# Plebański-Demiański solutions in Quadratic gravity with conformally coupled
scalar fields

Adolfo Cisterna<EMAIL_ADDRESS>Dipartimento di Fisica,
Universita di Trento, Via Sommarive 14, 38123 Povo (TN), Italy.  TIFPA - INFN,
Via Sommarive 14, 38123 Povo (TN), Italy.  Anibal Neira-Gallegos
<EMAIL_ADDRESS>Departamento de Física, Universidad de Concepción, Casilla,
160-C, Concepción, Chile.  Julio Oliva<EMAIL_ADDRESS>Departamento de
Física, Universidad de Concepción, Casilla, 160-C, Concepción, Chile.
Scarlett C. Rebolledo-Caceres<EMAIL_ADDRESS>Departamento de Física,
Universidad de Concepción, Casilla, 160-C, Concepción, Chile.

###### Abstract

We show that the Plebański-Demiański spacetime persists as a solution of
General Relativity when the theory is supplemented with both, a conformally
coupled scalar theory and with quadratic curvature corrections. The quadratic
terms are of two types and are given by quadratic combinations of the Riemann
tensor as well as a higher curvature interaction constructed with a scalar
field which is conformally coupled to quadratic terms in the curvature. The
later is built in terms of a four-rank tensor $S_{\mu\nu}^{\ \ \lambda\rho}$
that depends on the Riemann tensor and the scalar field, and that transforms
covariantly under local Weyl rescallings. Due to the generality of the
Plebański-Demiański family, several new hairy black hole solutions are
obtained in this higher curvature model. We pay particular attention to the
C-metric spacetime and the stationary Taub-NUT metric, which in the hyperbolic
case can be analytically extended leading to healthy, asymptotically AdS,
wormhole configurations. Finally, we present a new general model for higher
derivative, conformally coupled scalars, depending on an arbitrary function
and that we have dubbed Conformal K-essence. We also construct spherically
symmetric hairy black holes for these general models.

##  I Introduction

In gravity, described by General Relativity (GR) or in any of its extensions,
a fundamental angle to be investigated is their spectrum of solutions. Such
set includes gravitational waves TheLIGOScientific:2017qsa  and black holes as
well as other astrophysically relevant objects. Black holes are formed by
gravitational collapse Joshi:2012mk , a catastrophic event that may produce
singularities, pathological points in the spacetime where geodesic
completeness is broken and where the curvature may diverge Hawking:1973uf .
The classical validity of GR is guaranteed by the cosmic censorship hypothesis
Penrose:1969pc  which asserts that under generic conditions and for realistic
forms of matter the singularities are covered by a horizon, a surface from
which no information can escape to infinity. Ultimately, a black hole is the
region inside the event horizon, a region that is causally disconnected with
future null infinity and whose boundary in the spacetime is given by the event
horizon itself. It has been argued that, after the gravitational collapse
takes place, the formed black hole can only be described by a well-defined set
of parameters, its mass, electromagnetic charges and angular momentum, and
that no possible further relevant characteristics of the original matter that
produced the black hole, such as baryonic or leptonic numbers for example,
survive the process. In some case this conjecture can be proven and leads to
no-hair theorems Ruffini:1971bza , where hair is used to name all possible
characteristics of black holes that make them not bald, those quantities that
do not hold a Gauss law, and as a consequence are not conserved at infinity.
In spite of these results, it is possible to construct hairy black holes in
General Relativity with a sensible matter content. The first attempts in this
direction were carried out with the backreaction of a conformally coupled
scalar field BBMB ; Bekenstein:1974sf . A backreacting, minimally coupled,
static, spherically symmetric massless scalar field inevitably leads to a
singular spacetime, which can be integrated analytically Janis:1970kn ;
Janis:1968zz . Such pathology remains even beyond spherical symmetry in the
static case Bekenstein:1998aw 111Recently these no-hair results have been
revisited in the context of stationary, axisymmetric spacetimes endowed with
reasonable matter fields that do not share the same symmetries of the metric
Canfora:2013osa ; Herdeiro:2014goa .. Nevertheless, the introduction of a
conformal coupling with the curvature allows to by-pass the no-hair results
providing a healthy spacetime, but with a scalar field that diverges at the
horizon. Notwithstanding this result, as shown in Bekenstein:1975ts  such
singular behavior does not imply infinite tidal forces on a falling object.
The presence of a cosmological constant and a quartic self-interaction pushes
the scalar field singularity inside the event horizon Martinez:2002ru ;
Martinez:2004nb  and a rich family of causal structures can be obtained in
this setup Martinez:2005di . Those features are also exhibited by higher-
dimensional rotating solutions even when the coupling factor is not conformal
and a self-interacting potential is present Erices:2017izj . In this line,
including a self-interacting potential that induces a deviation from conformal
symmetry by the presence of linear and cubic terms, allows to construct
regular black holes and wormholes in (A)dS spacetime Kolyvaris:2009pc ,
Anabalon:2012tu , while the most general scalar field potential consistent
with a Petrov Type-D background has been found in Anabalon:2012ta . A charged
generalization of these solutions in the static case permits the presence of a
mass term for the scalar Ayon-Beato:2015ada .

When GR is embedded in a UV complete theory, higher curvature corrections
inevitably emerge. In four dimensions, quadratic higher curvature corrections
are particularly interesting since they provide a power counting
renormalizable theory, at the cost of introducing a ghost mode Stelle:1976gc .
Due to the topological nature of the quadratic Euler combination in four
dimensions, one has only two arbitrary couplings in the quadratic sector.
Unitarity is recovered if one restricts oneself to the model of the form
$R^{2}$, since this term is dual to the Einstein-Hilbert Lagrangian with a
scalar field. Consequently, it is natural to wonder how these higher curvature
corrections modify the spectrum of black holes. In vacuum one can prove that,
static, asymptotically flat black holes exist, only when the quadratic
combination reduces to the Weyl square term of four-dimensional Conformal
Gravity and the static black holes must be integrated numerically Lu:2015cqa
(see Lin:2016kip  for the asymptotically (A)dS case). Remarkably, it has been
found recently that the static black hole solutions of this model can be
constructed as a power series around the horizon, leading to a recurrence
equation that can be solved in a closed form Podolsky:2018pfe ; Svarc:2018coe
; Podolsky:2019gro ; Pravda:2020zno .
Since the presence of a conformally coupled scalar has proven to be fruitful
in the construction of analytic hairy black holes, it is also natural to
wonder about how to couple in a Weyl invariant manner a scalar field with
these higher curvature terms. The simplest way to perform such a task is to
make use of the following tensor

$S^{\mu\nu}_{\phantom{\mu\nu}\lambda\rho}=\phi^{2}R^{\mu\nu}_{\phantom{\mu\nu}\lambda\rho}-4\phi\delta^{[\mu}_{[\lambda}\nabla^{\nu]}\nabla_{\rho]}\phi+8\delta^{[\mu}_{[\lambda}\nabla^{\nu]}\phi\nabla_{\rho]}\phi-2\delta^{\mu\nu}_{\lambda\rho}\nabla_{\alpha}\phi\nabla^{\alpha}\phi\
,$ (1)

which was introduced in Oliva:2011np  in order to provide a higher dimensional
generalization of the standard conformally coupled scalar field to Euler
densities of higher degree (see also the works Dengiz:2011ig -Dengiz:2012jb
for the use of a Weyl gauging in the construction of conformal higher
curvature couplings with a vector field as well as a scalar). This tensor has
the same symmetry properties as the Riemann tensor, and under the local Weyl
rescalling

$g_{\mu\nu}\rightarrow\Omega(x)^{2}g_{\mu\nu}\ ,\
\phi\rightarrow\Omega(x)^{-1}\phi\ ,$ (2)

it transforms into

$S^{\mu\nu}_{\ \ \lambda\rho}\rightarrow\Omega(x)^{-2}S^{\mu\nu}_{\ \
\lambda\rho}\ .$ (3)

Conveniently, the standard Lagrangian for a conformally coupled scalar field
is, up to a boundary term, equal to the second trace of the tensor (1),
$S=g^{\mu\lambda}g^{\nu\rho}S_{\mu\nu\lambda\rho}$. Here after we also define
its first trace as $S_{\nu\rho}=g^{\mu\lambda}S_{\mu\nu\lambda\rho}$.
In this work we will consider the following model

$I[g,\phi]=\int\left(\frac{R-2\Lambda}{2\kappa}+\alpha_{1}R^{2}+\alpha_{2}C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}-\lambda\phi^{4}-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{12}R\phi^{2}+\beta\phi^{-4}S^{2}\right)\sqrt{-g}d^{4}x$
(4)

which can be dubbed Conformal Quadratic Gravity. We use $\kappa=8\pi G$. This
theory is the natural generalization of the standard Einstein-Hilbert
Lagrangian with a conformally coupled scalar field, including now conformal
couplings with quadratic terms in the curvature. For $\alpha_{1}=\beta=0$,
this model has been considered for example in Salvio:2017qkx  and
Salvio:2019wcp  in the context of a cosmological model for the early universe,
which may hold up to arbitrarily high energies, and has also been revisited in
second  in the realm of black holes. For non-vanishing $\alpha_{1}$ and
$\beta$, new static, spherically symmetric, asymptotically (A)dS black holes
were constructed in Caceres:2020myr . The main purpose of the present work is
to extend such explorations beyond staticity.
There are three algebraically independent quadratic combinations of the
Riemann tensor and the tensor $S_{\mu\nu\lambda\rho}$, namely
$\left(R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho},R_{\mu\nu}R^{\mu\nu},R^{2}\right)$
and
$\left(\phi^{-4}S_{\mu\nu\lambda\rho}S^{\mu\nu\lambda\rho},\phi^{-4}S_{\mu\nu}S^{\mu\nu},\phi^{-4}S^{2}\right)$,
where each of the elements of the later set transforms homogeneously, with a
weight $\Omega^{-4}\left(x\right)$ under a local Weyl rescaling (2). Using the
topological nature of the Euler combination in four dimensions, it is useful
to consider $R^{2}$ and $C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$ as the
basis for quadratic Lagrangians in four dimensions. In the conformal matter
sector, it can also be proven that
$\phi^{-4}\left(S_{\mu\nu\lambda\rho}S^{\mu\nu\lambda\rho}-4S_{\mu\nu}S^{\mu\nu}+S^{2}\right)$
is a boundary term. Additionally, denoting by
$W_{\mu\nu\lambda\rho}\left(S\right)$ the trace free part of the tensor
$S_{\mu\nu\lambda\rho}$, one can prove that
$\phi^{-4}W_{\mu\nu\lambda\rho}\left(S\right)W^{\mu\nu\lambda\rho}\left(S\right)=C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$.
Consequently, the Lagrangian (4) is the most general one containing up to
quadratic terms in both curvatures. It is interesting to notice that the
tensor $S_{\mu\nu\lambda\rho}$ can be obtained from a change of frame of the
Riemann tensor, where the scalar field emerges as a Weyl compensator. The
presence of non-conformally invariant terms in the action removes the pure
gauge nature of the scalar in such construction, and allows to introduce a new
conformal degree of freedom.
This paper is organized as follows: In Section II we provide the field
equations of the model, and review some of its properties including the
existence of static black hole solutions. In Section III, we construct new
solutions for this system within the C-metric ansatz as well as the Plebański-
Demiański family of spacetimes. Section IV is devoted to the analysis of the
causal structures of some particular cases containing Taub-NUT spacetimes as
well as asymptotically AdS wormholes. In Section V, we introduce a new family
of conformally invariant interactions which we dubbed Conformal K-essence, and
we show that they also possess, generically, hairy black holes. Our conclusion
is given in Section VI.

##  II The model and field equations

Stationary variations of the action (4) with respect to the metric and the
scalar provide the following set of field equations

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}$ $\displaystyle=\kappa
T_{\mu\nu}$ (5) $\displaystyle\left(\square-\frac{1}{6}R\right)\phi$
$\displaystyle=4\lambda\phi^{3}-4\beta\left(-S^{2}\phi^{-5}+\phi^{-3}RS-3S\phi^{-4}\square\phi-3\square\left(\phi^{-3}S\right)\right)$
(6)

where

$\displaystyle T_{\mu\nu}=-2\alpha_{1}\left(2g_{\mu\nu}\Box
R-2\nabla_{\nu}\nabla_{\mu}R+2RR_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{2}\right)-4\alpha_{2}\left(\nabla^{\alpha}\nabla^{\beta}C_{\alpha(\mu\nu)\beta}+R^{\alpha\beta}C_{\alpha(\mu\nu)\beta}\right)$
$\displaystyle-$ $\displaystyle
2\beta\left(\left(2R_{\mu\nu}-2\nabla_{\mu}\nabla_{\nu}+2g_{\mu\nu}\Box\right)(\phi^{-2}S)+12\nabla_{(\mu}(\phi^{-3}S)\nabla_{\nu)}\phi-6g_{\mu\nu}\nabla^{\alpha}(\phi^{-3}S\nabla_{\alpha}\phi)-\frac{1}{2}g_{\mu\nu}\phi^{-4}S^{2}\right)$
$\displaystyle+\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}(\partial\phi)^{2}+\frac{1}{6}\left(G_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu}\Box\right)\phi^{2}-\lambda
g_{\mu\nu}\phi^{4}\ .$ (7)

Due to the conformal invariance of the matter sector, the trace of the energy-
momentum tensor is proportional to the field equation for the scalar and
therefore, on-shell one has

$6\alpha_{1}\left(\square-\frac{1}{6\alpha_{1}}\right)\left(R-4\Lambda\right)=0\
.$ (8)

As usual this equation signals the presence of a scalar degree of freedom
coming from the quadratic gravity terms. Such d.o.f. has an effective mass of
$m_{eff}^{2}=\left(6\alpha_{1}\right)^{-1}$, and therefore, from the
perspective of flat spacetime one must impose $\alpha_{1}>0$. Using the
standard trick of multiplying and then integrating on the spacelike section of
the domain of outer communications of a would be stationary black hole, one
can prove that this equation equation (8) implies the constraint $R=4\Lambda$.
This argument also applies in some supergravity models with $R^{2}$ terms
Cisterna:2015iza .
As a matter of fact and also for simplicity, hereafter we focus on such sector
of the space of solutions of this higher curvature theory.
In the absence of the term $C_{\mu\nu\lambda\sigma}C^{\mu\nu\lambda\sigma}$ in
the action, i.e. when $\alpha_{2}=0$, it was proven in Caceres:2020myr  that
the model has the following configuration as a solution

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{\gamma}^{2}\
,\phi=\phi(r)\ ,$ (9)

being

$f(r)=-\frac{\Lambda}{3}r^{2}+\gamma\left(1-\frac{\mu}{r}\right)^{2}\ ,\
\phi(r)=\sqrt{\frac{3\mu^{2}(1+128\pi
G\Lambda(\alpha_{1}+\beta))}{4G\pi}}\frac{1}{r-\mu}\ ,$ (10)

and where $\gamma$ is the curvature of the horizon and the following relation
between the couplings must be met

$\lambda=-\frac{2\pi G\Lambda}{9(1+128\pi G\Lambda(\alpha_{1}+\beta))}\ .$
(11)

This metric, originally constructed in the model without higher derivative
terms, contains a rich set of causal structures Martinez:2002ru ;
Martinez:2004nb ; Martinez:2005di , which include asymptotically de Sitter
black holes with a scalar field regular on and outside the horizon, as well as
asymptotically locally AdS black holes with hyperbolic horizons. Including the
higher derivative terms, and assuming staticity, this is the only solution
within the ansatz with a single blackening factor. The solution can be charged
both electrically and magnetically Caceres:2020myr .

##  III The Plebański-Demiański family

The Plebański-Demiański spacetime is the most general Petrov Type-D metric
which solves the Einstein-Lambda field equations. In spite of its complexity,
in Anabalon:2009qt  it was shown that such stationary spacetime does support a
conformal hair on black holes as well as gravitational stealths (non-trivial
scalar field profiles with vanishing energy-momentum tensor AyonBeato:2004ig ;
AyonBeato:2005tu ) in the prototypical second order scalar-tensor model
$\alpha_{1}=\alpha_{2}=\beta=0$, while the the static C-metric ansatz was also
shown to lead to non-trivial solutions in this system in Charmousis:2009cm .
The ansatz leading to the Plebański-Demiański solution in General Relativity
Debever  can be written as Griffiths:2005qp

$ds^{2}=\frac{1}{\left(1-\alpha
xy\right)^{2}}\left[-\frac{X\left(x\right)\left(d\tau-\omega
y^{2}d\sigma\right)^{2}}{x^{2}+\omega^{2}y^{2}}+\frac{Y\left(y\right)\left(\omega
d\tau+x^{2}d\sigma\right)^{2}}{x^{2}+\omega^{2}y^{2}}+\left(x^{2}+\omega^{2}y^{2}\right)\left(\frac{dy^{2}}{Y\left(y\right)}+\frac{dx^{2}}{X\left(x\right)}\right)\right]\
.$ (12)

Here, $\alpha$ and $\omega$ are parameters introduced in Griffiths:2005qp ,
which facilitate taking different limits for physically relevant cases. Since
our matter field is conformally invariant, the trace of the field equations
leads to equation (8), and inspired by the no-hair result described above, we
solve such an equation by imposing

$R=4\Lambda\ .$ (13)

This equation is in particular solved by quartic polynomials on their
variables for $X$ and $Y$, while in this sector, the remaining field equations
imply the following non-trivial solution of the system

$\displaystyle X\left(x\right)$
$\displaystyle=-\left(\alpha^{2}y_{0}+\frac{1}{3}\Lambda_{0}\right)x^{4}-y_{2}x^{2}+x_{0}\
,$ (14) $\displaystyle Y\left(y\right)$
$\displaystyle=-\left(x_{0}\alpha^{2}+\frac{1}{3}\Lambda_{0}\omega^{2}\right)y^{4}+y_{2}y^{2}+y_{0}\
,$ (15) $\displaystyle\phi\left(x,y\right)$
$\displaystyle=\frac{B\left(1-\alpha xy\right)}{\left(1+\alpha xy\right)}\ ,$
(16)

which solves the field equations provided

$\kappa
B^{2}=6\left(1+16\kappa\Lambda_{0}\left(\alpha_{1}+\beta\right)\right)\ \
\text{and}\ \ \lambda=-\frac{\Lambda_{0}}{6B^{2}}.$ (17)

This configuration extends to the realm of Conformal Quadratic Gravity the
hairy Plebański-Demiański solution constructed in Anabalon:2009qt  as well as
the C-metric found in Charmousis:2009cm . As in Anabalon:2009qt , there are
also extra branches that lead to non-trivial scalar field profiles with
vanishing energy-momentum tensor on locally constant curvature spacetimes.
Nevertheless, the configuration defined by (14)-(16) defines a spacetime with
non-trivial curvature. Here, $x_{0},y_{0}$ and $y_{2}$ are integration
constants. Along the lines of Griffiths:2005qp  one can analyze the different
causal structures contained in (12). Below, we obtain the C-metric as well as
the topological Taub-NUT spacetime integrating the equations from scratch,
also we provide comments on their causal structures.

###  III.1 C-metric black holes

The metric

$ds^{2}=\frac{1}{\left(y-Ax\right)^{2}}\left[\frac{dx^{2}}{X\left(x\right)}+\frac{dy^{2}}{Y\left(y\right)}-Y\left(y\right)dt^{2}+X\left(x\right)d\sigma^{2}\right]\
,$ (18)

with

$\displaystyle X(x)$
$\displaystyle=-A^{2}y_{4}x^{4}-Ay_{3}x^{3}-y_{2}x^{2}-\frac{y_{3}(4y_{2}y_{4}-y_{3}^{2})}{8Ay_{4}^{2}}x+x_{0}\
,$ (19) $\displaystyle Y(x)$
$\displaystyle=y_{4}y^{4}+y_{3}y^{3}+y_{2}y^{2}+\frac{y_{3}(4y_{2}y_{4}-y_{3}^{2})}{8y_{4}^{2}}y-A^{2}x_{0}-\frac{\Lambda_{0}}{3}\
,$ (20) $\displaystyle\phi(x,y)$
$\displaystyle=4y_{4}\sqrt{\frac{3}{2\kappa}\left(1+16\kappa\Lambda_{0}\left(\alpha_{1}+\beta\right)\right)}\frac{y-Ax}{2y_{4}\left(Ax+y\right)+y_{3}}\
,$ (21)

is a solution of the higher-derivative action (4), provided the couplings of
the theory are related as

$\lambda=-\frac{\Lambda_{0}\kappa}{36\left(1+16\kappa\Lambda_{0}\left(\alpha_{1}+\beta\right)\right)}\
.$ (22)

Here $x_{0},y_{2},y_{3}$ and $y_{4}$ are constants. Depending on the structure
of zeros of the polynomial $X\left(x\right)$ one can have constant $t$
surfaces with different topologies. Our solution is an extension to the higher
derivative regime of the hairy C-metric found in Charmousis:2009cm  and
Anabalon:2009qt . The sign of the integration constants is not fixed a priori,
in contraposition to what occurs with a term of the form $Q^{2}$ in the
metric, when $Q$ is the electric charge. When $X\left(x\right)$ has two zeros
at $x=x_{\max}$ and $x=x_{\min}$, and it is positive between such points, one
can restrict the coordinate $x$ to such regions and identify the coordinate
$\sigma$. In general, this introduces conical singularities at $x=x_{\max}$
and $x=x_{\min}$, both of which can be remarkably removed in this case due to
the backreaction of the conformal scalar, leading to the cohomogeneity-two
black holes reported in Anabalon:2009qt . As shown above, such structure
persists when higher curvature terms are included.

##  IV Topological Taub-NUT with a conformal hair

It is known from Anabalon:2009qt  that a limit can be taken from the
Plebański-Demiański spacetime which leads to Taub-NUT spacetime in the second
order theory defined by (4) with $\alpha_{1}=\alpha_{2}=\beta=0$. Even more,
such spacetimes can be conveniently obtained by direct integration
Bhattacharya:2013hvm  and by using a generalization of the Ernst generating
technique Bardoux:2013swa ; Astorino:2014mda . In presence of the higher
derivative terms, when the relation (22) holds, the configurations are
corrected and lead to

$ds^{2}=-f\left(r\right)\left(dt+n\left\\{\begin{array}[c]{c}4\sin^{2}\left(\frac{\theta}{2}\right)\\\
\rho^{2}\\\
4\sinh^{2}\left(\frac{\theta}{2}\right)\end{array}\right\\}d\phi\right)^{2}+\frac{dr^{2}}{f\left(r\right)}+\left(r^{2}+n^{2}\right)\left(\left\\{\begin{array}[c]{c}d\theta^{2}\\\
d\rho^{2}\\\
d\psi^{2}\end{array}\right\\}+\left\\{\begin{array}[c]{c}\sin^{2}\theta\\\
\rho^{2}\\\ \sinh^{2}\psi\end{array}\right\\}d\phi^{2}\right)\ ,$ (23)

with

$\displaystyle f\left(r\right)$
$\displaystyle=-\frac{\Lambda_{0}}{3}\left(r^{2}+n^{2}\right)+\left(\gamma-\frac{4}{3}n^{2}\Lambda_{0}\right)\frac{\left(r-M\right)^{2}}{r^{2}+n^{2}}$
(24) $\displaystyle\phi\left(r\right)$
$\displaystyle=2\sqrt{\frac{3\left(1+16\kappa\Lambda_{0}\left(\alpha_{1}+\beta\right)\right)}{2\kappa}}\frac{\sqrt{M^{2}+n^{2}}}{r-M}$
(25)

where $\gamma=+1,0,-1$, corresponds to the first, second and third line of
(23), respectively, $M$ is the mass and $n$ the NUT parameter (notice that
Taub-NUT/Bolt solutions with scalar hair have also been recently obtained in
Ref. Arratia:2020hoy ).
Restricting to the hyperbolic case ($\gamma=-1$), one can perform the
following formal transformation on such a configuration

$t=ui+nT\text{, }\psi=\theta+\frac{\pi}{2}i\text{ and }\phi=T\ ,$ (26)

which leads to

$ds^{2}=\frac{dr^{2}}{f\left(r\right)}+f\left(r\right)\left(du+2n\sinh\theta
dT\right)^{2}+\left(r^{2}+n^{2}\right)\left(-\cosh^{2}\theta
dT^{2}+d\theta^{2}\right)\ .$ (27)

with $f(r)$ and the scalar field given by (24) and (25), respectively. For
negative cosmological constant, when $f\left(r\right)>0$ in the domain
$-\infty<r<\infty$, a condition that is achieved in an open set of the
parameter space $(n,M)$, this configuration describes the hairy extension of
the asymptotically locally AdS wormhole found in Anabalon:2018rzq . When a
Maxwell field is included, this spacetime can be embedded in $\mathcal{N}=2$
gauged supergravity and admits non-trivial Killing spinors Anabalon:2020loe .
It is interesting to notice, that the formal transformation (26) is not a
double Wick rotation but still it leads to a Lorentzian geometry with a new
causal structure. It is known that the hyperbolic Taub-NUT AdS is devoid of
closed timelike curves, and the wormhole geometry and its hairy extension
presented here maintain this property, which was proven in Anabalon:2018rzq
along the lines of the analysis performed in Coussaert:1994tu  for a
particular quotient of AdS3 spacetime.

##  V A general family of conformally invariant Lagrangians: Conformal
K-essence

Since under a local Weyl rescalling the tensor $S_{\ \ \lambda\rho}^{\mu\nu}$
transforms as $S_{\ \ \lambda\rho}^{\mu\nu}\rightarrow\Omega^{-4}S_{\ \
\lambda\rho}^{\mu\nu}$, the matter theory

$I_{\text{matter}}=\int d^{4}x\sqrt{-g}\phi^{4}F\left(\phi^{-4}S\right)\ ,$
(28)

is invariant under conformal transformations for an arbitrary function
$F\left(Y\right)=F\left(\phi^{-4}S\right)$. When $F$ is a constant function,
the action (28) corresponds to the conformal potential, while for $F$ being
the identity function, (28) reduces to the standard conformally coupled scalar
field, up to a boundary term. The combination
$Y=\phi^{-4}S=\phi^{-4}\left(\phi^{2}R-6\phi\square\phi\right)$ is indeed
conformally invariant. The action defined by (28) can be thought of as a
conformally invariant extension of a K-essence Lagrangian, namely a
conformally invariant version of the action $K\left(\phi\square\phi\right)$.
Coupling this conformal invariant action to GR

$I=\int\sqrt{-g}d^{4}x\left[R-2\Lambda+\phi^{4}F\left(\phi^{-4}S\right)\right]\
,$ (29)

leads to the field equations

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}$ $\displaystyle=T_{\mu\nu}\ $
(30) $\displaystyle
2\phi^{3}F\left(Y\right)-R\phi\frac{dF}{dY}+9\square\phi\frac{dF}{dY}-3\square\left(\phi\frac{dF}{dY}\right)$
$\displaystyle=0\ $ (31)

where

$\displaystyle T_{\mu\nu}$
$\displaystyle=\frac{1}{2}g_{\mu\nu}\phi^{4}F\left(Y\right)-\frac{dF}{dY}\left(\phi^{2}R_{\mu\nu}-6\phi\nabla_{\mu}\nabla_{\nu}\phi\right)+\nabla_{\mu}\nabla_{\nu}\left(\frac{dF}{dY}\phi^{2}\right)$
$\displaystyle-3\left(\nabla_{\mu}\left(\frac{dF}{dY}\phi\nabla_{\nu}\phi\right)+\nabla_{\nu}\left(\frac{dF}{dY}\phi\nabla_{\mu}\phi\right)\right)-\square\left(\frac{dF}{dY}\phi^{2}\right)g_{\mu\nu}+3\nabla^{\lambda}\left(\frac{dF}{dY}\phi\nabla_{\lambda}\phi\right)g_{\mu\nu}\
.$ (32)

One can check that, as expected from conformal invariance, the trace of the
energy-momentum tensor (32) vanishes by virtue of the equation for the scalar
field (31). The configuration

$\displaystyle ds^{2}$
$\displaystyle=-\left(-\frac{\Lambda_{0}}{3}r^{2}+\left(1-\frac{M}{r}\right)^{2}\right)dt^{2}+\frac{dr^{2}}{-\frac{\Lambda_{0}}{3}r^{2}+\left(1-\frac{M}{r}\right)^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\psi^{2}\right)$ (33) $\displaystyle\phi\left(r\right)$
$\displaystyle=\frac{A}{r-M}\ ,$ (34)

leads to a constant Weyl invariant combination

$Y=\phi^{-4}S=\phi^{-4}\left(\phi^{2}R-6\phi\square\phi\right)=\frac{4\Lambda_{0}M^{2}}{A^{2}}\
.$ (35)

Here we focus on the $\gamma=1$ case. Specifying the function $F$ in the
action (29), the field equations on the configuration defined by (33) and (34)
lead to the following two constraints

$\displaystyle 2M^{4}\Lambda_{0}+A^{4}F\left(Y\right)$ $\displaystyle=0\ ,\ $
(36) $\displaystyle
A^{2}F\left(Y\right)-2\Lambda_{0}M^{2}\frac{dF\left(Y\right)}{dY}$
$\displaystyle=0\ .$ (37)

These constraints can be interpreted as follows: the constraint (36) fixes the
value of the constant appearing on the scalar field $A$, while the constraint
(36) fixes the different parameters that appear in the action functional
through the election of the function $F\left(Y\right)$. For example, choosing
$F\left(Y\right)=-\frac{1}{12}Y-\lambda$ in (29) leads to the standard
conformally coupled scalar field with a conformal potential coupling $\lambda$
such that

$\phi\left(r\right)=2\sqrt{3}\frac{M}{r-M}\text{ and
}\lambda=-\frac{\Lambda_{0}}{72}\ ,$ (38)

which reduces to the solution found in Martinez:2002ru  with the normalization
$16\pi G=1$. We have therefore proved that provided the constraints (36) and
(37) are fulfilled, the black hole configuration (33) and (34) is a solution
of General Relativity supported by a general Conformal K-essence field (28).

##  VI Further comments

We have shown that the Plebański-Demiański spacetime, and therefore all Type-D
metrics, as for example the C-metric and topological Taub-NUT spacetimes can
be embedded as solutions of a quadratic gravity model, conformally coupled to
a scalar field with higher derivative corrections. Some of these spacetimes
have already been studied in the presence of a conformally coupled scalar
field with second order dynamics and we expect that in the black hole case,
the higher derivative terms may induce modifications on the thermodynamics, as
well as in the perturbative stability properties of the solutions. Our
solutions can also be extended to include a Maxwell field including both
electric and magnetic sources, which in the case of static black holes with
hyperbolic horizons in AdS may describe the causal structure of a black hole
inside a black hole Martinez:2005di .
The tensor $S_{\ \ \lambda\rho}^{\mu\nu}$ is the building block that allowed
to construct conformal couplings of a scalar to Euler densities in arbitrary
dimension Oliva:2011np . In such setup, one can also construct black hole
solutions with simple scalar field profiles, provided the couplings of the
different terms in the matter sector fulfil a relation Giribet:2014bva ;
Chernicoff:2016jsu ; Chernicoff:2016uvq ; Chernicoff:2016qrc , leading to an
interesting phase structure Giribet:2014fla ; Galante:2015voa ;
Hennigar:2015wxa ; Hennigar:2016xwd . Such constraint is of a different sort
than the one imposed by (22) which relates the gravitational and the matter
couplings. Using the same tensor, it would be interesting to construct new
higher curvature and derivative couplings in higher dimensions and classify
them. One may even consider complete contractions of tensor products of Weyl
tensor $C_{\ \ \lambda\rho}^{\mu\nu}$ and copies of the tensor $S_{\ \
\lambda\rho}^{\mu\nu}$. Such combinations will transform homogeneously under
conformal transformations and therefore it will lead to conformally invariant
Lagrangians by the introduction of a compensating power of the scalar field.
In order to go beyond the standard conformally coupled scalar field, but still
within the realm of a second order theory, one is forced to introduce a second
scalar leading to a Bi-scalar Extensions of Horndeski Theories
Charmousis:2014zaa , which may admit some of the stationary solutions we have
found in this paper.
Finally, we have introduced a general conformally invariant model that depends
on an arbitrary function, which we have dubbed Conformal K-essence. It is
tempting to conjecture that such general family of theories do also admit
stationary solutions in the Plebański-Demiański family. To study different
application of such Conformal K-essence it would be interesting to study first
the potential formation of caustic as was done for example in Babichev:2016hys
for the standard K-essence theory. Work along these lines is in progress.

##  VII Acknowledgments

We thank Andrés Anabalón, Nicolás Cáceres, José Figueroa, Nicolás Mora,
Marcelo Oyarzo and Ricardo Stuardo for valuable discussions. We would like to
express our gratitude to Carmen Lagler for her detailed proofreading of this
manuscript. This work is partially funded by Beca Chile de Postdoctorado
74200012 and FONDECYT grant 1181047 and 1210500. J.O. also thanks the support
of Proyecto de Cooperación Internacional 2019/13231-7 FAPESP/ANID.

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#  Metallic structures on the tangent bundle of a P-Sasakian manifold

Shahroud Azami  Department of Mathematics, Faculty of Sciences, Imam Khomeini
International University, Qazvin, Iran<EMAIL_ADDRESS>
###### Abstract.

In this article, we introduce some metallic structures on the tangent bundle
of a P-Sasakian manifold by complete lift, horizontal lift and vertical lift
of a P-Sasakian structure $(\phi,\eta,\xi)$ on tangent bundle. Then we
investigate the integrability and parallelity of these metallic structures.

###### Key words and phrases:

P-Sasakian manifold, complete lift, metallic structure, integrability.

###### 2010 Mathematics Subject Classification:

53C15, 53C25, 53C21

##  1\. Introduction

The lift of geometrical objects, vector fields, forms, has an important role
in differential geometry. By the method of lift, we can generalize to
differentiable structure on any manifold to its tangent bundle and any other
bundles on manifold [4, 5, 7]. In this paper we study the metallic structures
on tangent bundle of a P-Sasakian Riemannian manifold. The metallic structure
is a generalization of the almost product structure. A metallic structure is a
polynomial structure as defined by Goldenberg et al. in [1, 2]. In [3],
introduced the notation of metallic structure on a Riemannian manifold.
Suppose that $p$ and $q$ are two positive integers. The positive solution of
the equation $x^{2}-px-q=0$ is called members of the metallic means family.
These number showed by $\sigma_{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$, where it is
a generalization of golden proportions.

######  Definition 1.1.

Let $M$ be a manifold. A metallic structure on $M$ is an $(1,1)$ tensor field
$J$ which satisfies the equation $J^{2}=pJ+qI$, where $p,q$ are positive
integers and $I$ is the identity operators on the Lie algebra $\mathcal{X}(M)$
of vector fields on $M$. If $g$ is a Riemannian metric on $M$, then we say
that $g$ is $J$\- compatible whenever

(1.1) $g(JX,Y)=g(X,JY),\,\,\,\,\,\,\forall X,Y\in\mathcal{X}(M)$

or equivalently

(1.2) $g(JX,JY)=pg(X,JY)+qg(X,Y),\,\,\,\,\,\,\forall X,Y\in\mathcal{X}(M).$

In this case $(M,J,g)$ is named a metallic Riemannian manifold.

Let $J$ be a metallic structure on $M$. Then the Nijenhuise tensor $N_{J}$ of
$J$ is a tensor field of type $(1,2)$ given by

$N_{J}(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]+J^{2}[X,Y]$

for $X,Y\in\mathcal{X}(M)$.

On the other hand, at first time, Sato in [6] introduced the P-Sasakian
structure on a manifolds and studied several properties of these manifolds. An
$n$-dimensional smooth manifold $M$ is called an almost paracontact manifold
if it admits an almost paracontact structure $(\phi,\eta,\xi)$, consisting of
a $(1,1)$ tensor field $\phi$, a $1$-form $\eta$ and a vector field $\xi$
which satisfy the condition

(1.3)
$\phi^{2}=I-\eta\otimes\xi,\,\,\,\,\eta(\xi)=1,\,\,\,\,\,\phi\xi=0,\,\,\,\,\,\eta\circ\phi=0.$

Let $g$ be a Riemannian metric compatible with $(\phi,\eta,\xi)$ i.e.

(1.4) $g(X,Y)=g(\phi X,\phi Y)+\eta(X)\eta(Y),\,\,\,\,\,\forall
X,Y\in\mathcal{X}(M),$

or equivalently

(1.5) $g(X,\phi Y)=g(\phi X,Y),\,\,\,\,\,\,g(X,\xi)=\eta(X),\,\,\,\,\,\forall
X,Y\in\mathcal{X}(M),$

where $\mathcal{X}(M)$ is the collection of all smooth vector field on $M$.
Then $M$ is said to be an almost paracontact Riemannian manifold.
An almost paracontact Riemannian manifold $(M,g)$ is called a P-Sasakian
manifold if it satisfies

(1.6) $(\nabla_{X}\phi)(Y)=-g(X,Y)\xi-\eta(Y)X+2\eta(X)\eta(Y)\xi$

where $\nabla$ is the Levi-Civita connection of the Riemannain manifold. We
have

(1.7) $\nabla_{X}\xi=\phi X,\,\,\,\,\,\,\,(\nabla_{X}\eta)(Y)=g(\phi
X,Y)=(\nabla_{Y}\eta)(X),\,\,\,\,\,\,\,\forall X,Y\in\mathcal{X}(M).$

##  2\. Lifts of geometric structure on tangent bundle

Let $(M,g)$ be a smooth $n$-dimensional Riemannian manifold and denote its
tangent bundle by $TM$. We denote by $\pi:TM\to M$ the natural projection,
where it defines the natura bundle structure of $TM$ over $M$ and denote by
$T_{l}^{k}(M)$ the set of all tensor fields of the type $(k,l)$ in $M$. For
any point $(x,y)\in TM$, Let $V_{y}=\ker\\{\pi_{*}(y):T_{y}(TM)\to T_{x}M\\}$
and $VTM=\cup_{y\in TM}V_{y}$. Also, suppose that $HTM$ be a complement of
$VTM$ in $TM$, that is

$TTM=VTM\oplus HTM.$

$VTM$ and $HTM$ are called vertical distribution and horizontal distribution,
respectively. Suppose that the space $M$ is covered by a system of coordinate
neighbourhoods $(U,\varphi)=(U,x^{1},x^{2},...,x^{n})$, then the corresponding
induced local chart on $TM$ is
$(\pi^{-1}(U),x^{1},x^{2},...,x^{n},y^{1},y^{2},...,y^{n})$. If in any point
of $x\in M$, $\Gamma_{ki}^{h}(x)$ be the Christoffel symbols of $g$, then the
sets of vector fields $\\{\frac{\partial}{\partial
y^{1}},...,\frac{\partial}{\partial y^{n}}\\}$ and $\\{\frac{\delta}{\delta
x^{1}},...,\frac{\delta}{\delta x^{n}}\\}$ on $\pi^{-1}(U)$ define local frame
fields for $VTM$ and $HTM$, respectively, where $\frac{\delta}{\delta
x^{i}}=\frac{\partial}{\partial
x^{i}}-y^{k}\Gamma_{ki}^{l}\frac{\partial}{\partial y^{l}}$. Notice that the
set $\\{\frac{\partial}{\partial y^{i}},\frac{\delta}{\delta x^{j}}\\}$
defines a local frame on $TM$. In following from [4, 7], we recall some lifts
of geometrical objects of a manifold to its tangent bundle.

###  2.1. Vertical lifts

Let $f$ be a function on $M$. Then the vertical lift of $f$ to $TM$ is the
function $f^{v}$ on $TM$ given by $f^{v}=f\circ\pi$. For any vector field
$X\in\mathcal{X}(M)$, we define a vector field $X^{v}$ in $TM$ by
$X^{v}(\iota\omega)=(\omega(X))^{v}$, where $\omega$ is an arbitrary $1$-form
in $M$, We call $X^{v}$ the vertical lift of $X$. Notice, $X^{v}\in VTM$ and
for all function on $M$ we define $X^{v}(df)=X.f$. Let $F$ be a tensor field
of type $(1,r)$ or $(0,r)$, $r\geq 1$ on $M$. Then the vertical lift of $F$ on
$TM$ defined by

$F_{y}^{v}(\tilde{X}_{1},\tilde{X}_{2},...,\tilde{X}_{r})=\Big{(}F_{y}\big{(}\pi_{\star}(\tilde{X}_{1}),\pi_{\star}(\tilde{X}_{2}),...,\pi_{\star}(\tilde{X}_{r})\big{)}\Big{)}^{v}$

where $\tilde{X}_{1},\tilde{X}_{2},...,\tilde{X}_{r}\in T_{y}(TM),\,\,\,y\in
T_{x}M,\,\,\,x\in M$. Hence for any $X_{1},...,X_{r}\in\mathcal{X}(M)$ we have

$F^{v}({X}_{1}^{v},{X}_{2}^{v},...,{X}_{r}^{v})=0,\,\,F^{v}({X}_{1}^{c},{X}_{2}^{c},...,{X}_{r}^{c})=\Big{(}F({X}_{1},{X}_{2},...,{X}_{r}\big{)}\Big{)}^{v}.$

###  2.2. Complete lifts

If $f$ is a function in $M$, then the complete lift of $f$ is the function
$f^{c}$ on $TM$ and defined by

(2.1) $f^{c}(x,y)=df(x)(y),\,\,\,y\in T_{x}M,\,\,x\in M.$

Also the complete lift of vector field $X=X^{i}\frac{\partial}{\partial
x^{i}}$ on $M$ defined by

$X^{c}=X^{i}\frac{\partial}{\partial x^{i}}-y^{j}\frac{\partial
X^{i}}{\partial x^{j}}\frac{\partial}{\partial y^{i}}.$

Therefore we obtain $(\frac{\partial}{\partial
x^{i}})^{c}=\frac{\partial}{\partial x^{i}}$ and $X^{c}f^{c}=(Xf)^{c}$ for any
function $f$ on $M$. Suppose that $\omega$ is a $1$-form on $M$. The complete
lift of $\omega$ on $TM$ defined by
$\omega^{c}(X^{c})=(\omega(X))^{c},\,\,\,\,\omega^{c}(X^{v})=(\omega(X))^{v},\,\,\,X\in\mathcal{X}(M)$.
In general case, the complete lift of a tensor field $F$ of type $(1,r)$ or
$(0,r)$, $r\geq 1$, on $M$ defined by
$F^{c}(X_{1}^{c},...,X_{r}^{c})=(F(X_{1},...,X_{r}))^{c}$ for any
$X_{1},...,X_{r}\in\mathcal{X}(M)$. Then the complete lift of a Riemannian
metric $g$ defined by

$g^{c}=\left(\begin{array}[]{cc}y^{k}\frac{\partial g_{ij}}{\partial
x^{k}}&g_{ij}\\\ g_{ij}&0\\\ \end{array}\right).$

From [4] and [7], we have

######  Proposition 2.1.

Let $M$ be a manifold with a Riemannain metric $g$. For any
$X,Y\in\mathcal{X}(M),\,f\in C^{\infty}(M)$, and $(1,1)$ tensor field $F$ we
have :

  * •

$X^{v}f^{v}=0,\,\,X^{v}f^{c}=X^{c}f^{v}=(Xf)^{v},\,\,X^{c}f^{c}=(Xf)^{c}$,

  * •

$F^{c}(X^{v})=(F(X))^{v}$,

  * •

$g^{c}$ is semi-Riemannian metric and

$g^{c}(X^{v},Y^{c})=g^{c}(X^{c},Y^{v})=(g(X,Y))^{v},\,\,g^{c}(X^{v},Y^{v})=0,\,\,g^{c}(X^{c},Y^{c})=(g(X,Y))^{c}$

,

  * •

If $P(x)$ is a polynomial in one variable $x$ then $P(F^{c})=(P(F))^{c}$.

We define the complete lift of a linear connection $\nabla$ to $TM$ as the
unique linear connection $\nabla^{c}$ on $TM$ as
$\nabla^{c}_{X^{c}}Y^{c}=(\nabla_{X}Y)^{c}$ for $X,Y\in\mathcal{X}(M)$,
therefore

$\displaystyle\nabla^{c}_{\frac{\partial}{\partial
x^{i}}}\frac{\partial}{\partial x^{j}}$ $\displaystyle=$
$\displaystyle\Gamma_{ij}^{k}\frac{\partial}{\partial
x^{k}}+y^{l}\frac{\partial\Gamma_{ij}^{k}}{\partial
x^{l}}\frac{\partial}{\partial y^{k}},\,\,\nabla^{c}_{\frac{\partial}{\partial
y^{i}}}\frac{\partial}{\partial y^{j}}=0,$
$\displaystyle\nabla^{c}_{\frac{\partial}{\partial
x^{i}}}\frac{\partial}{\partial y^{j}}$ $\displaystyle=$
$\displaystyle\Gamma_{ij}^{k}\frac{\partial}{\partial
y^{k}},\,\,\nabla^{c}_{\frac{\partial}{\partial
y^{i}}}\frac{\partial}{\partial x^{j}}=\Gamma_{ij}^{k}\frac{\partial}{\partial
y^{k}}.$

######  Proposition 2.2 ([4, 7]).

Let $T$ and $R$ be the torsion and curvature tensor of $\nabla$, respectively.
Then $T^{c}$ and $R^{c}$ are the torsion and curvature tensor of $\nabla^{c}$,
respectively, and

  * •

$\nabla$ is symmetric if and only if $\nabla^{c}$ is symmetric.

  * •

$\nabla$ is flat if and only if $\nabla^{c}$ is flat.

######  Proposition 2.3 ([4, 7]).

Let $F$ be a tensor field of type $(1,r)$ or $(0,r)$, $r\geq 1$, on $M$ and
$X,Y\in\mathcal{X}(M)$, we have

$\displaystyle\nabla^{c}_{X^{v}}Y^{v}$ $\displaystyle=$
$\displaystyle(\nabla_{X}Y)^{v},\,\,\nabla^{c}_{X^{c}}Y^{c}=(\nabla_{X}Y)^{c},\,\,\nabla^{c}_{X^{v}}Y^{c}=\nabla^{c}_{X^{c}}Y^{v}=(\nabla_{X}Y)^{v},$
$\displaystyle\nabla^{c}F^{v}$ $\displaystyle=$ $\displaystyle(\nabla
F)^{v},\,\,\nabla^{c}F^{c}=(\nabla F)^{c}.$

###  2.3. Horizontal lifts

The horizontal lift $f^{h}$ of a function $f$ on $M$ is given by
$f^{h}=f^{c}-\nabla_{\gamma}f$ where $\nabla_{\gamma}f=\gamma(\nabla f)$,
where for any tensor field $F$ of type $(1,r)$ or $(0,r)$, $r\geq 1$, on $M$,
$\gamma_{X}F=(F_{X})^{v}$ and
$F_{X}(X_{1},...,F_{r-1})=F(X_{1},...,X_{r-1},X)$. For any vector field
$X=X^{i}\frac{\partial}{\partial x^{i}}$ on $M$ there exists a unique vector
$X^{h}\in HVM$ such that $\pi_{*}X^{h}=X$, that is if
$X=X^{i}\frac{\partial}{\partial x^{i}}$ then
$X^{h}=X^{i}\frac{\partial}{\partial
x^{i}}-y^{j}X^{i}\Gamma_{ij}^{l}\frac{\partial}{\partial y^{l}}$. We call
$X^{h}$ the horizontal lift of $X$ in the point $(x,y)\in TM$. Let $\omega$ be
a $1$-form on $M$. Then the horizontal lift $\omega^{h}$ of $\omega$ is
defined by $\omega^{h}=\omega^{c}-\nabla_{\gamma}\omega$. Then for any
$X\in\mathcal{X}(M)$ we have
$\omega^{h}(X^{h})=0,\,\,\omega^{h}(X^{v})=(\omega(X))^{v}$. The horizontal
lift of a $(1,1)$ tensor field $F$ on $M$ defined by
$F^{h}(X^{h})=(FX)^{h},\,\,F^{h}({X^{v}})=(FX)^{v}$. The horizontal lifts of a
Riemannian metric $g$ defined by
$g^{h}=g_{ij}\theta^{i}\otimes\eta^{j}+g_{ij}\eta^{i}\otimes\theta^{j}$ where
$\theta^{i}=dx^{i},\,\,\eta^{i}=y^{j}\Gamma_{jk}^{i}dx^{k}+dy^{i}$.
From [4, 7], we have

######  Proposition 2.4.

Let $M$ be a manifold with a Riemannain metric $g$. For any
$X,Y\in\mathcal{X}(M),\,f\in C^{\infty}(M)$, and $(1,1)$ tensor field $F$ we
have :

  * •

$g^{h}$ is semi-Riemannian metric and $g^{h}(X^{v},Y^{h})=(g(X,Y))^{v},$
$g^{h}(X^{v},Y^{v})=g^{h}(X^{h},Y^{h})=0$,

  * •

If $P(x)$ is a polynomial in one variable $x$ then $P(J^{h})=(P(J))^{h}$,

  * •

$g^{h}=g^{c}$ if and only if $\nabla g=0$,

  * •

$[X^{v},Y^{v}]=0,\,\,[X^{v},Y^{c}]=[X,Y]^{v},\,\,[X^{c},Y^{c}]=[X,Y]^{c}$,
$[X^{v},Y^{h}]=-(\nabla_{Y}X)^{v},\,\,[X^{h},Y^{h}]=[X,Y]^{h}-\gamma R(X,Y)$
where $R$ is curvature tensor of $g$ and the vertical vector lift $\gamma F$
defined by $(\gamma F)(y)=(F(y))^{v}$.

Let $\nabla$ be a linear connection on $M$, then we define the horizontal lift
of $\nabla$ to $TM$ as the unique linear connection $\nabla^{h}$ on $TM$ given
by

$\nabla^{h}_{X^{v}}Y^{v}=\nabla^{h}_{X^{v}}Y^{h}=0,\,\,\nabla^{h}_{X^{h}}Y^{v}=(\nabla_{X}Y)^{v},\,\,\nabla^{h}_{X^{h}}Y^{h}=(\nabla_{X}Y)^{h},$

for $X,Y\in\mathcal{X}(M)$. Hence
$\nabla^{h}_{X^{c}}Y^{c}=(\nabla_{X}Y)^{c}-\gamma R(.\,,X,Y)$ where
$R(.\,,X,Y)Z=R(Z,X,Y)$.

##  3\. Metallic structures on the tangent bundle of a P-Sasakian manifold

Let $M$ be an $n$-dimensional P-Sasakian manifold with structure tensor
$(\phi,\eta,\xi,g)$. In this part of this section we study the metallic
structure induced on $TM$ by the complete lift of a P-Sasakian structure.

######  Proposition 3.1.

On the tangent bundle of P-Sasakian manifold with structure tensor
$(\phi,\eta,\xi,g)$, there exists a metallic structure given by

(3.1)
$J=\frac{p}{2}I-(\frac{2\sigma_{p,q}-p}{2})(\phi^{c}+\eta^{v}\otimes\xi^{v}+\eta^{c}\otimes\xi^{c}).$

###### Proof.

From the definition of the almost paracontact structure of P-Sasakian
manifold, we obtain the following relations

$\displaystyle(\phi^{c})^{2}=(\phi^{2})^{c}=I-\eta^{c}\otimes\xi^{v}-\eta^{v}\otimes\xi^{c},$
$\displaystyle\eta^{v}(\xi^{c})=\eta^{c}(\xi^{v})=1,\,\,\,\,\,\,\eta^{v}(\xi^{v})=\eta^{c}(\xi^{c})=0,$
$\displaystyle\phi^{c}(\xi^{v})=\phi^{c}(\xi^{c})=0,\,\,\,\,\,\,\eta^{v}\circ\phi^{c}=\eta^{c}\circ\phi^{c}=0.$

Therefore for any $\tilde{X}\in\mathcal{X}(TM)$ we have

$\displaystyle
J(\xi^{v})=\frac{p}{2}\xi^{v}-\frac{2\sigma_{p,q}-p}{2}\xi^{c},\,\,\,\,\,\,J(\xi^{c})=\frac{p}{2}\xi^{c}-\frac{2\sigma_{p,q}-p}{2}\xi^{v},$
$\displaystyle
J(\phi^{c}\tilde{X})=\frac{p}{2}\phi^{c}\tilde{X}-\frac{2\sigma_{p,q}-p}{2}(\tilde{X}-\eta^{c}(\tilde{X})\xi^{v}-\eta^{c}(\tilde{X})\xi^{c}).$

Now, we obtain

$J(\tilde{X})=\frac{p}{2}\tilde{X}-\frac{2\sigma_{p,q}-p}{2}(\phi^{c}\tilde{X}+\eta^{v}(\tilde{X})\xi^{v}+\eta^{c}(\tilde{X})\xi^{c})$

and

$J^{2}(\tilde{X})=\frac{p}{2}J(\tilde{X})-\frac{2\sigma_{p,q}-p}{2}(J(\phi^{c}\tilde{X})+\eta^{v}(\tilde{X})J(\xi^{v})+\eta^{c}(\tilde{X})J(\xi^{c}))=pJ(\tilde{X})+q\tilde{X},$

and it complete the proof. ∎

######  Proposition 3.2.

If $M$ is a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$ and
$J$ is defined by (3.1) then we have

(3.2)
$g^{c}(J\tilde{X},J\tilde{Y})=pg^{c}(\tilde{X},J\tilde{Y})+qg^{c}(\tilde{X},\tilde{Y}),\,\,\,\,\,\forall\tilde{X},\tilde{Y}\in\mathcal{X}(TM).$

###### Proof.

For any $X,Y\in\mathcal{X}(M)$, we have

$\displaystyle
g^{c}(X^{v},Y^{v})=0,\,\,\,g^{c}(X^{v},\xi^{c})=(g(X,\xi))^{v}=(\eta(X))^{v},$
$\displaystyle g^{c}((\phi X)^{v},\xi^{c})=(g(\phi
X,\xi))^{v}=(g(X,\phi\xi))^{v}=0,$ $\displaystyle
g^{c}(\xi^{c},\xi^{c})=(g(\xi,\xi))^{c}=0.$

Therefore

$g^{c}(JX^{v},JY^{v})=-\frac{(2\sigma_{p,q}-p)p}{2}(\eta X)^{v}(\eta Y)^{v},$

and

$g^{c}(X^{v},JY^{v})=-\frac{2\sigma_{p,q}-p}{2}(\eta X)^{v}(\eta Y)^{v}.$

Thus

$g^{c}(JX^{v},JY^{v})=pg^{c}(X^{v},JY^{v})+qg^{c}(X^{v},Y^{v}).$

Also, using $g^{c}(X^{v},Y^{c})=(g(X,Y))^{v}$ and
$g^{c}(X^{c},Y^{c})=(g(X,Y))^{c}$ we have

$g^{c}(JX^{v},JY^{c})=(\frac{p^{2}}{2}+q)(g(X,Y))^{v}-\frac{(2\sigma_{p,q}-p)p}{2}[(g(X,\phi
Y))^{v}-(\eta X)^{v}(\eta Y)^{v}],$

and

$g^{c}(X^{v},JY^{c})=\frac{p}{2}(g(X,Y))^{v}-\frac{(2\sigma_{p,q}-p)p}{2}[(g(X,\phi
Y))^{v}-(\eta X)^{v}(\eta Y)^{v}].$

Hence

$g^{c}(JX^{v},JY^{c})=pg^{c}(X^{v},JY^{c})+qg^{c}(X^{v},Y^{c}).$

The other cases are similar. ∎

######  Theorem 3.3.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$ and
$J$ be defined by (3.1) then the metallic structure $J$ is integrable.

###### Proof.

The $1$-form $\eta$ defines on $(n-1)$-dimensional distribution $\mathcal{D}$
follows

(3.3) $\forall p\in M,\,\,\,\,\,\,\,\mathcal{D}_{p}=\\{v\in
T_{p}M:\eta(v)=0\\},$

the complement of $\mathcal{D}$ is $1$-dimensional distribution spanned by
$\xi$. Suppose that

$N^{1}=N_{\phi}-2d\eta\otimes\xi,\,\,N^{2}(X,Y)=(L_{\phi X}\eta)Y-(L_{\phi
Y}\eta)X,\,\,N^{3}=L_{\xi}\phi,\,\,N^{4}=L_{\xi}\eta.$

By Proposition 2.4 for any $X,Y$ of $C^{\infty}(M)$-module of all sections of
distribution $\mathcal{D}$, we have

$\displaystyle N_{J}(X^{v},Y^{v})$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle N_{J}(X^{v},Y^{c})$ $\displaystyle=$ $\displaystyle
A\Big{(}[N^{1}(X,Y)]^{v}+N^{2}(X,Y)\xi^{c}\Big{)},$ $\displaystyle
N_{J}(X^{c},Y^{c})$ $\displaystyle=$ $\displaystyle
A\Big{(}[N^{1}(X,Y)]^{c}+N^{2}(X,Y)\xi^{v}\Big{)},$ $\displaystyle
N_{J}(X^{v},\xi^{v})$ $\displaystyle=$ $\displaystyle
A\Big{(}-(N^{3}(X))^{v}+N^{4}(X)\xi^{c}\Big{)},$ $\displaystyle
N_{J}(X^{v},\xi^{c})$ $\displaystyle=$ $\displaystyle
A\Big{(}[\phi(N^{3}(X))-N^{4}(X)\xi]^{v}+N^{2}(X,\xi)\xi^{c}\Big{)},$
$\displaystyle N_{J}(X^{c},\xi^{v})$ $\displaystyle=$ $\displaystyle
A\Big{(}-(N^{3}(X))^{c}+(\phi(N^{3}(X)))^{v}-[N^{4}(\phi
X)-N^{4}(X)]^{c}\xi^{c}\Big{)},$ $\displaystyle N_{J}(X^{c},\xi^{c})$
$\displaystyle=$ $\displaystyle
A\Big{(}-(N^{3}(X))^{v}+N^{4}(X)+N^{2}(X,\xi)]\xi^{c}+\phi(N^{3}(X))-N^{4}(X)\xi]^{c}\Big{)},$
$\displaystyle N_{J}(\xi^{v},\xi^{v})$ $\displaystyle=$ $\displaystyle
N_{J}(\xi^{c},\xi^{c})=N_{J}(\xi^{v},\xi^{c})=0,$

where $A=(\frac{2\sigma_{p,q}-p}{2})^{2}$. But the tensor $N^{1}$ of a
P-Sasakian manifold vanishes. On the other hand if $N^{1}=0$ then also
$N^{2},N^{3}$ and $N^{4}$ vanish. Hence $N_{J}(\tilde{X},\tilde{Y})=0$ for all
$\tilde{X},\tilde{Y}\in\mathcal{X}(TM)$, that is $J$ is integrable. ∎

######  Theorem 3.4.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$ and
$J$ be defined by (3.1) then the metallic structure $J$ is never parallel with
respect to $\nabla^{c}$.

###### Proof.

We have

$\displaystyle(\nabla^{c}_{X^{c}}J)\xi^{c}$ $\displaystyle=$
$\displaystyle\nabla^{c}_{X^{c}}(J\xi^{c})-J(\nabla^{c}_{X^{c}}\xi^{c})$
$\displaystyle=$
$\displaystyle-\frac{2\sigma_{p,q}-p}{2}\left[\nabla^{c}_{X^{c}}\big{(}(\phi\xi)^{c}+\eta(\xi)^{v}\xi^{v}+\eta(\xi)^{c}\xi^{c}\big{)}-(\phi\nabla_{X}\xi)^{c}\right.$
$\displaystyle\left.-(\eta(\nabla_{X}\xi))^{v}\xi^{v}-(\eta(\nabla_{X}\xi))^{c}\xi^{c}\right].$

Using $\phi X=\nabla_{X}\xi$ we get

(3.5) $(\nabla^{c}_{X^{c}}J)\xi^{c}=-\frac{2\sigma_{p,q}-p}{2}[(\phi
X)^{v}-X^{c}]\neq 0,\,\,\,\,\forall X\in\mathcal{D}\setminus\\{0\\},$

where $\mathcal{D}$ is a distribution and defined by (3.3). ∎

######  Proposition 3.5.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$,
$\nabla\phi=0$ and $J$ be defined by (3.1). Then fundamental $2$-form $\Phi$,
given by

(3.6)
$\Phi(\tilde{X},\tilde{Y})=g^{c}(\tilde{X},J\tilde{Y})-\frac{p}{2}g^{c}(\tilde{X},\tilde{Y}),\,\,\,\,\,\,\tilde{X},\tilde{Y}\in\mathcal{X}(TM),$

is closed if and only if

(3.7) $g(\nabla_{Y}X,\phi Z)+g(\nabla_{Z}Y,\phi X)+g(\nabla_{X}Z,\phi
Y)=0,\,\,\,\,\,\,\forall X,Y,Z\in\mathcal{X}(M).$

###### Proof.

The cobundary formula for $d$ on a $2$-form $\Phi$ is

$\displaystyle 3d\Phi(\tilde{X},\tilde{Y},\tilde{Z})$ $\displaystyle=$
$\displaystyle\tilde{X}\Phi(\tilde{Y},\tilde{Z})+\tilde{Y}\Phi(\tilde{Z},\tilde{X})+\tilde{Z}\Phi(\tilde{X},\tilde{Y})$
$\displaystyle-\Phi([\tilde{X},\tilde{Y}],\tilde{Z})-\Phi([\tilde{Z},\tilde{X}],\tilde{Y})-\Phi([\tilde{Y},\tilde{Z}],\tilde{X})$

for any $\tilde{X},\tilde{Y},\tilde{Z}\in\mathcal{X}(TM)$. Hence for any
$X,Y,Z\in\mathcal{X}(M)$ we have

$\displaystyle 3d\Phi(X^{c},Y^{c},Z^{v})$ $\displaystyle=$ $\displaystyle
X^{c}g^{c}(Y^{c},JZ^{v})-\frac{p}{2}X^{c}g^{c}(Y^{c},Z^{v})+Y^{c}g^{c}(Z^{v},JX^{c})$
$\displaystyle-\frac{p}{2}Y^{c}g^{c}(Z^{v},X^{c})+Z^{v}g^{c}(X^{c},JY^{c})-\frac{p}{2}Z^{V}g^{c}(X^{c},Y^{c})$
$\displaystyle-g^{c}([X,Y]^{c},JZ^{v})+\frac{p}{2}g^{c}([X,Y]^{c},Z^{v})-g^{c}([Z,X]^{v},JY^{c})$
$\displaystyle+\frac{p}{2}g^{c}([Z,X]^{v},Y^{c})-g^{c}([Y,Z]^{v},JX^{c})+\frac{p}{2}g^{c}([Y,Z]^{v},X^{c}).$

On the other hand

$JZ^{v}=\frac{p}{2}Z^{v}-\frac{2\sigma_{p,q}-p}{2}\big{(}(\phi(Z))^{v}+(\eta(Z))^{v}\xi^{c}\big{)},$

and

$JX^{c}=\frac{p}{2}X^{c}-\frac{2\sigma_{p,q}-p}{2}\big{(}(\phi(X))^{v}+(\eta(X))^{v}\xi^{v}+(\eta(X))^{c}\xi^{c}\big{)}.$

Therefore,

$\displaystyle-\frac{6}{2\sigma_{p,q}-p}d\Phi(X^{c},Y^{c},Z^{v})$
$\displaystyle=$ $\displaystyle\left\\{Xg(Y,\phi Z)+Yg(Z,\phi X)+Zg(X,\phi
Y)\right.$ $\displaystyle\left.-g([X,Y],\phi Z)-g([Z,X],\phi Y)-g([Y,Z],\phi
X)\right\\}^{v}$
$\displaystyle+X^{c}[(\eta(Z))^{v}(\eta(Y))^{c}]+Y^{c}[(\eta(X))^{c}(\eta(Z))^{v}]$
$\displaystyle+Z^{v}[(\eta(Y))^{v}(\eta(X))^{v}]+Z^{v}[(\eta(Y))^{c}(\eta(X))^{c}]$
$\displaystyle-(\eta(Z))^{v}g([X,Y],\xi)^{c}-(\eta(Y))^{c}g([Z,X],\xi)^{v}$
$\displaystyle-(\eta(X))^{c}g([Y,Z],\xi)^{v}.$

Since $\nabla\phi=0$ and $g(X,\phi Y)=g(Y,\phi X)$, we get

$Xg(Y,\phi Z)-g([X,Y],\phi Z)=g(\nabla_{Y}X,\phi Z)+g(\nabla_{X}Z,\phi Y).$

Also, $\nabla_{X}\xi=\phi X$ results that

$\displaystyle g([X,Y],\xi)$ $\displaystyle=$ $\displaystyle
g(\nabla_{X}Y,\xi)-g(\nabla_{Y}X,\xi)$ $\displaystyle=$ $\displaystyle
Xg(Y,\xi)-g(Y,\nabla_{X}\xi)-Yg(X,\xi)+g(X,\nabla_{Y}\xi)$ $\displaystyle=$
$\displaystyle X(\eta(Y))-g(Y,\phi X)-Y(\eta(X))+g(X,\phi Y)$ $\displaystyle=$
$\displaystyle X(\eta(Y))-Y(\eta(X)).$

Thus

$\displaystyle-\frac{6}{2\sigma_{p,q}-p}d\Phi(X^{c},Y^{c},Z^{v})$
$\displaystyle=$ $\displaystyle 2\left\\{g(\nabla_{Y}X,\phi
Z)+g(\nabla_{Z}Y,\phi X)+g(\nabla_{X}Z,\phi Y)\right\\}$
$\displaystyle+2\left\\{(\eta(Y))^{c}(X\eta(Z))^{v}+(\eta(Z))^{v}(Y\eta(X))^{c}\right.$
$\displaystyle\left.+(\eta(X))^{c}(Z\eta(Y))^{v}\right\\}$

Now, if $X,Y,Z\in\mathcal{D}$ , then $d\Phi(X^{c},Y^{c},Z^{v})=0$ is
equivalent with (3.7), where $\mathcal{D}$ is a distribution and defined by
(3.3). Also, if $X=\xi$ or $Z=\xi$ , we get the same result. The other cases
reducible to (3.7). ∎

######  Note 3.6.

$\frac{p}{2}I-(\frac{2\sigma_{p,q}-p}{2})(\phi^{c}\pm\eta^{v}\otimes\xi^{v}\pm\eta^{c}\otimes\xi^{c})$
are also metallic structures. For these structures we can obtain the similar
results as for the metallic structure (3.1).

In the following we study a metallic structure on $TM$ induced by the
horizontal lift.

######  Proposition 3.7.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$.
Then there exists a metallic structure on its tangent bundle, given by

(3.8)
$F=\frac{p}{2}I-(\frac{2\sigma_{p,q}-p}{2})(\phi^{h}+\eta^{h}\otimes\xi^{h}+\eta^{v}\otimes\xi^{v}).$

###### Proof.

By definition vertical lift and horizontal lift of the almost paracontact
structure of P-Sasakian manifold, we have

$\displaystyle(\phi^{h})^{2}=(\phi^{2})^{h}=I-\eta^{h}\otimes\xi^{v}-\eta^{v}\otimes\xi^{h},$
$\displaystyle\eta^{v}(\xi^{h})=\eta^{h}(\xi^{v})=1,\,\,\,\,\,\,\eta^{v}(\xi^{v})=\eta^{h}(\xi^{h})=0,$
$\displaystyle\phi^{h}(\xi^{v})=\phi^{h}(\xi^{h})=0,\,\,\,\,\,\,\eta^{v}\circ\phi^{h}=\eta^{h}\circ\phi^{h}=0.$

Therefore for any $\tilde{X}\in\mathcal{X}(TM)$ we have

$\displaystyle
F(\xi^{v})=\frac{p}{2}\xi^{v}-\frac{2\sigma_{p,q}-p}{2}\xi^{h},\,\,\,\,\,\,J(\xi^{h})=\frac{p}{2}\xi^{h}-\frac{2\sigma_{p,q}-p}{2}\xi^{v},$
$\displaystyle
F(\phi^{h}\tilde{X})=\frac{p}{2}\phi^{h}\tilde{X}-\frac{2\sigma_{p,q}-p}{2}(\tilde{X}-\eta^{h}(\tilde{X})\xi^{v}-\eta^{v}(\tilde{X})\xi^{h}).$

Now, we obtain

$F(\tilde{X})=\frac{p}{2}\tilde{X}-\frac{2\sigma_{p,q}-p}{2}(\phi^{h}\tilde{X}+\eta^{v}(\tilde{X})\xi^{v}+\eta^{h}(\tilde{X})\xi^{h})$

and

$F^{2}(\tilde{X})=\frac{p}{2}F(\tilde{X})-\frac{2\sigma_{p,q}-p}{2}(F(\phi^{h}\tilde{X})+\eta^{v}(\tilde{X})F(\xi^{v})+\eta^{h}(\tilde{X})F(\xi^{h}))=pF(\tilde{X})+q\tilde{X},$

and it finish the proof. ∎

######  Definition 3.8 (Sasakian metric ).

Let $(M,g)$ be a Riemannian manifold. The Sasakian metric on $TM$ defined as
follows

(3.9)
$G(X^{v},Y^{h})=0,\,\,G(X^{v},Y^{v})=[g(X,Y)]^{v},\,\,\,G(X^{h},Y^{h})=[g(X,Y)]^{v},$

for any $X,Y\in\mathcal{X}(M)$.

######  Proposition 3.9.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$. If
$F$ defined by (3.8) on $TM$ and $G$ be Sasakian metric, then we have

(3.10)
$G(F\tilde{X},F\tilde{Y})=pG(\tilde{X},F\tilde{Y})+qG(\tilde{X},\tilde{Y}).$

for any $\tilde{X},\tilde{Y}\in\mathcal{X}(TM)$.

###### Proof.

For any $X,Y\in\mathcal{X}(M)$, we have

$\eta^{h}(X^{v})=\eta^{v}(X^{v})=0,\,\,\eta^{h}X^{h}=\eta^{v}X^{h}=(\eta(X))^{h},\,\,\phi^{h}X^{v}=(\phi
X)^{v},\,\,\phi^{h}X^{h}=(\phi X)^{h}.$

Hence

$FX^{v}=\frac{p}{2}X^{v}-\frac{2\sigma_{p,q}-p}{2}(\phi X)^{v}.$

Now by definition of Sasakian metric we get

$G(FX^{v},FY^{v})=\left\\{(q+\frac{p^{2}}{2})g(X,Y)-\frac{2\sigma_{p,q}-p}{2}pg(X,\phi
Y)\right\\}^{v}.$

and

$G(X^{v},FY^{v})=\left\\{\frac{p}{2}g(X,Y)-\frac{2\sigma_{p,q}-p}{2}g(X,\phi
Y)\right\\}^{v}.$

Therefore

$G(FX^{v},FY^{v})=pG(X^{v},FY^{v})+qG(X^{v},Y^{v}).$

the other cases are similar.

∎

######  Definition 3.10.

Let $(M,g)$ be a Riemannian manifold and $\nabla$ the Levi-Civita connection
with respect to the Riemannian metric $g$. Let $\mathcal{D}$ be a distribution
and defined by (3.3). Connection $\nabla$ is called $\mathcal{D}$-flat if
$\nabla_{X}Y\in\mathcal{D}$ for all $X,Y\in\mathcal{D}$.

######  Theorem 3.11.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$.
Then the metallic structure $F$ defined by (3.8) on $TM$ is integrable if and
only if $\nabla$ is $\mathcal{D}$-flat and

(3.11) $R(\phi X,\phi Y)+R(X,Y)-\phi\\{R(\phi X,Y)+R(X,\phi Y)\\}=0,$

where $R$ is curvature tensor of $M$.

###### Proof.

Let $X,Y\in\mathcal{D}$ and $U\in TM$. We have

$\displaystyle-\frac{2}{2\sigma_{p,q}-p}N_{F}(X^{h},Y^{h})U$ $\displaystyle=$
$\displaystyle[N^{1}(X,Y)]^{h}U+\\{\eta R(\phi X,Y)U+\eta R(X,\phi
Y)U\\}\xi^{h}$ $\displaystyle-\\{R(\phi X,\phi Y)U+R(X,Y)U-\phi R(\phi X,Y)U$
$\displaystyle+R(X,\phi Y)U\\}^{v}+N^{2}(X,Y)\xi^{v}.$

Also

$\displaystyle-\frac{2}{2\sigma_{p,q}-p}N_{F}(X^{h},Y^{v})$ $\displaystyle=$
$\displaystyle\big{(}\nabla_{\phi X}\phi Y-\phi\nabla_{\phi
X}Y-\phi\nabla_{X}\phi Y+\nabla_{X}Y\big{)}^{v}$
$\displaystyle-\\{\eta(\nabla_{\phi X}Y)+\eta(\nabla_{X}\phi Y)\\}\xi^{h},$
$\displaystyle N_{F}(X^{v},Y^{v})$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle-\frac{2}{2\sigma_{p,q}-p}N_{F}(X^{h},\xi^{h})U$
$\displaystyle=$ $\displaystyle\big{(}\nabla_{\phi
X}\xi-\phi(\nabla_{X}\xi)+\phi(N^{3}(X))\big{)}^{h}U+[\eta R(\phi
X,\xi)U]\xi^{h}$
$\displaystyle+\\{N^{2}(X,\xi)-\eta(\nabla_{X}\xi)\\}\xi^{v}+[\phi R(\phi
X,\xi)U]^{v},$ $\displaystyle-\frac{2}{2\sigma_{p,q}-p}N_{F}(X^{h},\xi^{v})U$
$\displaystyle=$ $\displaystyle-[N^{3}(X)]^{h}U+\\{N^{2}(X,\xi)+\eta
R(X,\xi)U\\}\xi^{h}$ $\displaystyle-\left\\{R(\phi X,\xi)U-\phi
R(X,\xi)U+\phi(\nabla_{\phi X}\xi)-\nabla_{X}\xi\right\\}^{v}$
$\displaystyle-N^{4}(X)\xi^{v},$

and

$-\frac{2}{2\sigma_{p,q}-p}N_{F}(X^{v},\xi^{v})=-(\nabla_{\xi}\phi
X-\phi\nabla_{\xi}X)^{v}+\eta(\nabla_{\xi}X)\xi^{h}.$

Hence $N_{F}=0$ if and only if (3.11) is true and

(3.12) $\nabla_{\phi X}\phi Y-\phi\nabla_{\phi X}Y-\phi\nabla_{X}\phi
Y+\nabla_{X}Y=0.$

From (1.6), (3.12) is equivalent with $\eta(\nabla_{X}Y)=0$, for any
$X,Y\in\mathcal{D}$ ,i.e. $\nabla$ is $\mathcal{D}$-flat. ∎

######  Theorem 3.12.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$.
Then the metallic structure $F$ defined by (3.8) on $TM$ is never parallel
with respect to $\nabla^{h}$.

###### Proof.

We have

$(\nabla_{X^{h}}^{h}F)\xi^{h}=\nabla_{X^{h}}^{h}(F\xi^{h})-F(\nabla_{X^{h}}^{h}\xi^{h})=-\frac{2\sigma_{p,q}-p}{2}[(\phi
X)^{v}-(\phi^{2}X)^{h}].$

If $X\in\mathcal{D}\setminus\\{0\\}$ then $(\nabla_{X^{h}}^{h}F)\xi^{h}\neq
0$, where $\mathcal{D}$ is defined by (3.3). ∎

We define the fundamental $2$-form $\Phi^{\prime}$ by

$\Phi^{\prime}(\tilde{X},\tilde{Y})=G(\tilde{X},F\tilde{Y})-\frac{p}{2}G(\tilde{X},\tilde{Y}),\,\,\,\,\,\tilde{X},\tilde{Y}\in\mathcal{X}(TM).$

######  Proposition 3.13.

Let $M$ be a P-Sasakian manifold with structure tensor $(\phi,\eta,\xi,g)$ and
$F$ be defined by (3.8). Then fundamental $2$-form $\Phi^{\prime}$, is never
closed.

###### Proof.

Let $X\in\mathcal{D}$ be an unit vector field i.e. $g(X,X)=1$, where
$\mathcal{D}$ is defined by (3.3). Then by similar proof of Proposition 3.5,
we have

$\displaystyle-\frac{6}{2\sigma_{p,q}-p}d\Phi^{\prime}(X^{h},X^{v},\xi^{v})$
$\displaystyle=$ $\displaystyle-G([\xi^{v},X^{h}],(\phi
X)^{v})=-g(\nabla_{X}\xi,\phi X)^{v}$ $\displaystyle=$
$\displaystyle-g(X,X)^{v}=-1.$

∎

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