
(43)

supplemented with a k-essence and a cubic galileon action given by

$\displaystyle\int
d^{4}x\sqrt{-g}\,\frac{2\lambda_{1\phi}}{X}(X\Box\phi-\phi_{\mu\nu}\phi^{\mu}\phi^{\nu})=\int
d^{4}x\sqrt{-g}\,\left((2X\beta_{X}+\beta)\Box\phi+\beta_{\phi}X\right)\ ,$
(44)

with $X\beta=\lambda_{1\phi}$. The identity (44) follows from the relation

$\displaystyle 2\int
d^{4}x\sqrt{-g}\,\beta_{X}\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}=-\int
d^{4}x\sqrt{-g}\,\left(X\beta_{\phi}+\beta\Box\phi\right)\,,$ (45)

for any function $\beta(\phi,X)$.

###  A.3 Geometric formulation of the cubic Galileon

As we have shown that a quadratic DHOST action can be rewritten in a more
geometrical way, it is natural to look for a way to replace the cubic galileon
term $Q(\phi,X)\Box\phi$ by a combination of $K_{\mu\nu}$ and $a_{\mu}$ as
well. The only combination of these objects that is linear in $\phi$ is the
trace of $K_{\mu\nu}$. Therefore, we now consider the action

$S[g_{\mu\nu},\phi]=\int d^{4}x\sqrt{-g}\left(\nu_{0}+\nu_{1}K\right)\,,$ (46)

where $\nu_{0}$ and $\nu_{1}$ are two arbitrary functions of $X$ and $\phi$.
Using (34), this action can be written as

$S[g_{\mu\nu},\phi]=\int
d^{4}x\sqrt{-g}\left[\nu_{0}+\alpha(\phi^{\mu}\phi^{\nu}\phi_{\mu\nu}-X\Box\phi)\right]\,,\quad\alpha\equiv-\frac{\varepsilon\nu_{1}}{{|X|^{3/2}}}\,.$
(47)

Using (45), we can show that, for any arbitrary function $\alpha(\phi,X)$,

$\int
d^{4}x\sqrt{-g}\,\alpha(\phi^{\mu}\phi^{\nu}\phi_{\mu\nu}-X\Box\phi)=-\frac{1}{2}\int
d^{4}x\sqrt{-g}\left[{X}A_{\phi}+\left({A}+2\alpha X\right)\Box\phi\right]\,,$
(48)

where $A$ is such that $A_{X}=\alpha$. As a consequence, the action (46) can
be written as

$S[g_{\mu\nu},\phi]=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left(2\pi-{X}A_{\phi}-\left({A}+2\alpha
X\right)\Box\phi\right)\,.$ (49)

One recovers the well-known k-essence and cubic galileon terms associated with
the functions $P$ and $Q$ given by,

$\displaystyle P(\phi,X)=\nu_{0}-\frac{X}{2}A_{\phi}\,,\qquad
Q(\phi,X)=-\frac{1}{2}\left(A+2\alpha X\right)\,.$ (50)

If the quadratic action 3 is also taken into account, it gives a linear
contribution given by 44. One must therefore replace $\alpha$ in (47) by

$\alpha=-\frac{\varepsilon\nu_{1}}{|X|^{3/2}}+\frac{2\lambda_{1\phi}}{X}\,.$
(51)

##  Appendix B Equations of motion in the geometric frame

The expression and geometrical interpretation of DHOST theories are much
simpler in this novel geometric frame than in the Horndeski frame. The
equations of motion are also simpler and can be formulated in geometrical
terms which could help finding solutions.

In this appendix, we compute the equations of motion of the Class I action
(18)

$\displaystyle S=\int d^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm
P}^{2}\,{{}^{(4)}\\!{R}}+\lambda_{1}\,{}^{(3)}\\!{R}+\nu_{1}\,K\,+\,\nu_{0}\right)\,,$
(52)

to which we added (for purposes of generality) a k-essence term $S_{\nu_{0}}$
associated with the function $\nu_{0}(\phi,X)$ and a K-term $S_{\nu_{1}}$
associated with the function $\nu_{1}(\phi,X)$ (46). As we proved in the
appendix A, the K-term can be equivalently reformulated as a cubic Galileon up
to k-essence terms.

Using the formulae

$\displaystyle\delta X$ $\displaystyle=$ $\displaystyle
2\phi^{\mu}\partial_{\mu}\delta\phi-\phi^{\mu}\phi^{\nu}\delta g_{\mu\nu}\,,$
(53) $\displaystyle\delta h_{\mu\nu}$ $\displaystyle=$ $\displaystyle\delta
g_{\mu\nu}+\frac{\delta
X}{X^{2}}\phi_{\mu}\phi_{\nu}-\frac{\phi_{\mu}\partial_{\nu}\delta\phi+\phi_{\nu}\partial_{\mu}\delta\phi}{X}\,,$
(54)

for the infinitesimal variations of $X$ and $h_{\mu\nu}$ together with the
relation $h=Xg$ (41) between the determinants $h=\text{det}(h_{\mu\nu})$ and
$g=\text{det}(g_{\mu\nu})$, we show by a direct calculation that the equation
of motion for the metric takes the standard form,

$\displaystyle\frac{M_{\rm P}^{2}}{2}\,{}^{(4)}{G}_{\mu\nu}$ $\displaystyle=$
$\displaystyle
T_{\mu\nu}^{(\lambda_{1})}+T_{\mu\nu}^{(\nu_{1})}+T_{\mu\nu}^{(\nu_{0})}$ (55)

where $T_{\mu\nu}^{(\nu_{0})}$ and $T_{\mu\nu}^{(\nu_{1})}$ are the usual
stress-energy tensors associated to the actions $S_{\nu_{0}}$ and
$S_{\nu_{1}}$ respectively,

$\displaystyle
T_{\mu\nu}^{(\nu_{0})}=\frac{\nu_{0}}{2}g_{\mu\nu}-\nu_{0X}\phi_{\mu}\phi_{\nu}\,,\qquad
T_{\mu\nu}^{(\nu_{1})}=\frac{1}{2q}\left(\nu_{1\phi}-2\nu_{1X}\Box\phi\right)\phi_{\mu}\phi_{\nu}\,,$
(56)

where $q\equiv{|X|^{1/2}}$, while $T_{\mu\nu}^{(\lambda_{1})}$ is the stress-
tensor energy associated to the three-dimensional Ricci term in (18)

$\displaystyle T_{\mu\nu}^{(\lambda_{1})}$ $\displaystyle=$
$\displaystyle-q\left[\mu_{1}\,{}^{(3)}\\!G_{\mu\nu}+\mu_{1X}{}^{(3)}\\!{R}\,\phi_{\mu}\phi_{\nu}+(h_{\mu\nu}{}^{3}\Box-{}^{3}\nabla_{\mu}{}^{3}\nabla_{\nu})\mu_{1}\right]\,,$
(57)

where ${}^{3}\nabla_{\mu}$ is the 3-dimensional covariant derivative with
respect to $h_{\mu\nu}$ and we introduced the notation
$\mu_{1}\equiv\lambda_{1}/q$ for simplicity.

Even though the equation of motion for the scalar field is redundant as it can
be deduced from the previous one, it is nonetheless useful to give its
expression which takes the form

$\displaystyle\nabla_{\mu}J^{\mu}+\Phi=0\,,$ (58)

where the curent $J^{\mu}$ and the source $\Phi$ are given by,

$\displaystyle
J^{\mu}=J^{\mu}_{(\lambda_{1})}+J^{\mu}_{(\nu_{1})}+J^{\mu}_{(\nu_{0})}\,,\qquad\Phi=\Phi_{(\lambda_{1})}+\Phi_{(\nu_{1})}+\Phi_{(\nu_{0})}\,,$
(59)

and each components of $J^{\mu}$ and $\Phi$ are given by,

$\displaystyle
J^{\mu}_{(\lambda_{1})}=\lambda_{1X}{}^{(3)}\\!{R}\phi^{\mu}\,,\quad
J^{\mu}_{(\nu_{1})}=\frac{1}{2q}(2q\nu_{1X}K-{\nu_{1\phi}})\phi^{\mu}+q\nu_{1X}a^{\mu}\,,\quad
J^{\mu}_{(\nu_{0})}=\nu_{0X}\phi^{\mu}\,,$ (60)
$\displaystyle\Phi_{(\lambda_{1})}=-\frac{\lambda_{1\phi}}{2}\,{}^{(3)}\\!{R}\,,\quad\Phi_{(\nu_{1})}=\frac{1}{2q}\left[X\nu_{1\phi\phi}+2X\nu_{1\phi
X}(\Box\phi-qK)\right]\,,\quad\Phi_{(\nu_{0})}=-\frac{\nu_{0\phi}}{2}\,.$ (61)

If the theory is shift symmetric, i.e. the functions $\lambda_{1}$, $\nu_{1}$
and $\nu_{0}$ do not depend on $\phi$, we recover the well-known fact that the
equation of the scalar field reduces to a conservation equation for the curent
$J^{\mu}$.

Notice that the equation for the metric involves third order derivatives of
the scalar field and, similarly, the equation for the scalar field involves
third order derivatives of the metric components. This is expected as the
action is not formulated in the Horndeski frame. However, in the case where
$\varepsilon=-1$, these higher order terms are all spatial derivatives which
is consistent with the fact that there is no ghost propagating in the theory.
These equations have a simple form compared to Horndeski theories. This might
help us in finding solutions.

Let us see how the conditions for having stealth solutions (see
Babichev:2013cya  for the first stealth solutions in Horndeski theories) are
formulated in the geometric frame. We assume that $X$ is a constant $X_{0}$.
Thus, the equations of motion simplify drastically and we have (for the metric
only)

$\displaystyle{M_{\rm
P}^{2}}\,{}^{(4)}\\!{G}_{\mu\nu}-\nu_{0}\,g_{\mu\nu}+2q\,\mu_{1}\,{}^{(3)}\\!G_{\mu\nu}+2\left(q\,\mu_{1X}\,{}^{(3)}\\!{R}+\nu_{0X}+\frac{\nu_{1X}}{q}\Box\phi\right)\phi_{\mu}\phi_{\nu}=0\,.$
(62)

Now, we see that, when the theory satisfies the following conditions888Notice
that these conditions are consistent with those obtained in Takahashi:2020hso
for DHOST theories when their actions are written in the usual form (4)
(supplemented with a k-essence term and a cubic galileon term),
$\displaystyle P+2\Lambda f=0\,,\quad
P_{X}+\Lambda(4f_{X}-X\alpha_{1X})=0\,,\quad
Q_{X}=0\,,\quad\alpha_{1}=0\,,\quad\alpha_{3}+2\alpha_{1X}=0\,.$ (63) When we
replace the coefficients $\alpha_{A}$ by their expressions (14) in terms of
$\lambda_{1}$ and also, $P$ and $Q$ by their expressions in terms of $\nu_{0}$
and $\nu_{1}$ (50), we recover immediately the stealth conditions (64). In
particular, we show that $Q_{X}=\nu_{1X}/q$, which immediately implies the
equivalence between the conditions $Q_{X}=0$ and $\nu_{1X}=0$.,

$\displaystyle\mu_{1}(X_{0})=0\,,\quad\mu_{1X}(X_{0})=0\,,\quad\nu_{0X}(X_{0})=0\,,\quad\nu_{1X}(X_{0})=0\,,$
(64)

it admits stealth solutions which satisfy the usual Einstein equation for
general relativity,

$\displaystyle{}^{(4)}\\!{G}_{\mu\nu}+\Lambda g_{\mu\nu}=0\,,\qquad M_{\rm
P}^{2}\Lambda+{\nu_{0}(X_{0})}=0\,.$ (65)

The conditions (64) are sufficient but not necessary for the existence of
stealth solutions. The theory could admit stealth Schwarzschild solutions for
instance without satisfying these conditions as shown in Takahashi:2020hso
(and references therein) for instance .

##  Appendix C Degeneracy conditions for quadratic DHOST theories

In this section, we recall the degeneracy conditions for quadratic DHOST
theories derived in Langlois:2015cwa , and then give their expression in terms
of the parameters $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$. The three
degeneracy conditions for the quadratic DHOST theories are

$D_{0}(X)=0\,,\qquad D_{1}(X)=0\,,\qquad D_{2}(X)=0\,,$ (66)

with

$\displaystyle D_{0}(X)$
$\displaystyle=-4(\alpha_{1}+\alpha_{2})\left[Xf(2\alpha_{1}+X\alpha_{4}+4f_{X})-2f^{2}-8X^{2}f_{X}^{2}\right]\,,$
(67) $\displaystyle D_{1}(X)$ $\displaystyle=\begin{aligned}
&4\left[X^{2}\alpha_{1}(\alpha_{1}+3\alpha_{2})-2f^{2}-4Xf\alpha_{2}\right]\alpha_{4}+4X^{2}f(\alpha_{1}+\alpha_{2})\alpha_{5}\\\
&+8X\alpha_{1}^{3}-4(f+4Xf_{X}-6X\alpha_{2})\alpha_{1}^{2}-16(f+5Xf_{X})\alpha_{1}\alpha_{2}\\\
&+4X(3f-4Xf_{X})\alpha_{1}\alpha_{3}-X^{2}f\alpha_{3}^{2}+32f_{X}(f+2Xf_{X})\alpha_{2}\\\
&-16ff_{X}\alpha_{1}-8f(f-Xf_{X})\alpha_{3}+48ff_{X}^{2}\,,\end{aligned}$
$\displaystyle D_{2}(X)$ $\displaystyle=\begin{aligned}
&4\left[2f^{2}+4Xf\alpha_{2}-X^{2}\alpha_{1}(\alpha_{1}+3\alpha_{2})\right]\alpha_{5}+4\alpha_{1}^{3}+4(2\alpha_{2}-X\alpha_{3}-4f_{X})\alpha_{1}^{2}\\\
&+3X^{2}\alpha_{1}\alpha_{3}^{2}-4Xf\alpha_{3}^{2}+8(f+Xf_{X})\alpha_{1}\alpha_{3}-32f_{X}\alpha_{1}\alpha_{2}\\\
&+16f_{X}^{2}\alpha_{1}+32f_{X}^{2}\alpha_{2}-16ff_{X}\alpha_{3}\,.\end{aligned}$

These expressions simplify quite a lot when they are written in term of
$\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$, substituting (14):

$\displaystyle D_{0}(X)$
$\displaystyle=\frac{2\lambda_{2}}{X}\left[\left({M_{\rm
P}^{2}}+2\lambda_{1}\right)\left({M_{\rm
P}^{2}}+2\lambda_{1}-\lambda_{3}-8X\lambda_{1X}\right)+16X^{2}\lambda_{1X}^{2}\right]\,,$
(68) $\displaystyle D_{1}(X)$ $\displaystyle=-\frac{2M_{\rm
P}^{4}}{X^{2}}\lambda_{3}+\frac{2\lambda_{2}}{X^{2}}\left[8\lambda_{1}^{2}-5M_{\rm
P}^{2}\lambda_{3}+4\lambda_{1}(2M_{\rm
P}^{2}-\lambda_{3}-8X\lambda_{1X})+2(M_{\rm
P}^{2}-4X\lambda_{1X})^{2}\right]\,,$ $\displaystyle D_{2}(X)$
$\displaystyle=-\frac{2M_{\rm
P}^{4}}{X^{2}}\lambda_{3}+\frac{2\lambda_{2}}{X^{2}}\left[4\lambda_{1}^{2}-4M_{\rm
P}^{2}\lambda_{3}+2\lambda_{1}(2M_{\rm
P}^{2}-\lambda_{3}-8X\lambda_{1X})+(M_{\rm
P}^{2}-4X\lambda_{1X})^{2}\right]\,.$

Weakly degenerate theories satisfy the single condition Langlois:2015cwa

$D_{0}(X)-XD_{1}(X)+X^{2}D_{2}(X)=0\,.$ (69)

##  Appendix D Formulae for the disformal transformations

In this appendix, we recall how a quadratic higher-order scalar-tensor action
transforms under disformal transformations. Notations are based on those
introduced in the paper Achour:2016rkg . Notice that the importance of
disformal transformations in higher-order scalar-tensor theories has been
realized first in Zumalacarregui:2013pma .

We consider the quadratic higher-order scalar-tensor action

$\tilde{S}[\tilde{g}_{\mu\nu},\phi]=\int
d^{4}x\sqrt{-\tilde{g}}\left(\tilde{f}(\tilde{X})\,{}^{(4)}\\!{R}+\sum_{A=1}^{5}\tilde{\alpha}_{A}(\tilde{X})\,L_{A}\right)\,,$
(70)

where $L_{A}$ are the elementary quadratic Lagrangians (4) and
$\tilde{X}\equiv\tilde{g}^{\mu\nu}\phi_{\mu}\phi_{\nu}$. For simplicity, we
assume that the theory is shift-symmetric.

Disformal transformations of the metric 1 induce a disformal transformation of
the action according to

$\displaystyle\tilde{S}[\tilde{g}_{\mu\nu},\phi]\;\longrightarrow\;S[g_{\mu\nu},\phi]=\tilde{S}[A(X)g_{\mu\nu}+B(X)\phi_{\mu}\phi_{\nu}]\,,$
(71)

where the quadratic part of $S[g_{\mu\nu},\phi]$ is of the form,

$\int
d^{4}x\sqrt{-g}\left(f(X)\,{}^{(4)}\\!{R}+\sum_{A=1}^{5}\alpha_{A}(X)\,L_{A}\right)\,,$
(72)

with

$\tilde{X}=\frac{X}{A+BX}\,,$ (73)

while the functions $f$ and $\alpha_{A}$ can be expressed in terms of
$\tilde{f}$ and $\tilde{\alpha}_{A}$ as follows,

$\displaystyle
f=\frac{J}{A}\tilde{f},\quad\alpha_{1}=-h+JT_{11}\tilde{\alpha}_{1},\quad\alpha_{2}=h+JT_{22}\tilde{\alpha}_{2},$
(74)
$\displaystyle\alpha_{3}=2h_{X}+J(\tilde{f}\gamma_{3}-2\tilde{f}_{X}\delta_{3}+T_{13}\tilde{\alpha}_{1}+T_{23}\tilde{\alpha}_{2}+T_{33}\tilde{\alpha}_{3}),$
$\displaystyle\alpha_{4}=-2h_{X}+J(\tilde{f}\gamma_{4}-2\tilde{f}_{X}\delta_{4}+T_{14}\tilde{\alpha}_{1}+T_{44}\tilde{\alpha}_{4}),$
$\displaystyle\alpha_{5}=J(\tilde{f}\gamma_{5}-2\tilde{f}_{X}\delta_{5}+T_{15}\tilde{\alpha}_{1}+T_{25}\tilde{\alpha}_{2}+T_{35}\tilde{\alpha}_{3}+T_{45}\tilde{\alpha}_{4}+T_{55}\tilde{\alpha}_{5}).$

We have introduced the notations,

$\displaystyle J=A^{3/2}\sqrt{A+BX},\quad h=-\frac{BJ\tilde{f}}{A(A+BX)},$
$\displaystyle T_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{(A+BX)^{2}}\,,\quad
T_{13}=\frac{2A_{X}}{A(A+BX)^{2}}\,,$ $\displaystyle T_{14}$ $\displaystyle=$
$\displaystyle\frac{2\left(X\left(A_{X}+XB_{X}\right){}^{2}-A\left(2\left(A_{X}+XB_{X}\right)+B\right)\right)}{A(A+BX)^{3}},$
$\displaystyle T_{15}$ $\displaystyle=$
$\displaystyle\frac{3B^{2}X^{2}A_{X}^{2}}{A^{2}(A+BX)^{4}}-\frac{2B^{2}XA_{X}}{A(A+BX)^{4}}+\frac{B^{2}}{(A+BX)^{4}}-\frac{2BX^{3}B_{X}^{2}}{A(A+BX)^{4}}-\frac{X^{2}B_{X}^{2}}{(A+BX)^{4}}$
$\displaystyle-$
$\displaystyle\frac{4BX^{2}A_{X}B_{X}}{A(A+BX)^{4}}+\frac{4BXA_{X}^{2}}{A(A+BX)^{4}}-\frac{2XA_{X}B_{X}}{(A+BX)^{4}}+\frac{4BXB_{X}}{(A+BX)^{4}}+\frac{2A_{X}^{2}}{(A+BX)^{4}}+\frac{2AB_{X}}{(A+BX)^{4}}\,,$
$\displaystyle T_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{(A+BX)^{2}}\,,\quad
T_{23}=-\frac{2\left(A\left(-2A_{X}+XB_{X}+B\right)-3BXA_{X}\right)}{A(A+BX)^{3}}\,$
$\displaystyle T_{25}$ $\displaystyle=$
$\displaystyle\frac{\left(A\left(-2A_{X}+XB_{X}+B\right)-3BXA_{X}\right){}^{2}}{A^{2}(A+BX)^{4}}\,,$
$\displaystyle T_{33}$ $\displaystyle=$
$\displaystyle\frac{A-X\left(A_{X}+XB_{X}\right)}{(A+BX)^{4}}\,,$
$\displaystyle T_{35}$ $\displaystyle=$
$\displaystyle-\frac{\left(A\left(-2A_{X}+XB_{X}+B\right)-3BXA_{X}\right)\left(A-X\left(A_{X}+XB_{X}\right)\right)}{A(A+BX)^{5}},$
$\displaystyle T_{44}$ $\displaystyle=$
$\displaystyle\frac{\left(A-X\left(A_{X}+XB_{X}\right)\right){}^{2}}{A(A+BX)^{4}},\quad
T_{45}=-\frac{B\left(A-X\left(A_{X}+XB_{X}\right)\right){}^{2}}{A(A+BX)^{5}}\,,$
$\displaystyle T_{55}$ $\displaystyle=$
$\displaystyle\frac{\left(A-X\left(A_{X}+XB_{X}\right)\right){}^{2}}{(A+BX)^{6}}\,,$
$\displaystyle\gamma_{3}=-\frac{B\left(BXA_{X}+A\left(2A_{X}+XB_{X}+B\right)\right)}{A^{2}(A+BX)^{2}},$
$\displaystyle\gamma_{4}=\frac{A^{2}\left(2BA_{X}+4XA_{X}B_{X}+6A_{X}^{2}+B^{2}+BXB_{X}\right)+2B^{2}X^{2}A_{X}^{2}+ABXA_{X}\left(8A_{X}+4XB_{X}+B\right)}{A^{3}(A+BX)^{2}},$
$\displaystyle\gamma_{5}=-\frac{2A_{X}\left(BA_{X}+2AB_{X}\right)}{A^{3}(A+BX)},$
$\displaystyle\delta_{3}=\frac{B}{A(A+BX)},\quad\delta_{4}=\frac{-4BXA_{X}-A\left(6A_{X}+2XB_{X}+B\right)}{A^{2}(A+BX)},\quad\delta_{5}=\frac{2\left(2BA_{X}+AB_{X}\right)}{A^{2}(A+BX)}\,.$

All functions are evaluated in $X$ and not $\tilde{X}$ (and derivatives are
with respect to $X$ and not $\tilde{X}$).

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