# On curvature related geometric properties of Hayward black hole spacetime

Absos Ali Shaikh∗1, Shyamal kumar Hui2, Biswa Ranjan Datta3 and Mousumi
Sarkar4
1,2,3,4 Department of Mathematics,
University of Burdwan,
Golapbag, Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the study of curvature properties of Hayward black
hole (briefly, HBH) spacetime, which is a solution of Einstein field equations
(briefly, EFE) having non-vanishing cosmological constant. We have proved that
the HBH spacetime is an Einstein manifold of level $2$, $2$-quasi Einstein,
generalized quasi-Einstein and Roter type manifold. Also, it is shown that the
nature of the HBH spacetime is pseudosymmetric and it obeys several types of
pseudosymmetries, such as, pseudosymmetry due to concircular, conformal and
conharmonic curvature (i.e., $F\cdot F=\mathcal{L}Q(g,F)$ for $F=W,C,K$ with a
smooth scalar function $\mathcal{L}$), and it also possesses the relation
$R\cdot R-\mathcal{L}Q(g,C)=Q(S,R)$. It is engrossing to mention that the
nature of energy momentum tensor of the HBH spacetime is pseudosymmetric. On
the basis of curvature related properties, we have made a comparison among
Reissner-Nordström spacetime, interior black hole spacetime and HBH spacetime.
Also, it is shown that the HBH spacetime admits an almost $\eta$-Ricci soliton
as well as an almost $\eta$-Ricci-Yamabe soliton. Finally, an elegant
comparative study is delineated between the HBH spacetime and the point-like
global monopole spacetime with respect to different kinds of symmetry, such
as, motion, curvature collineation, curvature inheritance etc.

###### Key words and phrases:

Hayward metric, Einstein field equation, pseudosymmetric type curvature
condition, Weyl conformal curvature tensor, Roter type manifold, Ein$(2)$

###### 2020 Mathematics Subject Classification:

53B20, 53B30, 53B50, 53C15, 53C25, 53C35, 83C15

##  1\. Introduction

Let us consider a semi-Riemannian manifold $M$ of dimension $n\geq 3$ such
that $\nabla$ is the Levi-Civita connection of the semi-Riemannian metric $g$
with signature $(t,n-t)$, $0\leq t\leq n$ and $R$, $S$, $\kappa$ are
respectively the Riemann, Ricci, scalar curvature of $M$. A connected
4-dimensional manifold $M$ with Lorentzian signature (1,3) or (3,1) is a
spacetime.

The curvature carries an enormous significance to acquire the shape of a
space. In fact, the geometry of a space can be described by curvature
explicitly as the relation $\nabla R=0$ defines the notion of locally
symmetric manifolds (see, Cartan [24]). Cartan [25] introduced the notion of
semisymmetric manifolds defined as $R\cdot R=0$ (see also, [122, 123, 124])
and the concept of pseudosymmetric manifolds was introduced by Adamów and
Deszcz [18], which is known as Deszcz pseudosymmetric space. A large number of
physicists and mathematicians investigated the concept of locally symmetric
manifolds and introduced several generalized notions of symmetries, such as,
recurrent manifolds by Ruse [80, 81, 82], (see also [129]), different kinds of
generalized notion of recurrent manifolds by Shaikh et al. [108, 109, 112,
111, 91, 110, 106, 113], curvature $2$-forms of recurrent manifolds by Besse
[22, 69, 74, 75, 76], pseudosymmetric manifolds by Chaki [26, 27], weakly
symmetric manifolds by Tamássy and Binh [126, 127] etc. Haesen and Verstraelen
[59, 60, 61] exhibited the geometrical and physical significance of various
pseudosymmetries. We mention that the Deszcz pseudosymmetry achieved a great
importance during last four decades due to its applications in the study of
general relativity and cosmology as numerous spacetimes (see, [20, 47, 68, 84,
86, 95, 88, 52]) have been found to be pseudosymmetric. It is noteworthy to
mention that pseudosymmetries in the sense of Deszcz and Chaki are not
equivalent (see, [97]).

In 1982, during the study of compact $3$-dimensional manifolds with positive
Ricci curvature, Hamilton [63] established a process of evolving a Riemanninan
metric over time, called Ricci flow. The self-similar solutions of the Ricci
flow are known as Ricci solitons, which are natural generalizations of
Einstein metrics [22, 23, 117, 83]. The notion of Ricci soliton has been
generalized in different ways, e.g. almost Ricci soliton, $\eta$-Ricci
soliton, almost $\eta$-Ricci soliton etc.

If the Ricci curvature $S$ and the metric tensor $g$ of a Riemannian manifold
$M$ realize

$\frac{1}{2}\pounds_{\xi}g+S-\mu g=0$

for a constant $\mu$, then $M$ is said to be a Ricci soliton, where
$\pounds_{\xi}$ is the Lie derivative in the direction of the soliton vector
field $\xi$. It is expanding, steady or shrinking according to the condition
$\mu<0$, $\mu=0$ or $\mu>0$ respectively. It is called an almost Ricci soliton
[78] if $\mu$ is a non-constant smooth function. We mention that if the
corresponding soliton vector field $\xi$ of a Ricci soliton is Killing, then
the Ricci soliton turns into an Einstein manifold. Again, if a non-zero
$1$-form $\eta$ on $M$ satisfies the relation

$\frac{1}{2}\pounds_{\xi}g+S-\mu g+\lambda(\eta\otimes\eta)=0,$

$\mu,\lambda$ being constants, then $M$ is called an $\eta$-Ricci soliton
[16]. The $\eta$-Ricci soliton is said to be an almost $\eta$-Ricci soliton
[15] if $\mu,\lambda$ are allowed to be smooth functions.

On the other hand, simultaneously with the notion of Ricci flow, Hamilton [64]
introduced the notion of Yamabe flow. Recently, as a scalar combination of
Ricci and Yamabe flow, Güler and Cráşmareǎnu [58] established a new geometric
flow, which is called Ricci-Yamabe flow, and Ricci-Yamabe (resp., Yamabe)
solitions are the self-similar solutions of Ricci-Yamabe (resp., Yamabe) flow.
If in a Riemannian manifold $M$ the Ricci curvature $S$ and the metric tensor
$g$ realize the relation

$\frac{1}{2}\pounds_{\xi}g+\alpha_{1}S+\left(\mu-\frac{1}{2}\alpha_{2}\kappa\right)g=0,$

with the constants $\alpha_{1}$, $\alpha_{2}$, $\mu$, scalar curvature
$\kappa$ and the soliton vector field $\xi$, then $M$ is called a Ricci-Yamabe
soliton [118]. We note that if $\alpha_{1}=0,\alpha_{2}=2$ (resp.,
$\alpha_{1}=1,\alpha_{2}=0$), then it turns into Yamabe soliton (resp., Ricci
soliton). In addition, if $\alpha_{1}$, $\alpha_{2}$, $\mu$ are allowed to be
non-constant smooth functions, then $M$ is known as an almost Ricci-Yamabe
soliton [118]. Again, if there is a non-zero 1-form $\eta$ satisfying

$\frac{1}{2}\pounds_{\xi}g+\alpha_{1}S+\left(\mu-\frac{1}{2}\alpha_{2}\kappa\right)g+\lambda\eta\otimes\eta=0,$

with the constants $\alpha_{1}$, $\alpha_{2}$, $\mu$, $\lambda$, then $M$ is
called an $\eta$-Ricci-Yamabe soliton [118]. If the constants $\alpha_{1}$,
$\alpha_{2}$, $\mu$, $\lambda$ are allowed to be non-constant smooth
functions, then $M$ is called an almost $\eta$-Ricci-Yamabe soliton [118]. A
plenty of research papers (see, [14, 13, 1, 89] and the references therein) on
Ricci soliton, Yamabe soliton and their generalizations are appeared during
last three decades, and nowadays it is an abuzz topic of research in
differential geometry.

To construct gravitational potential, one can impose the symmetry in EFE and
hence the geometrical symmetries play a crucial role in the theory of general
relativity. Along a vector field, certain geometric quantity is preserved if
the Lie derivative of the corresponding tensor vanishes with respect to that
vector field, and the vanishing Lie derivative explains the geometrical
symmetries. The notions of motion, curvature collineation, Ricci collineation
etc. are the examples of such symmetries. Katzin et al. [66, 67] rigorously
investigated the role of curvature collineation in general relativity. In
1992, Duggal [17] introduced the notion of curvature inheritance generalizing
the concept of curvature collineation for the (1,3)-type curvature tensor.
During last three decades, a plenty of articles (see, [4, 2, 3, 5, 6, 7, 8, 9,
10, 11, 12, 90, 96]) appeared in the literature regarding the investigations
of such kinds of symmetries. Recently, during the investigation of geometric
properties of Robinson-Trautman spacetime, Shaikh and Datta [96] introduced
the concept of generalized curvature inheritance, which is a generalization of
curvature collineation as well as curvature inheritance for the (0,4)-type
curvature tensor. We note that the notions of curvature inheritance for the
(1,3)-type curvature tensor and for the (0,4)-type curvature tensor are not
equivalent [96]. In this paper, we have checked that the HBH spacetime does
not admit any of the curvature related symmetries. Finally, a worthy
comparison between the HBH spacetime and the point-like global monopole
spacetime in terms of such symmetries is exhibited.

In $2006,$ Hayward [65] modeled the famous exact regular black hole metric,
which is the solution of the EFE in spherical symmetry, and it is a simple and
singularity free black hole spacetime in general relativity. The line element
of HBH spacetime, in spherical coordinates $(t,r,\theta,\phi)$, is given by

$\displaystyle
ds^{2}=-\left(1-\frac{2mr^{2}}{r^{3}+2mb^{2}}\right)dt^{2}+\left(1-\frac{2mr^{2}}{r^{3}+2mb^{2}}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (1.1)

where the parameters $m$ and $b$ represents mass and length-scale
respectively. The metric (1.1) is non-singular, because if $r\to\infty,$ the
metric approaches to $1-\frac{2m}{r}$, and if $r\to 0$, it approaches unity
smoothly. The metric consists of the least number of free parameters ($b$
only) with the properties

  1. (1)

Schwarzschild asymptotic behavior at large radii and

  2. (2)

regularly at the center such that $F(r)=1-\frac{2mr^{2}}{r^{3}+2mb^{2}}\to
1+O(r^{2}),$.

Hence it is minimal. The importance of Hayward spacetime (1.1) is realized
from the several studies, such as, Chiba and Kimura [29] have obtained the
timelike geodesics and null geodesics equation of a particle in HBH spacetime
and Maluf and Neves [70] have studied certain thermodynamic quantities like
Hawking temperature, entropy and heat capacity of HBH spacetime. The stability
of the thin-shell wormholes constructed by the HBH spacetime are studied by
Halilsoy et al. [62]. However, several curvature properties of HBH spacetime
are yet to be investigated.

The purpose of the article is to focus and to determine of several geometric
properties of HBH spacetime. It is found that the HBH spacetime is neither
semisymmetric nor Ricci generalized pseudosymmetric, but it is pseudosymmetric
and satisfies several pseudosymmetric type curvature conditions, such as,
pseudosymmetry due to concircular, conharmonic and conformal curvature
tensors. Also, we have exhibited that both $Q(g,C)$ and $Q(S,C)$ are linearly
dependent on the difference $(C\cdot R-R\cdot C)$. It is also proved that the
HBH spacetime is an Einstein manifold of level $2$, $2$-quasi Einstein,
generalized quasi-Einstein and Roter type manifold. Moreover, the nature of
the stress energy momentum tensor of HBH spacetime is pseudosymmetric.

The article is embellished as follows: we discuss some definitions of
geometric structures in in Section $2$, which are essential throughout the
paper to investigate the geometric properties of HBH spacetime. Section $3$
deals with the study of the HBH spacetime and obtained some interesting
results. In Section $4$, certain geometric properties of energy momentum
tensor of HBH spacetime are determined. Based on the curvature properties of
HBH spacetime, an worthy comparison with the point-like global monopole
spacetime and Reissner-Nordström spacetime has been exhibited in Section $5$.
Section $6$ is concerned with the nature of Ricci soliton and Ricci-Yamabe
soliton admited by the HBH spacetime. Section $7$ is devoted to a comparative
study between between the HBH spacetime and the point-like global monopole
spacetime with respect to different kinds of symmetry, such as, motion,
curvature collineation and curvature inheritance.

##  2\. Preliminaries

This section consists of various rudimentary facts about various geometric
structures, Ricci soliton and symmetries (such as, motion, curvature
collineation (also, curvature inheritance) for the (1,3)-type curvature tensor
and for the (0,4)-type curvature tensor, Ricci collineation and Ricci
inheritance), which are necessary for investigating the geometric structures
on HBH spacetime.

The Kulkarni-Nomizu product $A\wedge U$ of two (0,2)-type symmetric tensors
$A$ and $U$ is defined as ([38, 55, 57, 68]):

$(A\wedge
U)_{pq\mu\nu}=A_{p\nu}U_{q\mu}-A_{p\mu}U_{q\nu}+A_{q\mu}U_{p\nu}-A_{q\nu}U_{p\mu}.$

For $j=1,2,3,4$ we will consider $\varpi,\varpi_{j}\in\chi(M),$ Lie algebra of
all smooth vector fields throughout the paper. For a symmetric $(0,2)$-type
tensor $Z$, the endomorphism $h_{1}\wedge_{Z}h_{2}$ can be defined as ([30,
31, 38])

$(\varpi_{1}\wedge_{Z}\varpi_{2})v=Z(\varpi_{2},v)\varpi_{1}-Z(\varpi_{1},v)\varpi_{2}.$

Now, the endomorphisms $\mathcal{R},\mathcal{W},\mathcal{C},\mathcal{K}$ and
$\mathcal{P}$ can be defined on $M$ as follows: ([46, 93, 103, 105])

$\mathcal{R}(\varpi_{1},\varpi_{2})=[\nabla_{\varpi_{1}},\nabla_{\varpi_{2}}]-\nabla_{[\varpi_{1},\varpi_{2}]},$
$\mathcal{W}(\varpi_{1},\varpi_{2})=\mathcal{R}(\varpi_{1},\varpi_{2})-\frac{\kappa}{n(n-1)}\varpi_{1}\wedge_{g}\varpi_{2},$
$\mathcal{C}(\varpi_{1},\varpi_{2})=\mathcal{R}(\varpi_{1},\varpi_{2})-\frac{1}{n-2}\left(\varpi_{1}\wedge_{g}\mathcal{J}\varpi_{2}+\mathcal{J}\varpi_{1}\wedge_{g}\varpi_{2}-\frac{\kappa}{n-1}\varpi_{1}\wedge_{g}\varpi_{2}\right),$
$\mathcal{K}(\varpi_{1},\varpi_{2})=\mathcal{R}(\varpi_{1},\varpi_{2})-\frac{1}{n-2}\left(\varpi_{1}\wedge_{g}\mathcal{J}\varpi_{2}+\mathcal{J}\varpi_{1}\wedge_{g}\varpi_{2}\right),$
$\mathcal{P}(\varpi_{1},\varpi_{2})=\mathcal{R}(\varpi_{1},\varpi_{2})-\frac{1}{n-1}\varpi_{1}\wedge_{S}\varpi_{2},$

where $\mathcal{J}$ is the Ricci operator defined as
$S(\varpi_{1},\varpi_{2})=g(\varpi_{1},\mathcal{J}\varpi_{2})$. Now, we define
the $(0,4)$-type tensor field $T$ corresponding to the endomorphism
$\mathscr{T}(\varpi_{1},\varpi_{2})$ on $M$ as

$T(\varpi_{1},\varpi_{2},\varpi_{3},\varpi_{4})=g(\mathscr{T}(\varpi_{1},\varpi_{2})\varpi_{3},\varpi_{4}).$

If the endomorphism $\mathscr{T}$ is replaced by $\mathcal{R}$ (resp.,
$\mathcal{W}$, $\mathcal{K}$, $\mathcal{C}$ and $\mathcal{P}$) in above, we
obtain the Riemann (resp., concircular, conharmonic, conformal and projective)
curvature tensor $R$ (resp., $W$, $K$, $C$ and $P$) of type $(0,4)$. These
tensors are locally given by

$R_{pq\mu\nu}=g_{p\alpha}(\partial_{n}\Gamma^{\alpha}_{q\mu}-\partial_{\mu}\Gamma^{\alpha}_{q\nu}+\Gamma^{\beta}_{q\mu}\Gamma^{\alpha}_{\beta\nu}-\Gamma^{\beta}_{q\nu}\Gamma^{\alpha}_{\beta\mu}),$
$C_{pq\mu\nu}=R_{pq\mu\nu}+\frac{\kappa}{2(n-1)(n-2)}(g\wedge
g)_{pq\mu\nu}-\frac{1}{n-2}(g\wedge S)_{pq\mu\nu},$
$W_{pq\mu\nu}=R_{pq\mu\nu}-\frac{\kappa}{2n(n-1)}(g\wedge g)_{pq\mu\nu},$
$K_{pq\mu\nu}=R_{pq\mu\nu}-\frac{1}{n-2}(g\wedge S)_{pq\mu\nu}\ and\ $
$P_{pq\mu\nu}=R_{pq\mu\nu}-\frac{1}{n-1}(g_{p\nu}S_{q\mu}-g_{q\nu}S_{p\mu}),$

where $\partial_{\alpha}=\frac{\partial}{\partial x^{\alpha}}$ and
$\Gamma^{\alpha}_{q\mu}$ denotes the Christoffel symbols of $2$nd kind.

Let $H$ be a $(0,k)$-type ($k\geq 1$) tensor on $M$. Then the $(0,k+2)$-type
tensor $T\cdot H$ is given by [36, 45, 100]

$(T\cdot H)_{q_{{}_{1}}q_{{}_{2}}\cdots q_{{}_{k}}\nu\mu}=-g^{pr}[T_{\mu\nu
q_{{}_{1}}r}H_{pq_{{}_{2}}\cdots q_{{}_{k}}}+\cdots+T_{\mu\nu
q_{{}_{k}}r}H_{q_{{}_{1}}q_{{}_{2}}\cdots p}].$

Again, for a symmetric $(0,2)$-type tensor field $Z$, the Tachibana tensor
$Q(Z,H)$ of type $(0,k+2)$ is obtained as follows: [43, 97, 125]

$Q(Z,H)_{q_{{}_{1}}q_{{}_{2}}\cdots q_{{}_{k}}\nu\mu}=Z_{\mu q_{{}_{1}}}H_{\nu
q_{{}_{2}}\cdots q_{{}_{k}}}+\cdots+Z_{\mu
q_{{}_{k}}}H_{q_{{}_{1}}q_{{}_{2}}\cdots\nu}-Z_{\nu q_{{}_{1}}}H_{\mu
q_{{}_{2}}\cdots q_{{}_{k}}}-\cdots-Z_{\nu
q_{{}_{k}}}H_{q_{{}_{1}}q_{{}_{2}}\cdots\mu}.$

######  Definition 2.1.

[18, 25, 32, 33, 39, 40, 87, 100, 104, 122, 123, 124] Let $M$ be a semi-
Riemannian manifold. $M$ is called a H-semisymmetric type manifold due to T if
$M$ possesses the relation $T\cdot H=0$. Further, $M$ is said to be
H-pseudosymmetric type manifold due to T if the relation $T\cdot
H=\mathcal{L}_{H}Q(Z,H)$ holds for a smooth function $\mathcal{L}_{H}$ on
$\\{x\in M:Q(Z,H)\neq 0\ at\ x\\}$ (i.e., the tensors $T\cdot H$ and $Q(Z,H)$
are linearly dependent).

In the above definition, if we replace $T$ = $R$ and $H$ = $R$ (resp., $P$,
$K$, $W$, $C$ and $S$), then the $H$-semisymmetric type manifold due to $T$
turns into semisymmetric (resp., projectively, conharmonically, concircularly,
conformally, Ricci semisymmetric) manifold and if $T=R$, $H=R$ and $Z=g$
(resp., $S$), then the $H$-pseudosymmetric type manifold due to $T$ becomes
Deszcz pseudosymmetric (resp., Ricci generalized pseudosymmetric) manifold.
Also, if we replace $T$ = $W$, $C$, $P$ and $K$, then we obtain several
pseudosymmetric type curvature conditions.

######  Definition 2.2.

$($[40, 41, 44, 83, 99, 105, 117]$)$ $M$ is called quasi-Einstein (resp.,
Einstein and $2$-quasi-Einstein) manifold if for a scalar $\alpha$ the rank of
$(S-\alpha g)$ is $1$ (resp., $0$ and $2$). In particular, for $\alpha=0$ the
quasi-Einstein manifold turns into Ricci simple. A generalized quasi-Einstein
manifold (in the sense of Chaki [28]) is defined as

$S=\alpha g+\beta\Pi\otimes\Pi+\gamma(\Pi\otimes\phi+\phi\otimes\Pi)$

where $\alpha$, $\beta$ and $\gamma$ are scalars and $\Pi$, $\phi$ are
$1$-forms.

It may be mentioned that Robertson Walker spacetime [19, 77, 120] is quasi-
Einstein, Kaigorodov spacetime [95] is Einstein, Kantowski-Sachs spacetime
[93] and Som-Raychaudhuri spacetime [102] are $2$-quasi-Einstein, Vaidya
metric [107], Gödel spacetime [46] and Morris-Thorne spacetime [51] are also a
Ricci simple manifold.

######  Definition 2.3.

If the Ricci tensor $S$ of a semi-Riemannain manifold $M$ satisfies the
relation

$(\nabla_{\varpi_{1}}S)(\varpi_{2},\varpi_{3})+(\nabla_{\varpi_{2}}S)(\varpi_{3},\varpi_{1})+(\nabla_{\varpi_{3}}S)(\varpi_{1},\varpi_{2})=0,$
$(resp.,(\nabla_{\varpi_{1}}S)(\varpi_{2},\varpi_{3})=(\nabla_{\varpi_{2}}S)(\varpi_{1},\varpi_{3}))$

then it is known as cyclic parallel Ricci tensor (see, [54, 92, 114, 115])
(resp., Codazzi type Ricci tensor (see [53, 119])).

It may be noted that the Ricci tensor of $(t-z)$-type plane wave spacetime
[50] is of Codazzi type and the Ricci tensor of cyclic parallel has been found
in Gödel spacetime [46].

######  Definition 2.4.

$($[22, 100, 105]$)$ A semi-Riemannian manifold $M$ is an Einstein manifold of
level $4$ (resp., $3$ and $2$) if it satisfies

$\vartheta_{1}g+\vartheta_{2}S+\vartheta_{3}S^{2}+\vartheta_{4}S^{3}+S^{4}=0,$

(resp., $\vartheta_{5}g+\vartheta_{6}S+\vartheta_{7}S^{2}+S^{3}=0$ and
$\vartheta_{8}g+\vartheta_{9}S+S^{2}=0$), where $\vartheta_{i}$ $(1\leq i\leq
9)$ are smooth functions on $M$.

We mention that Vaidya-Bonner spacetime [94] and Lifshitz spacetime [116] are
Ein$(3)$ while Siklos spacetime [95] and Nariai spacetime [87] are Ein$(2)$
manifolds.

######  Definition 2.5.

If the Riemann tensor $R$ can be written in the form

$\displaystyle
R=S^{2}\wedge(\varsigma_{6}S^{2})+S\wedge(\varsigma_{4}S+\varsigma_{5}S^{2})+g\wedge(\varsigma_{1}g+\varsigma_{2}S+\varsigma_{3}S^{2})$

for some scalars $\varsigma_{i}$ $(1\leq i\leq 6)$, then $M$ is called
generalized Roter type manifold [34, 37, 41, 42, 101, 105]. Further, $M$ is
known as a Roter type manifold [34, 35, 44, 48, 56] if $g\wedge g,$ $S\wedge
S$ and $g\wedge S$ are linearly dependent on $R$ (i.e.,
$\varsigma_{3}=\varsigma_{5}=\varsigma_{6}=0$).

It may be noted that Nariai spacetime [87], Melvin magnetic metric [86] as
well as Robinson-Trautman spacetime [84] are Roter type, while Lifshitz metric
[116] and Vaidya-Bonner metric [94] are generalized Roter type manifolds.

######  Definition 2.6.

[126, 127] A weakly $T$-symmetric manifold $M$ is defined by the equation

$\displaystyle(\nabla_{\varpi}T)(\varpi_{1},\varpi_{2},\varpi_{3},\varpi_{4})$
$\displaystyle=$ $\displaystyle\Pi(\varpi)\otimes
T(\varpi_{1},\varpi_{2},\varpi_{3},\varpi_{4})+\Omega_{1}(\varpi_{1})\otimes
T(\varpi,\varpi_{2},\varpi_{3},\varpi_{4})$ $\displaystyle+$
$\displaystyle\Omega_{1}(\varpi_{2})\otimes
T(\varpi_{1},\varpi,\varpi_{3},\varpi_{4})+\Omega_{2}(\varpi_{3})\otimes
T(\varpi_{1},\varpi_{2},\varpi,\varpi_{3})$ $\displaystyle+$
$\displaystyle\Omega_{2}(\varpi_{4})\otimes
T(\varpi_{1},\varpi_{2},\varpi_{3},\varpi),$

where $\Pi$, $\Omega_{1}$, $\Omega_{2}$ are $1$-forms on $M$. In particular,
$M$ reduces to a recurrent [80, 81, 129] (resp., Chaki pseudosymmetric [26,
27]) manifold if $\Omega_{1}$=$\Omega_{2}$=$0$ (resp.,
$\Omega_{1}$=$\Omega_{2}=\Pi/2$).

######  Definition 2.7.

$($[31, 37, 71, 72, 73]$)$ Let $T$ be a $(0,4)$-type tensor field and
$\mathcal{Z}$ be the endomorphism corresponding to a tensor $Z$ of type
$(0,2)$ on $M$. Then, the tensor $Z$ is said to be $T$-compatible if $M$
admits

$\displaystyle\mathop{\mathcal{S}}_{\varpi_{1},\varpi_{2},\varpi_{3}}T(\mathcal{Z}\varpi_{1},\varpi,\varpi_{2},\varpi_{3})=0,$

where the cyclic sum over $\varpi_{1}$, $\varpi_{2}$ and $\varpi_{3}$ is
denoted by $\mathcal{S}$. Again, $T$-compatibility of an $1$-form $\zeta$ is
defied by the $T$-compatibility of $\zeta\otimes\zeta$.

In the above definition, if we replace $T$ by $R$ (resp., $P$, $K$, $W$ and
$C$) then we obtain Riemann (resp., projective, conharmonic, concircular and
conformal) compatibility of $Z$.

######  Definition 2.8.

Let $T$ be a $(0,4)$-type tensor on $M$ and $\mathcal{Z}$ be the endomorphism
corresponding to a $(0,2)$-type tensor $Z$ on $M$. If $M$ possesses the
relation

$\displaystyle\mathop{\mathcal{S}}_{\varpi_{1},\varpi_{2},\varpi_{3}}(\nabla_{\varpi_{1}}T)(\varpi_{2},\varpi_{3},\varpi_{4},\varpi_{5})=\mathop{\mathcal{S}}_{\varpi_{1},\varpi_{2},\varpi_{3}}\Sigma(\varpi_{1})T(\varpi_{2},\varpi_{3},\varpi_{4},\varpi_{5})$

for an $1$-form $\Sigma$, then the curvature $2$-forms $\Omega^{m}_{(T)l}$
[69] are recurrent [74, 75, 76]. Further, the $1$-forms $\Lambda_{(Z)l}$[121]
are recurrent if

$(\nabla_{\varpi_{1}}Z)(\varpi_{2},\varpi)-(\nabla_{\varpi_{2}}Z)(\varpi_{1},\varpi)=\Sigma(\varpi_{1})Z(\varpi_{2},\varpi)-\Sigma(\varpi_{2})Z(\varpi_{1},\varpi)$

holds on $M$ for an $1$-form $\Sigma$.

######  Definition 2.9.

$($[79, 128]$)$ Let T be a (0,4)-type tensor on $M$ and $L(M)$ be the set of
all 1-forms $\Pi$ on $M$ satisfying

$\displaystyle\mathop{\mathcal{S}}_{\varpi_{1},\varpi_{2},\varpi_{3}}\Pi(\varpi_{1})\otimes
T(\varpi_{2},\varpi_{3},\varpi_{4},\varpi_{5})=0$

with $dimL(M)\geq 1$. Then $M$ is called a $T$-space by Venzi.

Several notions of geometrical symmetries, such as, motion, curvature
collineation for (0,4)-type curvature tensor and for (1,3)-type curvature
tensor, curvature inheritance for (0,4)-type curvature tensor and for
(1,3)-type curvature tensor, Ricci collineation and Ricci inheritance, all of
which are originated from the Lie derivatives of different tensors, are
essential to be reviewed for the study of symmetry in the HBH spacetime.

######  Definition 2.10.

A manifold $M$ admits motion with respect to some vector field $\xi$ if
$\pounds_{\xi}g=0$. The vector field $\xi$ is also called Killing.

Katzin et al. [66, 67], in 1969, introduced the concept of curvature
collineation for the (1,3)-type curvature tensor by vanishing Lie derivative
of the (1,3)-type Riemann curvature tensor with respect to some vector field.
Again, in 1992, by introducing the notion of curvature inheritance for the
(1,3)-type curvature tensor, Duggal [17] generalizes the concept of curvature
collineation.

######  Definition 2.11.

([17]) A semi-Riemannian manifold $M$ admits curvature inheritance for the
(1,3)-type curvature tensor $\widetilde{R}$ if $M$ satisfies

$\pounds_{\xi}\widetilde{R}=\lambda\widetilde{R}$

for a non-Killing vector field $\xi$, where $\lambda$ is a scalar function and
the (1,3)-type curvature tensor $\widetilde{R}$ is related to the (0,4)-type
curvature tensor $R$ by
$R(v_{1},v_{2},v_{3},v_{4})=g(\widetilde{R}(v_{1},v_{2})v_{3},v_{4})$. In
particular, if $\lambda=0$, then it turns into curvature collineation [66, 67]
for the (1,3)-type curvature tensor $\widetilde{R}$ (i.e.,
$\pounds_{\xi}\widetilde{R}=0$).

######  Definition 2.12.

([17]) A semi-Riemannian manifold $M$ realizes Ricci inheritance if for some
vector field $\xi$ and for some scalar function $\lambda$, $M$ possesses the
relation

$\pounds_{\xi}S=\lambda S.$

. Further, if $\lambda=0$, it transforms into Ricci collineation (i.e.,
$\pounds_{\xi}S=0$).

Recently, generalizing the notion of curvature inheritance for (0,4)-type
curvature tensor $R$ Shaikh and Datta [96] introduced the concept of
generalized curvature inheritance for (0,4)-type curvature tensor $R$, which
is given as follows:

######  Definition 2.13.

([96]) A semi-Riemannian manifold $M$ admits generalized curvature inheritance
for (0,4)-type curvature tensor $R$ if there is a non-Killing vector field
$\xi$ which satisfies the relation

$\pounds_{\xi}R=\lambda R+\lambda_{1}g\wedge g+\lambda_{2}g\wedge
S+\lambda_{3}S\wedge S,$

where $\lambda,\lambda_{1},\lambda_{2},\lambda_{3}$ are the scalar functions.
In particular, if $\lambda_{i}=0$ for $i=1,2,3$, then $M$ admits curvature
inheritance for (0,4)-type curvature tensor $R$. Further, if
$\lambda=0=\lambda_{i}$ for $i=1,2,3$, then it becomes curvature collineation
for (0,4)-type curvature tensor $R$.

##  3\. Hayward black hole spacetime admitting geometric structures

In coordinates $(t,r,\theta,\phi)$, the metric tensor of HBH spacetime is
given by:

$g=\left(\begin{array}[]{cccc}-(1-\frac{2mr^{2}}{r^{3}+2mb^{2}})&0&0&0\\\
0&(1-\frac{2mr^{2}}{r^{3}+2mb^{2}})^{-1}&0&0\\\ 0&0&r^{2}&0\\\
0&0&0&r^{2}\sin^{2}\theta\end{array}\right).$

Now, the components of the metric $g$ are

$\begin{array}[]{c}g_{11}=-\left(1-\frac{2mr^{2}}{r^{3}+2mb^{2}}\right),\;g_{22}=\left(1-\frac{2mr^{2}}{r^{3}+2mb^{2}}\right)^{-1},\\\
g_{33}=r^{2},\;g_{44}=r^{2}\sin^{2}\theta,\;g_{ij}=0,\;\mbox{otherwise}.\end{array}$

Let $B=2b^{2}m+r^{2}(r-2m)$, $B_{1}=2b^{2}m+r^{3}$, $B_{2}=4b^{2}m-r^{3}$,
$B_{3}=b^{2}m-r^{3}$ and $B_{4}=10b^{2}m-r^{3}$. The non-vanishing components
of the Christoffel symbols $(\Gamma^{h}_{ij})$ of 2nd kind are calculated as
given below:

$\begin{array}[]{c}\Gamma^{2}_{11}=-\frac{mrBB_{2}}{B_{1}^{3}},\ \
\Gamma^{1}_{12}=-\frac{mrB_{2}}{BB_{1}}=-\Gamma^{2}_{22},\\\
\Gamma^{3}_{23}=\frac{1}{r}=\Gamma^{4}_{24},\ \
\Gamma^{2}_{33}=-r+\frac{2mr^{3}}{B_{1}},\\\ \Gamma^{4}_{34}=\cot\theta,\ \
\Gamma^{2}_{44}=-\frac{rB\sin^{2}\theta}{B_{1}},\\\
\Gamma^{3}_{44}=-\cos\theta\sin\theta.\end{array}$

The non-vanishing components of the Riemann-curvature $(R_{abcd})$ and Ricci
tensor $(S_{ab})$ and the scalar curvature $\kappa$ are calculated as given
below:

$\begin{array}[]{c}R_{1212}=-\frac{2m(2b^{2}m(2b^{2}m-7r^{3})+r^{6})}{B_{1}^{3}},\
\ \par
R_{1313}=\frac{mr^{2}BB_{2}}{B_{1}^{3}}=\frac{1}{\sin^{2}\theta}R_{1414},\\\
R_{2323}=\frac{mr^{2}B_{2}}{BB_{1}}=\frac{1}{\sin^{2}\theta}R_{2424},\ \
R_{3434}=\frac{2mr^{4}\sin^{2}\theta}{B_{1}};\\\ \end{array}$
$\begin{array}[]{c}S_{11}=\frac{24b^{2}m^{2}BB_{3}}{B_{1}^{4}},\ \
S_{22}=\frac{24b^{2}m^{2}(-b^{2}m+r^{3})}{BB_{1}^{2}},\\\
S_{33}=-\frac{12b^{2}m^{2}r^{2}}{B_{1}^{2}},\ \ S_{44}=\sin^{2}\theta
S_{33};\end{array}$
$\kappa=\frac{24b^{2}m^{2}(r^{3}-4b^{2}m)}{(2b^{2}m+r^{3})^{3}}.$

From the above calculation, one can obtain the following:

######  Proposition 3.1.

The HBH spacetime is neither Einstein nor quasi-Einstein but $(i)$ it is
$2$-quasi-Einstein for $\alpha=-\frac{12b^{2}m^{2}}{B_{1}^{2}}$ and $(ii)$ for
$\alpha=-\frac{12b^{2}m^{2}}{B_{1}^{2}}$, $\beta=1$, $\gamma=1$,
$\Pi$=$\left\\{-\frac{B}{B_{1}},1,0,0\right\\}$ and $\phi$=
$\left\\{\frac{36b^{2}m^{2}r^{3}+B_{1}^{2}B}{2B_{1}^{3}},\frac{18b^{2}m^{2}r^{3}}{B_{1}^{2}B}-\frac{1}{2},0,0\right\\}$,
it is generalized quasi-Einstein in the sense of Chaki.

Let $\mathcal{K}^{1}=(g\wedge g)$, $\mathcal{K}^{2}=(g\wedge S)$ and
$\mathcal{K}^{3}=(S\wedge S).$ Then the components other than zero of
$\mathcal{K}^{1}$, $\mathcal{K}^{2}$ and $\mathcal{K}^{3}$ are calculated as
given below:

$\begin{array}[]{c}\mathcal{K}^{1}_{1212}=2,\mathcal{K}^{1}_{1313}=\frac{2r^{2}B}{B_{1}}=\frac{1}{\sin^{2}\theta}\mathcal{K}^{1}_{1414},\\\
\mathcal{K}^{1}_{2323}=-\frac{2r^{2}B_{1}}{B}=\frac{1}{\sin^{2}\theta}\mathcal{K}^{1}_{2424},\
\ \mathcal{K}^{1}_{3434}=-2r^{4}\sin^{2}\theta;\end{array}$

$\begin{array}[]{c}\mathcal{K}^{2}_{1212}=-\frac{48b^{2}m^{2}B_{3}}{B_{1}^{3}},\
\
\mathcal{K}^{2}_{1313}=-\frac{12b^{2}m^{2}r^{2}BB_{2}}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}\mathcal{K}^{2}_{1414},\\\
\mathcal{K}^{2}_{2323}=\frac{12b^{2}m^{2}r^{2}B_{2}}{BB_{1}^{2}}=\frac{1}{\sin^{2}\theta}\mathcal{K}^{2}_{2424},\
\
\mathcal{K}^{2}_{3434}=\frac{24b^{2}m^{2}r^{4}\sin^{2}\theta}{B_{1}^{2}};\end{array}$

$\begin{array}[]{c}\mathcal{K}^{3}_{1212}=\frac{1152b^{4}m^{4}B_{3}^{2}}{B_{1}^{6}},\
\
\mathcal{K}^{3}_{1313}=\frac{576b^{4}m^{4}r^{2}B_{3}B}{B_{1}^{6}}=\frac{1}{\sin^{2}\theta}\mathcal{K}^{3}_{1414},\\\
\mathcal{K}^{3}_{2323}=-\frac{576b^{4}m^{4}r^{2}B_{3}}{B_{1}^{4}B}=\frac{1}{\sin^{2}\theta}\mathcal{K}^{3}_{2424},\
\
\mathcal{K}^{3}_{3434}=-\frac{288b^{4}m^{4}r^{4}\sin^{2}\theta}{B_{1}^{4}}.\\\
\end{array}$

From the above calculation, it follows that $S\wedge S$, $g\wedge S$, $g\wedge
g$, and $R$ are linearly dependent in HBH spacetime, and hence the Riemann
tensor $R$ can be explicitly given as follows:

$\displaystyle
R=\varsigma_{1}\mathcal{K}^{1}+\varsigma_{2}\mathcal{K}^{2}+\varsigma_{3}\mathcal{K}^{3}$
(3.1)

where $\varsigma_{1}=m(\frac{2}{3r^{3}}-\frac{1}{B_{1}})$,
$\varsigma_{2}=\frac{1}{36}(10+\frac{16b^{2}m}{r^{3}}+\frac{r^{3}}{b^{2}m})$
and $\varsigma_{3}=\frac{B_{2}B_{1}^{3}}{432b^{4}m^{3}r^{3}}$.
On contraction the relation (3.1) entails

$\displaystyle S^{2}+\vartheta_{1}S+\vartheta_{2}g=0$ (3.2)

where $\vartheta_{1}=\frac{12b^{2}m^{2}B_{2}}{B_{1}^{3}}$ and
$\vartheta_{2}=\frac{288b^{4}m^{4}B_{3}}{B_{1}^{5}}$.
From the relation (3.1) and (3.2), we can state the following:

######  Proposition 3.2.

The HBH spacetime is neither Ein$(3)$ nor generalized Roter type but it
fulfills $(i)$ Roter type and $(ii)$ Einstein manifold of level $2.$

The non-vanishing components $C_{abcd}$ of the conformal curvature tensor $C$
(upto symmetry) are calculated and given as below:

$\begin{array}[]{c}C_{1212}=\frac{2mr^{3}B_{2}}{B_{1}^{3}},\ \
C_{1313}=-\frac{mr^{5}B_{2}B}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}C_{1414},\\\
C_{2323}=\frac{mr^{5}B_{2}}{B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}C_{2424},\ \
C_{3434}=-\frac{2mr^{7}B_{2}\sin^{2}\theta}{B_{1}^{3}}.\\\ \end{array}$

If $\mathscr{D}_{abcd,f}=\nabla_{f}R_{abcd}$ and
$\mathscr{F}_{abcd,f}=\nabla_{f}C_{abcd}$, then the components other than zero
of $\nabla{R}$ and $\nabla{C}$ are obtained as follows:

$\begin{array}[]{c}\mathscr{D}_{1212,2}=\frac{6m(40b^{4}m^{2}r^{2}-32b^{2}mr^{5}+r^{8})}{B_{1}^{4}},\
\
\mathscr{D}_{1213,3}=\frac{3mr^{4}B_{4}B}{B_{1}^{4}}=\mathscr{D}_{1313,2},\\\
\mathscr{D}_{1214,4}=-\frac{3mr^{4}B_{4}B\sin^{2}\theta}{B_{1}^{4}}=\mathscr{D}_{1414,2},\
\
\par\mathscr{D}_{2323,2}=-\frac{3mr^{4}B_{4}}{B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}\mathscr{D}_{2424,2},\\\
\mathscr{D}_{2334,4}=\frac{3mr^{6}\sin^{2}\theta}{B_{1}^{2}}=-\mathscr{D}_{2434,3}=-\frac{1}{2}\mathscr{D}_{3434,2};\end{array}$

$\begin{array}[]{c}\mathscr{F}_{1212,2}=\frac{6mr^{2}(8b^{4}m^{2}-12b^{2}mr^{3}+r^{6})}{B_{1}^{4}}=-\frac{1}{r^{4}\sin^{2}\theta}\mathscr{F}_{3434,2},\\\
\mathscr{F}_{1213,3}=-\frac{3mr^{4}BB_{2}}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}\mathscr{F}_{1214,4},\
\
\par\mathscr{F}_{1313,2}=-\frac{3mr^{4}(8b^{4}m^{2}-12b^{2}mr^{3}+r^{6})B}{B_{1}^{5}}=\frac{1}{\sin^{2}\theta}\mathscr{F}_{1414,2},\\\
\mathscr{F}_{2323,2}=\frac{3mr^{4}(8b^{4}m^{2}-12b^{2}mr^{3}+r^{6})}{B_{1}^{3}B}=\frac{1}{\sin^{2}\theta}\mathscr{F}_{2424,2},\
\
\mathscr{F}_{2334,4}=-\frac{3mr^{6}B_{2}\sin^{2}\theta}{B_{1}^{3}}=-\mathscr{F}_{2434,3}.\end{array}$

From the above components we get the following proposition:

######  Proposition 3.3.

The HBH spacetime is not conformally recurrent but its $(i)$ conformal
$2$-form are recurrent for the $1$-forms
$\\{0,-\frac{6b^{2}m(8b^{2}m-5r^{3})}{8b^{4}m^{2}r+2b^{2}mr^{4}-r^{7}},0,0\\}$
and $(ii)$ the general form of $R$-compatible tensor and $C$-compatible tensor
are given by

$\left(\begin{array}[]{cccc}\mathscr{Z}_{11}&\mathscr{Z}_{12}&0&0\\\
\mathscr{Z}_{12}&\mathscr{Z}_{22}&0&0\\\
0&0&\mathscr{Z}_{33}&\mathscr{Z}_{34}\\\
0&0&\mathscr{Z}_{34}&\mathscr{Z}_{44}\end{array}\right)$

where $\mathscr{Z}_{ij}$ are arbitrary scalars.

Let $\mathcal{M}^{1}=R\cdot R$, $\mathcal{M}^{2}=R\cdot C$,
$\mathcal{M}^{3}=C\cdot R$, $\mathscr{P}^{1}=Q(g,R)$,
$\mathscr{P}^{2}=Q(S,R)$, $\mathscr{P}^{3}=Q(g,C)$ and
$\mathscr{P}^{4}=Q(S,C)$. Then the components of $\mathcal{M}^{1}$,
$\mathcal{M}^{2}$, $\mathcal{M}^{3}$, $\mathscr{P}^{1}$, $\mathscr{P}^{2}$,
$\mathscr{P}^{3}$ and $\mathscr{P}^{4}$, which do not vanish, are given upto
symmetry as follows:

$\begin{array}[]{c}\mathcal{M}^{1}_{1223,13}=-\frac{3m^{2}r^{5}(40b^{4}m^{2}-14b^{2}mr^{3}+r^{6})}{B_{1}^{5}}=-\mathcal{M}^{1}_{1213,23},\
\
\mathcal{M}^{1}_{1434,13}=-\frac{3m^{2}r^{7}BB_{2}\sin^{2}\theta}{B_{1}^{5}}=-\mathcal{M}^{1}_{1334,14},\\\
\mathcal{M}^{1}_{1224,14}=-\frac{3m^{2}r^{5}(40b^{4}m^{2}-14b^{2}mr^{3}+r^{6})\sin^{2}\theta}{B_{1}^{5}}=-\mathcal{M}^{1}_{1214,24},\
\
\mathcal{M}^{1}_{2434,23}=-\frac{3m^{2}r^{7}B_{2}\sin^{2}\theta}{B_{1}^{3}B}=-\mathcal{M}^{1}_{2334,24};\end{array}$

$\begin{array}[]{c}\mathcal{M}^{2}_{1223,13}=-\frac{3m^{2}r^{5}B_{2}^{2}}{B_{1}^{5}}=-\mathcal{M}^{2}_{1213,23},\
\
\mathcal{M}^{2}_{1434,13}=\frac{3m^{2}r^{7}BB_{2}\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{2}_{1334,14},\\\
\mathcal{M}^{2}_{1224,14}=-\frac{3m^{2}r^{5}B_{2}^{2}\sin^{2}\theta}{B_{1}^{5}}=-\mathcal{M}^{2}_{1214,24},\
\
\par\mathcal{M}^{2}_{2434,23}=\frac{3m^{2}r^{7}B_{2}^{2}\sin^{2}\theta}{B_{1}^{4}B}=-\mathcal{M}^{2}_{2334,24};\par\end{array}$

$\begin{array}[]{c}\mathcal{M}^{3}_{1223,13}=-\frac{3m^{2}r^{8}(40b^{4}m^{2}-14b^{2}mr^{3}+r^{6})}{B_{1}^{6}}=-\mathcal{M}^{3}_{1213,23},\\\
\mathcal{M}^{3}_{1434,13}=-\frac{3m^{2}r^{10}BB_{2}\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{3}_{1334,14},\\\
\mathcal{M}^{3}_{1224,14}=-\frac{3m^{2}r^{8}(40b^{4}m^{2}-14b^{2}mr^{3}+r^{6})\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{3}_{1214,24},\\\
\mathcal{M}^{3}_{2434,23}=-\frac{3m^{2}r^{10}B_{2}\sin^{2}\theta}{B_{1}^{4}B}=-\mathcal{M}^{3}_{2334,24};\end{array}$

$\begin{array}[]{c}\mathscr{P}^{1}_{1223,13}=\frac{3mr^{5}B_{4}}{B_{1}^{3}}=-\mathscr{P}^{1}_{1213,23},\
\
\mathscr{P}^{1}_{1434,13}=\frac{3mr^{7}B\sin^{2}\theta}{B_{1}^{3}}=-\mathscr{P}^{1}_{1334,14},\\\
\mathscr{P}^{1}_{1224,14}=\frac{3mr^{5}B_{4}\sin^{2}\theta}{B_{1}^{3}}=-\mathscr{P}^{1}_{1214,24},\
\
\mathscr{P}^{1}_{2434,23}=\frac{3mr^{7}\sin^{2}\theta}{B_{1}B}=-\mathscr{P}^{1}_{2334,24};\\\
\end{array}$

$\begin{array}[]{c}\mathscr{P}^{2}_{1223,13}=-\frac{216b^{4}m^{4}r^{5}}{B_{1}^{5}}=-\mathscr{P}^{2}_{1213,23},\
\
\mathscr{P}^{2}_{1434,13}=-\frac{36b^{2}m^{3}r^{7}B\sin^{2}\theta}{B_{1}^{5}}=-\mathscr{P}^{2}_{1334,14},\\\
\mathscr{P}^{2}_{1224,14}=-\frac{216b^{4}m^{4}r^{5}\sin^{2}\theta}{B_{1}^{5}}=-\mathscr{P}^{2}_{1214,24},\
\
\mathscr{P}^{2}_{2434,23}=\frac{36b^{2}m^{3}r^{7}\sin^{2}\theta}{B_{1}^{3}B}=-\mathscr{P}^{2}_{2334,24};\end{array}$

$\begin{array}[]{c}\mathscr{P}^{3}_{1223,13}=\frac{3mr^{5}B_{2}}{B_{1}^{3}}=-\mathscr{P}^{3}_{1213,23},\
\
\mathscr{P}^{3}_{1434,13}=-\frac{3mr^{7}BB_{2}\sin^{2}\theta}{B_{1}^{4}}=-\mathscr{P}^{3}_{1334,14},\\\
\mathscr{P}^{3}_{1224,14}=\frac{3mr^{5}B_{2}\sin^{2}\theta}{B_{1}^{3}}=-\mathscr{P}^{3}_{1214,24},\
\
\mathscr{P}^{3}_{2434,23}=-\frac{3mr^{7}B_{2}\sin^{2}\theta}{B_{1}^{2}B}=-\mathscr{P}^{3}_{2334,24};\end{array}$

$\begin{array}[]{c}\mathscr{P}^{4}_{1223,13}=-\frac{72b^{4}m^{4}r^{5}B_{2}}{B_{1}^{6}}=-\mathscr{P}^{4}_{1213,23},\\\
\mathscr{P}^{4}_{1434,13}=-\frac{36b^{2}m^{3}r^{7}(2b^{2}m-r^{3})B_{2}B\sin^{2}\theta}{B_{1}^{7}}=-\mathscr{P}^{4}_{1334,14},\\\
\mathscr{P}^{4}_{1224,14}=-\frac{72b^{4}m^{4}r^{5}B_{2}\sin^{2}\theta}{B_{1}^{6}}=-\mathscr{P}^{4}_{1214,24},\\\
\mathscr{P}^{4}_{2434,23}=\frac{36b^{2}m^{3}r^{7}(8b^{4}m^{2}-6b^{2}mr^{3}+r^{6})\sin^{2}\theta}{B_{1}^{5}B}=-\mathscr{P}^{4}_{2334,24}.\end{array}$

From the above components we get the following:

######  Proposition 3.4.

The HBH spacetime is not Ricci generalized pseodosymmetric but it is
pseudosymmetric and realizes several pseudosymmetric type curvature relations:

  1. (i)

$R\cdot R=-\frac{mB_{2}}{B_{1}^{2}}Q(g,R)$ and hence $R\cdot
C=-\frac{mB_{2}}{B_{1}^{2}}Q(g,C)$,

  2. (ii)

$C\cdot R=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,R)$ and hence $C\cdot
C=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,C),$

  3. (iii)

$R\cdot R-\frac{m(16b^{2}m-r^{3})}{B_{2}B_{1}}Q(g,C)=Q(S,R),$

  4. (iv)

$C\cdot R-R\cdot C=\mathcal{L}_{1}\ Q(g,R)+\mathcal{L}_{2}\ Q(S,R),$ where
$\mathcal{L}_{1}=-\frac{8b^{2}m^{2}B_{2}}{(16b^{2}m-r^{3})B_{1}^{2}}$ and
$\mathcal{L}_{2}=\frac{16b^{4}m^{2}-8b^{2}mr^{3}+r^{6}}{(r^{3}-16b^{2}m)B_{1}},$

  5. (v)

$C\cdot R-R\cdot C=\mathcal{L}_{3}\ Q(g,C)+\mathcal{L}_{4}\ Q(S,C),$ where
$\mathcal{L}_{3}=\frac{8b^{2}m^{2}B_{2}}{B_{1}^{3}}$ and $\mathcal{L}_{4}=1$.

The non-vanishing components $P_{abcd}$ of the projective curvature tensor $P$
(upto symmetry) of the HBH spacetime are calculated as follows:

$\begin{array}[]{c}P_{1212}=\frac{2mr^{3}B_{4}}{B_{1}^{3}}=-P_{1221},\ \
P_{1313}=-\frac{mr^{5}B_{4}B}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}P_{1414},\\\
P_{1331}=-\frac{mr^{5}B}{B_{1}^{3}}=\frac{1}{\sin^{2}\theta}P_{1441},\ \ \par
P_{2323}=\frac{mr^{5}B_{4}}{B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}P_{2424},\\\
P_{2332}=\frac{mr^{5}}{B_{1}B}=\frac{1}{\sin^{2}\theta}P_{2442},\ \
P_{3434}=\frac{2mr^{7}\sin^{2}\theta}{B_{1}^{2}}=-P_{3443}.\end{array}$

Let $\mathcal{M}^{4}=P\cdot S$ and $\mathscr{P}^{5}=Q(g,S)$. Then the non-
vanishing components of the tensor $\mathcal{M}^{4}$ and $\mathscr{P}^{5}$ are
obtained as follows:

$\begin{array}[]{c}\mathcal{M}^{4}_{13,13}=-\frac{36b^{2}m^{3}r^{5}B_{2}B}{B_{1}^{6}}=-\mathcal{M}^{4}_{13,31},\
\
\mathcal{M}^{4}_{14,14}=-\frac{36b^{2}m^{3}r^{5}B_{2}B\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{4}_{14,41},\\\
\mathcal{M}^{4}_{23,23}=\frac{36b^{2}m^{3}r^{5}B_{2}}{BB_{1}^{4}}=-\mathcal{M}^{4}_{23,32},\
\
\mathcal{M}^{4}_{24,24}=\frac{36b^{2}m^{3}r^{5}B_{2}\sin^{2}\theta}{BB_{1}^{4}}=-\mathcal{M}^{4}_{24,42};\end{array}$
$\begin{array}[]{c}\mathscr{P}^{5}_{1313}=\frac{36b^{2}m^{2}r^{5}B}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}\mathscr{P}^{5}_{1414},\
\
\par\mathscr{P}^{5}_{2323}=-\frac{36b^{2}m^{2}r^{5}}{B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}\mathscr{P}^{5}_{2424}.\end{array}.$

From the above components we get the following:

######  Proposition 3.5.

The HBH spacetime fulfills the curvature conditions
$(i)$ $R\cdot P=-\frac{mB_{2}}{B_{1}^{2}}Q(g,P)$ $(ii)$ $P\cdot
S=-\frac{mB_{2}}{B_{1}^{2}}Q(g,S)$ and $(iii)$ the general form of
$P$-compatible tensor is given by

$\left(\begin{array}[]{cccc}\mathscr{Z}_{11}&\mathscr{Z}_{12}&0&0\\\
\mathscr{Z}_{12}&\mathscr{Z}_{22}&0&0\\\
0&0&\mathscr{Z}_{33}&\mathscr{Z}_{34}\\\
0&0&\mathscr{Z}_{34}&\mathscr{Z}_{44}\end{array}\right)$

where $\mathscr{Z}_{ij}$ are arbitrary scalars.

The non-vanishing components of the concircular curvature tensor $W$ and
conharmonic curvature tensor $K$ (upto symmetry) are computed as follows:

$\begin{array}[]{c}W_{1212}=\frac{2mr^{3}(13b^{2}m-r^{3})}{B_{1}^{3}},\ \
W_{1313}=-\frac{mr^{5}B_{2}B}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}W_{1414},\\\
W_{2323}=\frac{mr^{5}B_{2}}{B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}W_{2424},\ \
W_{3434}=\frac{2mr^{7}(5b^{2}m+r^{3})\sin^{2}\theta}{B_{1}^{3}};\end{array}.$

$\begin{array}[]{c}K_{1212}=\frac{2mB_{2}}{B_{1}^{2}},\ \
K_{1313}=-\frac{mr^{2}B_{2}^{2}B}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}K_{1414},\\\
K_{2323}=\frac{mr^{2}B_{2}^{2}}{B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}K_{2424},\
\ K_{3434}=-\frac{2mr^{4}B_{2}\sin^{2}\theta}{B_{1}^{2}}.\end{array}.$

If $\mathcal{M}^{5}=W\cdot R$ and $\mathcal{M}^{6}=K\cdot R$, then the non-
vanishing components of the tensors $\mathcal{M}^{5}$ and $\mathcal{M}^{6}$
are given as below:

$\begin{array}[]{c}\mathcal{M}^{5}_{1223,13}=-\frac{3m^{2}r^{8}(40b^{4}m^{2}-14b^{2}mr^{3}+r^{6})}{B_{1}^{6}}=-\mathcal{M}^{5}_{1213,23},\\\
\mathcal{M}^{5}_{1434,13}=-\frac{3m^{2}r^{10}BB_{2}\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{5}_{1334,14},\\\
\mathcal{M}^{5}_{1224,14}=-\frac{3m^{2}r^{8}(40b^{4}m^{2}-14b^{2}mr^{3}+r^{6})\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{5}_{1214,24},\\\
\mathcal{M}^{5}_{2434,23}=-\frac{3m^{2}r^{10}B_{2}\sin^{2}\theta}{B_{1}^{4}B}=-\mathcal{M}^{5}_{2334,24}\end{array}$

$\begin{array}[]{c}\mathcal{M}^{6}_{1223,13}=\frac{3m^{2}r^{5}B_{4}B_{2}^{2}}{B_{1}^{6}}=-\mathcal{M}^{6}_{1213,23},\
\
\mathcal{M}^{6}_{1434,13}=-\frac{3m^{2}r^{7}B_{2}^{2}B\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{6}_{1334,14},\\\
\mathcal{M}^{6}_{1224,14}=\frac{3m^{2}r^{5}B_{4}B_{2}^{2}\sin^{2}\theta}{B_{1}^{6}}=-\mathcal{M}^{6}_{1214,24},\
\
\par\mathcal{M}^{6}_{2434,23}=\frac{3m^{2}r^{7}B_{2}^{2}\sin^{2}\theta}{B_{1}^{4}B}=-\mathcal{M}^{6}_{2334,24}.\end{array}$

From the above components we get the following:

######  Proposition 3.6.

The HBH spacetime fulfills the curvature conditions

$(i)\ W\cdot R=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,R),$

$(ii)$ $K\cdot R=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,R)$ and $(iii)$ the general
form of $W$and $K$-compatible tensor are given by

$\left(\begin{array}[]{cccc}\mathscr{Z}_{11}&\mathscr{Z}_{12}&0&0\\\
\mathscr{Z}_{12}&\mathscr{Z}_{22}&0&0\\\
0&0&\mathscr{Z}_{33}&\mathscr{Z}_{34}\\\
0&0&\mathscr{Z}_{34}&\mathscr{Z}_{44}\end{array}\right)$

where $\mathscr{Z}_{ij}$ are arbitrary scalars.

From the above propositions $(\ref{pr1})$-$(\ref{pr6})$, we can conclude the
curvature restricted geometric properties of HBH spacetime as follows:

######  Theorem 3.1.

The HBH spacetime admits the following curvature properties:

  1. (i)

$R\cdot R=-\frac{mB_{2}}{B_{1}^{2}}Q(g,R).$ Hence $R\cdot
S=-\frac{mB_{2}}{B_{1}^{2}}Q(g,S)$, $R\cdot
C=-\frac{mB_{2}}{B_{1}^{2}}Q(g,C)$, $R\cdot
P=-\frac{mB_{2}}{B_{1}^{2}}Q(g,P)$, $R\cdot W=-\frac{mB_{2}}{B_{1}^{2}}Q(g,W)$
and $R\cdot K=-\frac{mB_{2}}{B_{1}^{2}}Q(g,K)$;

  2. (ii)

$C\cdot R=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,R).$ Hence $C\cdot
S=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,S)$, $C\cdot
C=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,C)$, $C\cdot
P=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,P)$, $C\cdot
W=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,W)$ and $C\cdot
K=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,K)$;

  3. (iii)

$W\cdot R=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,R).$ Hence $W\cdot
S=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,S)$,
$W\cdot C=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,C)$, $W\cdot
P=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,P)$, $W\cdot
W=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,W)$ and $W\cdot
K=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,K)$;

  4. (iv)

$K\cdot R=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,R)$ . Hence $K\cdot
S=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,S)$, $K\cdot
C=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,C)$, $K\cdot
P=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,P)$, $K\cdot
W=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,W)$ and $K\cdot
K=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,K)$;

  5. (v)

it satisfies the pseudosymmetric type curvature conditions $R\cdot
R-\mathcal{L}Q(g,C)=Q(S,R)$, where
$\mathcal{L}=\frac{m(16b^{2}m-r^{3})}{B_{2}B_{1}}$;

  6. (vi)

the tensor $C\cdot R-R\cdot C$ depends linearly on the tensors $Q(g,C)$,
$Q(S,C)$, $Q(g,R)$ and $Q(S,R)$,

  7. (vii)

it is Ricci pseudosymmetric due to projective curvature i.e., $P\cdot
S=-\frac{mB_{2}}{B_{1}^{2}}Q(g,S)$ is satisfied,

  8. (viii)

its conformal $2$-forms are recurrent for the for $1$-forms
$\\{0,-\frac{6b^{2}m(8b^{2}m-5r^{3})}{8b^{4}m^{2}r+2b^{2}mr^{4}-r^{7}},0,0\\},$

  9. (ix)

it is a Roter type spacetime,

  10. (x)

it is Ein$(2)$ spacetime as it possesses $S^{2}+\psi_{1}S+\psi_{2}g=0$ for
$\psi_{1}=\frac{12b^{2}m^{2}B_{2}}{B_{1}^{3}}$ and
$\psi_{2}=\frac{288b^{4}m^{4}B_{3}}{B_{1}^{5}},$

  11. (xi)

it is $2$-quasi-Einstein as $\alpha=-\frac{12b^{2}m^{2}}{B_{1}^{2}},$

  12. (xii)

for $\alpha=-\frac{12b^{2}m^{2}}{B_{1}^{2}}$, $\beta=1$, $\gamma=1$, $\phi$=
$\left\\{\frac{36b^{2}m^{2}r^{3}+B_{1}^{2}B}{2B_{1}^{3}},\frac{18b^{2}m^{2}r^{3}}{B_{1}^{2}B}-\frac{1}{2},0,0\right\\}$
and $\Pi$=$\left\\{-\frac{B}{B_{1}},1,0,0\right\\}$, the HBH spacetime is
generalized quasi-Einstein in the sense of Chaki,

  13. (xiii)

the general form of $R$, $C$, $P$, $W$ and $K$-compatible tensors in HBH
spacetime are given by

$\left(\begin{array}[]{cccc}\mathscr{Z}_{11}&\mathscr{Z}_{12}&0&0\\\
\mathscr{Z}_{12}&\mathscr{Z}_{22}&0&0\\\
0&0&\mathscr{Z}_{33}&\mathscr{Z}_{34}\\\
0&0&\mathscr{Z}_{34}&\mathscr{Z}_{44}\end{array}\right)$

where $\mathscr{Z}_{ij}$ are arbitrary scalars,

  14. (xiv)

its Ricci tensor is compatible for $C$, $P$, $R$, $K$ and $W$.

######  Remark 3.1.

The HBH spacetime does not admit the following geometric structures:

  1. (i)

$\nabla P\neq 0$ and hence $\nabla R\neq 0$, $\nabla C\neq 0$, $\nabla K\neq
0$ and $\nabla W\neq 0$,

  2. (ii)

for any $1$-form $\Pi$, $\nabla P\neq\Pi\otimes P$ and hence it is not
recurrent for $P$, $R$, $W$, $K$ and $C$,

  3. (iii)

it does not satisfy the semi-symmetric type condition $R\cdot H=0$ where
$H=P,K,W,C,S$,

  4. (iv)

it is not Ricci generalized pseudosymmetric,

  5. (v)

it does not realize $P\cdot R=\mathcal{L}Q(g,R)$ for any smooth function
$\mathcal{L}$. Hence it is neither $P\cdot W=\mathcal{L}Q(g,W)$, $P\cdot
K=\mathcal{L}Q(g,K)$ nor $P\cdot C=\mathcal{L}Q(g,C),$

  6. (vi)

it is not $T$-space by Venzi for $T=C,R,P,W,K,$

  7. (vii)

it is neither Einstein nor quasi-Einstein,

  8. (viii)

the curvature $2$-forms for $R$, $K$, $W$ and $P$ are not recurrent,

  9. (ix)

the Ricci tensor of HBH spacetime is neither cyclic parallel nor Codazzi type,

  10. (x)

the HBH spacetime is neither weakly symmetric nor Chaki pseudosymmetric for
$P,$ $W,$ $K,$ $R$ and $C$.

##  4\. Energy momentum tensor of Hayward black hole spacetime

In Einstein field equation (briefly, EFE), the energy momentum tensor $T^{EM}$
in terms of curvature restrictions is presented as
$T^{EM}=\frac{1}{\nu}[S-\frac{\kappa}{2}g+\Lambda g],$ where $\Lambda$ is the
cosmological constant, $\nu=\frac{8\pi G}{c^{4}}$ ($G$ being the Newton’s
gravitational constant and c being the speed of light in vacuum). The
components other than zero of the energy momentum tensor $T^{EM}_{ab}$ are
given below:

$\begin{array}[]{c}T^{EM}_{11}=-\frac{3b^{2}m^{2}B}{2B_{1}^{3}},\ \
T^{EM}_{22}=-\frac{3b^{2}m^{2}}{2B_{1}B},\\\
T^{EM}_{33}=\frac{3b^{2}m^{2}r^{2}B_{3}}{B_{1}^{3}}=\frac{1}{\sin^{2}\theta}T^{EM}_{44}.\\\
\end{array}$

The non-vanishing components of the covariant derivative of energy momentum
tensor are calculated as follows:

$\begin{array}[]{c}T^{EM}_{11,2}=\frac{9b^{2}m^{2}r^{2}B}{B_{1}^{4}},\ \
T^{EM}_{22,2}=-\frac{9b^{2}m^{2}r^{2}}{B_{1}^{2}B},\\\
T^{EM}_{23,3}=\frac{9b^{2}m^{2}r^{4}}{2B_{1}^{3}}=\frac{1}{\sin^{2}\theta}T^{EM}_{24,4},\\\
T^{EM}_{33,2}=\frac{9b^{2}m^{2}r^{4}(-5b^{2}m+2r^{3})}{B_{1}^{4}}=\frac{1}{\sin^{2}\theta}T^{EM}_{44,2}.\end{array}$

Let $\mathcal{V}^{1}=R\cdot T^{EM}$, $\mathcal{V}^{2}=C\cdot T^{EM}$,
$\mathcal{V}^{3}=W\cdot T^{EM}$, $\mathcal{V}^{4}=K\cdot T^{EM}$, and
$\mathcal{U}^{1}=Q(g,T^{EM})$. Then the non-vanishing components of the
tensors $\mathcal{V}^{1}$, $\mathcal{V}^{2}$, $\mathcal{V}^{3}$,
$\mathcal{V}^{4}$ and $\mathcal{U}^{1}$ are obtained as follows:

$\begin{array}[]{c}\mathcal{V}^{1}_{1313}=-\frac{9b^{2}m^{3}r^{5}B_{2}B}{2B_{1}^{6}}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{1}_{1414},\
\
\mathcal{V}^{1}_{2323}=\frac{9b^{2}m^{3}r^{5}B_{2}}{2B_{1}^{4}B}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{1}_{2424};\end{array}$
$\begin{array}[]{c}\mathcal{V}^{2}_{1313}=-\frac{9b^{2}m^{3}r^{8}B_{2}B}{2B_{1}^{7}}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{2}_{1414},\
\
\mathcal{V}^{2}_{2323}=\frac{9b^{2}m^{3}r^{8}B_{2}}{2B_{1}^{5}B}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{2}_{2424};\end{array}$
$\begin{array}[]{c}\mathcal{V}^{3}_{1313}=-\frac{9b^{2}m^{3}r^{8}B_{2}B}{2B_{1}^{7}}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{3}_{1414},\
\
\mathcal{V}^{3}_{2323}=\frac{9b^{2}m^{3}r^{8}B_{2}}{2B_{1}^{5}B}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{3}_{2424};\end{array}$
$\begin{array}[]{c}\mathcal{V}^{4}_{1313}=-\frac{9b^{2}m^{3}r^{5}B_{2}^{2}B}{2B_{1}^{7}}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{4}_{1414},\
\
\mathcal{V}^{4}_{2323}=-\frac{9b^{2}m^{3}r^{5}B_{2}^{2}}{2B_{1}^{5}B}=\frac{1}{\sin^{2}\theta}\mathcal{V}^{4}_{2424};\end{array}$
$\begin{array}[]{c}\mathcal{U}^{1}_{1313}=\frac{9b^{2}m^{2}r^{5}B}{2B_{1}^{4}}=\frac{1}{\sin^{2}\theta}\mathcal{U}^{1}_{1414},\
\
\mathcal{U}^{1}_{2323}=-\frac{9b^{2}m^{2}r^{5}}{2B_{1}^{2}B}=\frac{1}{\sin^{2}\theta}\mathcal{U}^{1}_{2424}.\end{array}$

From the above components we get the following theorem:

######  Theorem 4.1.

The energy momentum tensor of the HBH spacetime admits the following geometric
properties:

  1. (i)

$R\cdot T^{EM}=-\frac{mB_{2}}{B_{1}^{2}}Q(g,T^{EM})$ i.e., the nature of the
energy momentum tensor is pseudosymmetric,

  2. (ii)

$C\cdot T^{EM}=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,T^{EM})$,

  3. (iii)

$W\cdot T^{EM}=-\frac{mr^{3}B_{2}}{B_{1}^{3}}Q(g,T^{EM}),$

  4. (iv)

$K\cdot T^{EM}=\frac{mB_{2}^{2}}{B_{1}^{3}}Q(g,T^{EM})$ and

  5. (v)

the energy momentum tensor is Riemann compatible, projective compatible,
conharmonic compatible, concircular compatible and conformal compatible.

##  5\. Hayward black hole spacetime Vs interior black hole spacetime and
Reissner-Nordström spacetime

The interior black hole spacetime [49, 98] is a spherically symmetric non-
static solution of EFE. Physically, it describes the empty spacetime in the
exterior region of a black hole. A comparison between HBH spacetime and
interior black hole spacetime in terms of their curvature properties is
delineated as follows:
Similarities:

  1. (i)

both the spacetimes are pseudosymmetric,

  2. (ii)

both the spacetimes are pseudosymmetric due to conharmonic, concircular as
well as conformal curvature,

  3. (iii)

both the spacetimes are Einstein manifolds of level $2$ and $2$-quasi Einstein
manifolds,

  4. (iv)

both the spacetimes are Roter type,

  5. (v)

Ricci tensor is Riemann compatible as well as Weyl compatible .

Again, the exterior gravitational field of a non-rotating charged body can be
described by Reissner-Nordström spacetime [68], which is a spherically
symmetric solution of EFE having cosmological constant zero. This solution is
more general than the Schwarzschild solution of EFE as the Reissner-Nordström
solution admits non-vanishing charges. An elegant comparison between HBH
spacetime and Reissner-Nordström spacetime based on the curvature properties
is described as follows:
Dissimilarities:

  1. (i)

the conharmonic $2$-forms of Reissner-Nordström spacetime are recurrent while
HBH spacetime does not admit such recurrence,

  2. (ii)

HBH spacetime does not vanish scalar curvature while for the Reissner-
Nordström spacetime the scalar curvature vanishes.

However, the HBH spacetime and the Reissner-Nordström spacetime have the
following similar properties:

  1. (i)

both spacetimes are Roter type,

  2. (ii)

both the spacetimes are Einstein manifolds of level $2$,

  3. (iii)

both are pseudosymmetric as well as pseudosymmetric due to Weyl conformal
tensor,

  4. (iv)

conformal $2$-forms for both the spacetimes are recurrent,

  5. (v)

both are $2$-quasi-Einstein manifold,

  6. (vi)

Ricci tensor of both the spacetimes are Riemann compatible as well as Weyl
compatible.

##  6\. Ricci soliton and symmetries on Hayward black hole spacetime

Let $\mathcal{K}(M)$ be the set of all Killing vector fields on $M$. Then
$\mathcal{K}(M)$ is a Lie subalgebra of the Lie algebra $\chi(M)$ of all
smooth vector fields on $M$ and $\mathcal{K}(M)$ contains at most $n(n+1)/2$
linearly independent Killing vector fields, and if $\mathcal{K}(M)$ consists
of exactly $n(n+1)/2$ linearly independent Killing vector fields, then $M$ is
known as a maximally symmetric space. We mention that $M$ is a maximally
symmetric space if $M$ is of constant scalar curvature. We note that the
scalar curvature $\kappa$ of HBH spacetime is not constant as shown in Section
3 by $\kappa=\frac{24b^{2}m^{2}(r^{3}-4b^{2}m)}{(2b^{2}m+r^{3})^{3}}$ and
hence it is not maximally symmetric. Now, we investigate some Killing and non-
Killing vector fields on HBH spacetime given as follows:

######  Proposition 6.1.

The vector fields ${\frac{\partial}{\partial t}}$ and
${\frac{\partial}{\partial\phi}}$ on the HBH spacetime are Killing (i.e.,
$\pounds_{\frac{\partial}{\partial
t}}g=0=\pounds_{\frac{\partial}{\partial\phi}}g$).

######  Corollary 6.1.

For each real number $\lambda_{1}$ and $\lambda_{2}$, the vector field
$\lambda_{1}{\frac{\partial}{\partial
t}}+\lambda_{2}{\frac{\partial}{\partial\phi}}$ on the HBH spacetime is also
Killing.

The vector field ${\frac{\partial}{\partial r}}$ is non-Killing, and if
$\mathcal{A}=\pounds_{\frac{\partial}{\partial r}}g$, then the non-zero
components of $\mathcal{A}$ are calculated as follows:

$\begin{array}[]{c}\mathcal{A}_{11}=\frac{2mrB_{2}}{B_{1}^{2}},\ \ \
\mathcal{A}_{22}=\frac{2mrB_{2}}{B^{2}},\\\ \mathcal{A}_{33}=2r,\ \ \
\mathcal{A}_{44}=2r\sin^{2}\theta.\end{array}$

Therefore, for the non-Killing vector field $\frac{\partial}{\partial r}$ and
the $1$-form $\eta=(0,1,0,0)$, the HBH spacetime possesses the following
relation:

$\displaystyle\pounds_{\frac{\partial}{\partial
r}}g+2\sigma_{1}S+2\sigma_{2}g-2\sigma_{3}\eta\otimes\eta=0,$

where $\sigma_{1},\sigma_{2},\sigma_{3}$ are given by

$\left.\begin{aligned}
\sigma_{1}&=\frac{B_{1}^{2}(4b^{4}m^{2}+4b^{2}mr^{3}-3mr^{5}+r^{6})}{36b^{2}m^{2}r^{4}B},\\\
\sigma_{2}&=\frac{4b^{4}m^{2}-2b^{2}mr^{3}+3mr^{5}-2r^{6}}{3r^{4}B},\\\
\sigma_{3}&=\frac{2mrB_{2}}{B^{2}}.\end{aligned}\ \ \right\\}$ (6.1)

This leads to the following:

######  Theorem 6.1.

The HBH spacetime realizes almost $\eta$-Ricci-Yamabe soliton for the non-
Killing soliton vector field $\frac{\partial}{\partial r}$ and the 1-form
$\eta=(0,1,0,0)$ provided $(2b^{2}m-2mr^{2}+r^{3})\neq 0$, i.e., for the
soliton vector field $\xi=\frac{\partial}{\partial r}$, the HBH spacetime
possesses

$\frac{1}{2}\pounds_{\xi}g+\sigma_{1}S+\left(\lambda-\frac{1}{2}\sigma_{4}\kappa\right)g+\sigma_{3}\eta\otimes\eta=0,$

where $\sigma_{4}=2$, $\lambda=\sigma_{2}+\kappa$, and
$\sigma_{1},\sigma_{2},\sigma_{3}$ are given in (6.1).

######  Theorem 6.2.

If
$(2b^{2}m+r^{3})^{2}(4b^{4}m^{2}+4b^{2}mr^{3}-3mr^{5}+r^{6})=36b^{2}m^{2}r^{4}(2b^{2}m-2mr^{2}+r^{3})$
with $(2b^{2}m-2mr^{2}+r^{3})\neq 0$, then for the soliton vector field
$\frac{\partial}{\partial r}$, the HBH spacetime admits an almost $\eta$-Ricci
soliton with the $1$-form $\eta=(0,1,0,0)$, i.e., for the vector field
$\xi=\frac{\partial}{\partial r}$, the HBH spacetime realizes

$\displaystyle\frac{1}{2}\pounds_{\xi}g+S+\sigma_{2}g-\sigma_{3}\eta\otimes\eta=0.$

where $\sigma_{2}$, $\sigma_{3}$ is given in (6.1).

Let $\mathcal{E}=\pounds_{\frac{\partial}{\partial r}}S$,
$\mathcal{G}=\pounds_{\frac{\partial}{\partial r}}\widetilde{R}$ and
$\mathcal{H}=\pounds_{\frac{\partial}{\partial r}}R$. Then the non-vanishing
components of $\mathcal{E}$, $\mathcal{G}$ and $\mathcal{H}$ are computed as
follows:

$\begin{array}[]{c}\mathcal{E}_{11}=-\frac{24b^{2}m^{2}r^{2}\\{2(7m-3r)r^{6}+b^{2}mr^{3}(3r-40m)+2b^{4}m^{2}(4m+15r)\\}}{B_{1}^{5}},\\\
\mathcal{E}_{22}=-\frac{24b^{2}m^{2}r\\{2b^{4}m^{2}(4m-15r)+b^{2}m(20m-3r)r^{3}+2r^{6}(3r-5m)\\}}{B_{1}^{3}B^{2}},\\\
\mathcal{E}_{33}=-\frac{48b^{2}m^{2}rB_{3}}{B_{1}^{3}}=\frac{1}{\sin^{2}\theta}\mathcal{E}_{44}\end{array}$

$\begin{array}[]{c}\mathcal{G}^{1}_{212}=\frac{2mr\left\\{16b^{6}m^{3}(2m-15r)+r^{9}(4m-3r)+24b^{4}m^{2}r^{3}(5m+3r)+6b^{2}mr^{6}(15r-26m)\right\\}}{B_{1}^{3}B}\\\
\mathcal{G}^{1}_{313}=\frac{mr(16b^{4}m^{2}-26b^{2}mr^{3}+r^{6})}{B_{1}^{3}}=-\mathcal{G}^{1}_{331}=-\frac{1}{\sin^{2}\theta}\mathcal{G}^{1}_{441}=\mathcal{G}^{2}_{323}\\\
=-\mathcal{G}^{1}_{332}=\frac{1}{\sin^{2}\theta}\mathcal{G}^{2}_{424}=-\frac{1}{\sin^{2}\theta}\mathcal{G}^{2}_{442},\\\
\mathcal{G}^{2}_{112}=\frac{2mr\left\\{(8m-3r)r^{9}+72b^{4}m^{2}r^{3}(5m+r)+6b^{2}mr^{6}(15r-38m)-16b^{6}m^{3}(2m+15r)\right\\}}{B_{1}^{5}}=-\mathcal{G}^{2}_{121},\\\
\mathcal{G}^{3}_{113}=-\frac{mr\left\\{(8m-3r)r^{6}+4b^{2}mr^{3}(6r-19m)+4b^{4}m^{2}(8m+15r)\right\\}}{B_{1}^{4}}=-\mathcal{G}^{3}_{131}=\mathcal{G}^{4}_{114}=-\mathcal{G}^{4}_{141},\\\
\mathcal{G}^{3}_{223}=\frac{mr\left\\{(4m-3r)r^{6}+4b^{2}mr^{3}(6r-11m)+4b^{4}m^{2}(15r-8m)\right\\}}{B_{1}^{2}B^{2}}=-\mathcal{G}^{3}_{232}=\mathcal{G}^{4}_{224}=-\mathcal{G}^{4}_{242},\\\
\mathcal{G}^{3}_{434}=\frac{2mrB_{2}\sin^{2}\theta}{B_{1}^{2}}=-\mathcal{G}^{3}_{443}=-\sin^{2}\theta\mathcal{G}^{4}_{334}=\sin^{2}\theta\mathcal{G}^{4}_{343}.\end{array}$

$\begin{array}[]{c}\mathcal{H}_{1212}=\frac{6m(40b^{4}m^{2}r^{2}-32b^{2}mr^{5}+r^{8})}{B_{1}^{4}}=-\mathcal{H}_{1221}=\mathcal{H}_{2121},\\\
\mathcal{H}_{1313}=\frac{mr\left\\{-32b^{6}m^{3}+(4m-r)r^{8}+4b^{2}mr^{5}(6r-17m)+4b^{4}m^{2}r^{2}(16m+9r)\right\\}}{B_{1}^{4}}=\mathcal{H}_{3131}=-\mathcal{H}_{1331}\\\
=\frac{1}{\sin^{2}\theta}\mathcal{H}_{4141}=\frac{1}{\sin^{2}\theta}\mathcal{H}_{1414}=-\frac{1}{\sin^{2}\theta}\mathcal{H}_{1441},\\\
\mathcal{H}_{2323}=\frac{mr\left\\{32b^{6}m^{3}-36b^{4}m^{2}r^{3}+12b^{2}m(3m-2r)r^{5}+r^{9}\right\\}}{B_{1}^{2}B^{2}}=-\mathcal{H}_{2332}=\frac{1}{\sin^{2}\theta}\mathcal{H}_{2424}\\\
=-\frac{1}{\sin^{2}\theta}\mathcal{H}_{2442}=\mathcal{H}_{3232}=\frac{1}{\sin^{2}\theta}\mathcal{H}_{4242},\\\
\mathcal{H}_{3434}=\frac{2mr^{3}(8b^{2}m+r^{3})\sin^{2}\theta}{B_{1}^{2}}=-\mathcal{H}_{3443}=\mathcal{H}_{4343}.\end{array}$

If $\mathcal{M}=\pounds_{\frac{\partial}{\partial\theta}}g$,
$\mathcal{N}=\pounds_{\frac{\partial}{\partial\theta}}S$,
$\mathcal{Q}=\pounds_{\frac{\partial}{\partial\theta}}\widetilde{R}$ and
$\mathcal{O}=\pounds_{\frac{\partial}{\partial\theta}}R$, then the non-zero
components of $\mathcal{M}$, $\mathcal{N}$, $\mathcal{Q}$ and $\mathcal{O}$
are given as follows:

$\begin{array}[]{c}\mathcal{Q}^{1}_{414}=\frac{mr^{2}B_{2}\sin
2\theta}{B_{1}^{2}}=-\mathcal{Q}^{1}_{441}=\mathcal{Q}^{2}_{424}=-\mathcal{Q}^{2}_{442},\
\ \mathcal{Q}^{3}_{434}=\frac{2mr^{2}\sin
2\theta}{B_{1}}=-\mathcal{Q}^{3}_{443},\end{array}$

$\begin{array}[]{c}\mathcal{O}_{1414}=\frac{mr^{2}B\sin
2\theta}{B_{1}^{3}}=-\mathcal{O}_{1441}=\frac{1}{B_{2}}\mathcal{O}_{4141},\ \
\mathcal{O}_{2424}=\frac{mr^{2}B_{2}\sin
2\theta}{B_{1}B}=-\mathcal{O}_{2442}=\mathcal{O}_{4242},\\\
\mathcal{O}_{3434}=\frac{2mr^{4}\sin
2\theta}{B_{1}}=-\mathcal{O}_{3443}=\mathcal{O}_{1343}.\end{array}$

From the above calculation of the Lie derivative of various curvature tensors
it can be easily checked that with respect to the non-Killing vector fields
$\frac{\partial}{\partial r}$, $\frac{\partial}{\partial\theta}$ and
$\lambda_{1}\frac{\partial}{\partial
r}+\lambda_{2}\frac{\partial}{\partial\theta}$ ($\lambda_{1},\lambda_{2}$
being real numbers), the HBH spacetime admits

  1. (i)

neither Ricci collineation nor Ricci inheritance,

  2. (ii)

neither curvature collineation for (1,3)-type curvature tensor nor curvature
collineation for (0,4)-type curvature tensor,

  3. (iii)

neither curvature inheritance for (1,3)-type curvature tensor nor curvature
inheritance for (0,4)-type curvature tensor.

##  7\. Hayward black hole spacetime Vs point-like global monopole spacetime

The point-like global monopole spacetime [21, 85] is a static and spherically
symmetric solution of EFE. It is a heavy object characterized by divergent
mass and spherically symmetry, and against polar as well as spherical
perturbation it is expected to be stable. A comparative study between the HBH
spacetime and the point-like global monopole spacetime with respect to various
kind of symmetries and Ricci soliton is given as follows:
Similarities:

  1. (i)

both the spacetimes admits motion for the vector fields
$\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial\phi}$, i.e., the
vector fields $\frac{\partial}{\partial t}$ and
$\frac{\partial}{\partial\phi}$ are Killing in both the spacetimes,

  2. (ii)

the vector fields $\frac{\partial}{\partial r}$ and
$\frac{\partial}{\partial\theta}$ are non-Killing in both the spacetimes,

  3. (iii)

with respect to the non-Killing vector field
$\frac{\partial}{\partial\theta}$, both the spacetimes realize neither
curvature collineation nor curvature inheritance for (1,3)-type curvature
tensor,

  4. (iv)

with respect to the non-Killing vector field
$\frac{\partial}{\partial\theta}$, both the spacetimes possess neither Ricci
collineation nor Ricci inheritance.

Nevertheless, they have the following dissimilar properties:
Dissimilarities:

  1. (i)

with respect to the non-Killing vector field $\frac{\partial}{\partial r}$,
the point-like global monopole spacetime admits Ricci collineation as well as
curvature collineation for (1,3)-type curvature tensor, whereas HBH spacetime
does not admit such collineations,

  2. (ii)

for the non-Killing vector fields $\frac{\partial}{\partial r}$,
$\frac{\partial}{\partial\theta}$ and $\lambda_{1}\frac{\partial}{\partial
r}+\lambda_{2}\frac{\partial}{\partial\theta}$ ($\lambda_{1},\lambda_{2}$
being real numbers), the point-like global monopole spacetime possesses
curvature inheritance for the (0,4)-type curvature tensor, but HBH spacetime
does not realize such inheritance,

  3. (iii)

with respect to the soliton vector field $\frac{\partial}{\partial r}$, the
HBH spacetime admits both the almost $\eta$-Ricci soliton and almost
$\eta$-Ricci-Yamabe soliton for the 1-form $\eta=(0,1,0,0)$, but the point-
like global monopole spacetime realizes neither almost $\eta$-Ricci soliton
nor almost $\eta$-Ricci-Yamabe soliton with respect to the non-Killing vector
field $\frac{\partial}{\partial r}$.

##  8\. Acknowledgment

The third author is grateful to the Council of Scientific and Industrial
Research (CSIR File No.: 09/025(0253)/2018-EMR-I), Govt. of India, for the
award of SRF (Senior Research Fellowship). The fourth author greatly
acknowledges to The University Grants Commission, Government of India for the
award of Senior Research Fellow. All the algebraic computations of Section $3$
to $7$ are performed by a program in Wolfram Mathematica developed by the
first author (A. A. Shaikh).

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