# Electrical and thermal control of Fabry-Pérot cavities mediated by Casimir
forces

Lixin Ge<EMAIL_ADDRESS>School of Physics and Electronic Engineering,
Xinyang Normal University, Xinyang 464000, China  Bingzhong Li  School of
Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000,
China  Hao Luo  School of Physics and Electronic Engineering, Xinyang Normal
University, Xinyang 464000, China  Ke Gong  School of Physics and Electronic
Engineering, Xinyang Normal University, Xinyang 464000, China

###### Abstract

Dynamic tuning of optical cavities is highly desired in many photonic systems.
Here, we show that Fabry-Pérot(FP) cavities can be actively controlled by the
Casimir force. The optical FP cavities consist of a gold nanoplate confronted
to an electrical-connecting multi-layer substrate in a liquid environment. The
gold nanoplate can be stably suspended due to the balance of repulsive and
attractive Casimir forces. Moreover, the suspension distance are modulated
strongly by the electric gating or temperature of the system. As a result, we
could shift the resonant wavelengthes of the cavities with tens of nanometers
at optical frequencies. Finally, we analyze the influence of Brownian motion
on the equilibrium distances. Due to the high $Q$-factor of the FP cavities,
our proposed system offers a remarkable platform to experimentally investigate
the thermal Casimir effect at sub-micrometer separations

##  I Introduction

The Casimir force between two perfect metallic plates, predicted by Hendrik
Casimir in 1948, is a macroscopic quantum effect resulting from the zero-point
fluctuations in vacuum Casimir (1948). Latter, this quantum effect was
generalized by E. M. Lifshitz to include frequency-dependent dielectrics and
finite temperatures Lifshitz (1955). The Casimir forces between two objects
consisted by the same materials are generally attractive. In the past two
decades, great effort has been devoted to the search for Casimir repulsions in
the vacuum environmentWoods et al. (2016); Kenneth et al. (2002); Zhao et al.
(2009); Levin et al. (2010); Rodriguez-Lopez and Grushin (2014); Nie et al.
(2013); Li and Khandekar (2021); Camacho et al. (2022), yet lack of
experimental verifications due to the strict constrains. By contrast, the
Casimir repulsions have been experimentally achieved between two liquid-
separated objects (labelled 1 and 2) when the permittivity satisfies
$\varepsilon_{1}(i\xi)>\varepsilon_{\mathrm{liq}}(i\xi)>\varepsilon_{2}(i\xi)$
for a vast range of frequency Munday et al. (2009), where
$\varepsilon_{\mathrm{liq}}(i\xi)$ is the permittivity of the intervening
liquid evaluated with imaginary frequency $\omega=i\xi$. Interestingly, stable
suspensions mediated by Casimir repulsions were reported in different
configurationsZhao et al. (2019); Dou et al. (2014); Esteso et al. (2015); Liu
and Zhang (2016); Ye et al. (2018); Esteso et al. (2022).

Recently, an new concept for tunable FP cavities has been proposed by Esteso
et al.Esteso et al. (2019), based on the Casimir force. The FP cavities play a
crucial role in optical spectroscopy and find extensive applicationsVaughan
(2017). For instance, the FP cavities consisted of metal-insulator-metal have
been received considerable interest in nanophotonics, due to their excellent
performances on strong light-matter interactions Caligiuri et al. (2020); Liu
et al. (2010); Deng et al. (2015). In general, the resonances of FP cavities
are fixed once the samples are fabricated Esteso et al. (2019). The dynamic
tuning of optical FP cavities through the Casimir force, particularly using
external stimuli such as electric gating, and temperature, remains largely
unknown in this field.

Tunable Casimir forces can be realized by changing the dielectric response of
the materials through external stimuli, e.g., electric gating Ge et al.
(2020a); Gong et al. (2022), magnetic fields Jiang and Wilczek (2019); Zeng
and Yang (2011); Wang et al. (2006), optical lasers Chen et al. (2007); Chang
et al. (2011), etc. Another scheme to dynamically tune the Casimir forces is
based on the change of temperature Yampol’skii et al. (2008); Galkina et al.
(2009); Boström et al. (2018); Ge and Shi (2022). Generally, thermal effect on
the Casimir forces is weak Boström and Sernelius (2000); Sushkov et al.
(2011). For a vacuum gap, the thermal Casimir effect is observable only when
the separation is large (e.g., over three micrometers)Sushkov et al. (2011).
Such large separation severely affects its applications. Recently, a strong
thermal Casimir effect based on graphene sheets was revealed at sub-micrometer
scales Liu et al. (2021); Abbas et al. (2017); Bimonte et al. (2017);
Khusnutdinov et al. (2018). The temperature dependence of Casimir forces for
graphene is attributed to two different mechanisms. The first one is the
thermal fluctuation, as illustrated by the implicit term in Klimchitskaya and
Mostepanenko (2015). The second one relies on the fact that the dielectric
response of graphene is temperature dependence (the explicit term in
Klimchitskaya and Mostepanenko (2015)). This kind of temperature modulation
can be manifest at shorter separations.

In this study, we aim to dynamically tune FP cavities by manipulating the
Casimir forces. Our system comprises a gold nanoplate and a Teflon-coated
metal-oxide-semiconductor (MOS) substrate. Notably, the Casimir forces acting
on the suspended gold nanoplate exhibit a strong dependence on both the gating
voltage and the temperature. As a result, the equilibrium separation of the
gold nanoplate undergoes significant alterations. The resonant wavelength of
the optical FP cavities can be shifted for tens of nanometers. These
remarkable shifts can be accurately detected using state-of-the-art
experimental techniques. At the end, the Brownian motion of nanoplates is
taken into account, and our study presents an accurate approach for measuring
thermal Casimir forces at sub-micrometer separations via spectroscopic of the
cavities.

##  II Theoretical models

Figure 1: (color online) (a)Schematic view of the system. The incident light
is a plane wave, and the two parallel reflecting mirrors are consisted of the
gold nanoplate and the gold substrate. (b) The dielectric response of
materials evaluated in imaginary frequency. The plot depicts the permittivity
of the accumulation layer of ITO, where the gating voltage increases from 0 to
$V_{b}$ with a step size of $V_{b}/4$.

Figure 1(a) illustrates the schematic of the system under study. In this
setup, a gold nanoplate is suspended in a liquid of glycerol. The materials of
Teflon, indium tin oxide(ITO), silica, and glycerol, are almost transparency
at optical frequencies. Hence, the two parallel reflecting mirrors of the
optical FP cavities are the suspended gold nanoplate and the gold substrate.
There exists an electrical connection between the gold substrate and the ITO
layer, which can be controlled by a gating voltage.

Considering the in-plane dimension of nanoplate is much larger than the
separation, a proximity force approximation is applied for the calculations.
The Casimir force is calculated by $F_{c}=-\partial E_{c}(d)/\partial d$,
where $E_{c}(d)$ is the Casimir energy between the nanoplate and the
multilayer substrate:Zhao et al. (2019)

$\frac{E_{c}(d)}{A}=k_{b}T\overset{\infty}{\underset{n=0}{\sum}}^{\prime}\int\frac{d^{2}\mathbf{k_{\|}}}{(2\pi)^{2}}\log\det\left[1-\mathbf{R_{1}}\cdot\mathbf{R_{2}}e^{-2K_{n}d}\right],$
(1)

where $A$ represents the in-plane area, $k_{B}$ is the Boltzmann’s constant,
$T$ is the temperature,$d$ is the separation, the prime denotes a pre-factor
1/2 for $n$=0, $\mathbf{k_{\parallel}}$ is the parallel wavevector,
$K_{n}=\sqrt{k_{\parallel}^{2}+\varepsilon_{liq}(i\xi_{n})\xi_{n}^{2}/c^{2}}$
is the vertical wavevector in the liquid,
$\xi_{n}=2\pi\frac{k_{b}T}{\hbar}n(n=0,1,2,3\ldots)$ is the discrete Matsubara
frequencies, $\hbar$ is the reduced Planck’s constant, $c$ is the speed of
light in vacuum. $\mathbf{R}_{1,2}$ is the $2\times 2$ reflection matrix,
given by

$\mathbf{R_{j}}=\left(\begin{array}[]{cc}r_{j}^{s}&0\\\
0&r_{j}^{p}\end{array}\right),$ (2)

where $r_{j}^{\alpha}$ with $j$=1 and $j$=2 are the reflection coefficients
for the upper and lower layered structures, and the superscripts $\alpha=s$
and $p$ correspond to the polarization of transverse electric ($\mathbf{TE}$)
and transverse magnetic ($\mathbf{TM}$) modes, respectively. The reflection
coefficients are associated with the layer thicknesses and permittivity of
materials, which can be calculated by a transfer matrix method (TMM)Ge et al.
(2020b).

The generalized Drude-Lorentz model is applied for the goldGe et al. (2020b).
The dielectric models and parameters for the materials of Teflon, silica, and
glycerol are adopted from the recent literatures Moazzami Gudarzi and
Aboutalebi (2021); Sehmi et al. (2017). The permittivity for the ITO film is
evaluated by the Kramers-Kronig relationship:

$\varepsilon(i\xi_{n})=1+\frac{2}{\pi}\int_{0}^{\infty}\frac{\mathrm{Im}[\varepsilon(x)]x}{x^{2}+\xi_{n}^{2}}dx.$
(3)

The absorption of the ITO materials (imaginary part)is constructive by the sum
of the Drude model Krasavin and Zayats (2012) and the Tauc-Lorentz model
Jellison Jr and Modine (1996); Banishev et al. (2012). Under electrical-bias,
the ITO layer is divided into two distinct regions: the background layer and
the accumulation layer. The carrier densities in these two layers are
represented as $N_{b}$ and $N_{a}$, respectively. The magnitude of $N_{b}$ is
fixed upon fabrication, whereas the magnitude of $N_{a}$ can be adjusted by
varying the gating voltage. As given in previous literatures, the accumulation
layer possesses a homogeneous carrier density, and its thickness is determined
byGong et al. (2022); Yi et al. (2013):

$L_{a}=\frac{\pi}{\sqrt{2}}\sqrt{\frac{k_{B}T\varepsilon_{0}\varepsilon_{\mathrm{ITO}}}{N_{b}e^{2}}},$
(4)

where $e$ is the electron charge, $\varepsilon_{0}$ is the vacuum
permittivity, $\varepsilon_{\mathrm{ITO}}$=9.3 is the static permittivity of
ITO. In this work, The thickness of the ITO layer is fixed to be 5 nm, and the
background carrier density $N_{\mathrm{b}}$= $10^{19}$ cm-3. We have
$L_{\mathrm{a}}$=2.56 nm at $T$=300 K. On the other hand, the analytical
expression of $N_{\mathrm{a}}$ is given by Gong et al. (2022):

$N_{a}=N_{b}+\frac{\varepsilon_{0}\varepsilon_{s}V_{g}}{eL_{s}L_{a}},$ (5)

where $\varepsilon_{\mathrm{s}}$=3.9 denotes the static dielectric constant of
silica, $V_{g}$ is the applied voltage, and it should be smaller than the
breaking down voltage $V_{b}$=$E_{b}L_{s}$, where we assume the breakdown
field $E_{b}$=30 MV/cm Papadakis and Atwater (2015); Sire et al. (2007).

The permittivity evaluated in the imaginary frequency is presented in Fig.
1(b). Notably, the dielectric functions of the ITO in the accumulation layer
exhibit significant variations with increasing voltages in the infrared
frequency range. Teflon possesses the lowest permittivity, making it
particularly desirable for Casimir repulsions at small separations. Although
the permittivity of glycerol is close to that of silica at infrared and
visible frequencies, its static permittivity, about 42.4, at zero frequency is
much larger than that of silica.

Figure 2: (color online) (a) The Casimir pressures under different gating
voltages, where $\Delta_{\mathrm{v}}=V_{b}/3$, where the breakdown voltage is
450 V. The positive (negative) sign of the pressure corresponds to the
repulsive (attractive) force. The gray dashed line represents the pressure
generated from the gravity and buoyancy. (b)The equilibrium distances as a
function of applied voltages. The inset show the equilibrium distances as a
function of the thickness of silica layer.

Figure 3: (color online) (a) The Casimir pressures for different temperatures,
where there is no applied voltage. The gray dashed line represents the
pressure generated from the gravity and buoyancy. (b)The equilibrium distances
as a function of temperature.

##  III Results and discussions

The Casimir pressure of suspended gold nanoplate versus the separation is
shown in Fig. 2(a). The thickness of the gold nanoplate, Teflon, ITO, and
silica layers are set to be 40, 10, 5, and 150 nm, respectively. The results
demonstrate a significant modulation of the Casimir pressure by the gating
voltage. At small separations, the Casimir pressure is repulsive, while it
turns to be an attractive force for larger separations. A separation for zero
pressure, known as the Casimir equilibrium, is identified at a specific
separation. With increasing the voltages, the pressure tends to be attractive,
and the separation for the Casimir equilibrium decreases correspondingly. In
the absence of voltage, the Casimir equilibrium appears at approximately 85
nm, which decreases by about 18 nm as the voltage approaches to $V_{b}$. For a
gold nanoplate with thickness of 40 nm, the pressure resulting from gravity
and buoyancy is estimated to be 0.007 Pa Ge et al. (2020b), and is represented
by the dashed grey lines in Fig. 2(a). The equilibrium distance, denoted as
$d_{e}$, is established through the delicate balance among the Casimir force,
gravity, and buoyancy. The $d_{e}$ is slightly smaller than the separation of
Casimir equilibrium.

Figure 4: (color online) (a)The reflectance of FP cavity varies with the layer
thickness of silica, where the separation $d$ is fixed at 60 nm. (b)The
modulation of reflectance through the applied voltage and temperatures. (c)
and (d) show the reflection spectra in and out of the equilibria. The
equilibrium distances for $L_{s}$=150 nm and $L_{s}$=50 nm are 85 nm and 44
nm, respectively. The assumed error of the reflectance with $\pm$ 0.05 is
given for the equilibrium separation represented by the grey bars.

The equilibrium distance as a function of applied voltage is depicted in Fig.
2(b). The findings demonstrate a decline in the equilibrium distance as the
voltage increases, ranging from 0 up to $V_{b}$. The equilibrium distance is
also influenced by the thickness of the silica layer, as shown for
$L_{s}$=100, 150 and 200 nm. The inset of Fig. 2(b) reveals that increasing
the layer thickness, would results in elevated equilibrium distances, and the
difference of $d_{e}$ between $V_{g}=0$ and $V_{g}=V_{b}$ expands. At $L_{s}$
= 300 nm, the difference of $d_{e}$ for the $V_{g}=0$ and $V_{g}=V_{b}$
reaches nearly 36 nm. The calculations suggest that the equilibrium distance,
so as the resonant length, of the FP cavities can be effectively modulated by
the tunable Casimir forces through electrical gating.

The Casimir pressure is also strongly dependent on the temperature, as shown
in Fig. 3(a). The layer thicknesses of the materials are kept the same as
those in Fig. 2(a). To manifest the temperature effect on the Casimir force,
the applied voltage is assumed to be zero. We find that the Casimir pressure
tends to be more repulsive as the temperature increases, and the variation of
$d_{e}$ is near 10 nm when the temperature increases from 300 to 400 K. The
Casimir pressure as a function of temperature is shown in Fig. 3(b) under
different $L_{s}$. Again, the value of $d_{e}$ increases when the layer
thickness $L_{s}$ increases from 100 to 200 nm. The variation of $d_{e}$ with
respect to $T$ is almost linear as reported in Esteso et al. (2016). Such
effective modulation of $d_{e}$ due to the temperature has been proposed for
detection of thermal Casimir effect Rodriguez et al. (2010). However, the
optical resonances have not been employed in the literatureEsteso et al.
(2016); Rodriguez et al. (2010).

The thickness of silica needs to be carefully designed. The contour plot of
the reflectance via silica thickness and wavelength $\lambda$ is shown in Fig.
4(a), wherein the separation $d$ is fixed at 60 nm. The reflectance is
calculated by the TMM in the real frequency Zhan et al. (2013). The results
reveal that different resonant modes are excited when the thickness $L_{s}$
varies from 50 to 400 nm. A fundamental mode with $m$=1 is excited in the
visible regime when the layer thickness $L_{s}$ is in proper range. As the
thickness increases to a higher value, e.g., 300 nm, other high-order modes
are presented.

Figure 5: (color online) The probability of suspension positions due to the
Brownian effect with (a) $L_{\mathrm{s}}$=300 nm and (b) $L_{\mathrm{s}}$=150
nm. The increasing voltages are 0, Vb/2, and Vb. The increasing temperatures
are 300, 350 and 400 K. The dashed lines represent the initialing state with
$V_{g}$=0 and $T$=300 K. The insets in (a) and (b) show the corresponding
contour maps of the reflectance.

Figure 4(b) show the reflection spectra under different external stimuli,
where the nanoplate is suspended at the equilibrium distance $d_{e}$. The
resonance of the FP cavities is modulated efficiently by the applied voltage
and temperature. The equilibrium distances are about 85, 75 and 67 nm for the
applied voltage are 0, $V_{b}/2$ and $V_{b}$, respectively. On the other hand,
the equilibrium distances increase to 90 and 95 nm for the temperatures 350
and 400 K, respectively. Hence, the resonant dip has a blue shift for
increasing the voltages, while it has a red shift for increasing the
temperature. The results show that the giant shifting of the resonances over
tens of nanometers, is achieved by electrical gating or temperatures.

In a real configuration, the imperfection of the experiments (e.g.,
misalignment, surface roughness, electrostatic forces…) may exist, and the
errors of the reflectance with $\pm$ 0.05 (gray lines) should be introduced as
indicated in Esteso et al. (2019). Here, the high $Q$-factor of optical
cavities provides an avenue for accurate spectroscopic measurements. For
instance, the resonant wavelength without the electrical gating appears at
about 810 nm, where the equilibrium separation is 85 nm. When the separation
is out of equilibrium with $d$=82 nm, the resonant wavelength can be clarify
by the spectroscopy, as shown in Fig. 4(c). The calculated results indicate
that such small errors have limited influences as long as the spectra having
high $Q$-factors. For thickness $L_{s}$=50 nm, the equilibrium separation is
44 nm, and the shift of the reflection spectrum could not clarify in Fig.
4(d). This is because the resonance of the cavities appear at the lossy regime
($\lambda$ is smaller than about 550 nm) with $L_{s}$=50 nm, the change of
reflection spectrum due to the the variation of separation could not be
detected by the spectroscopy at such configuration.

The Brownian motion should be considered when the suspended nanoplate is
finite size. The random Brownian motion happens at both lateral and vertical
directions. The lateral Brownian motion does not affect the optical resonances
of FP cavities. However, the vertical Brownian motion could make the stable
suspension of nanoplates out of equilibrium. The normalized probability of the
suspension distance due to the vertical Brownian effect is given by Varela et
al. (2011):

$\rho(d,T)=\frac{\text{Exp}{[-U(d)/k_{B}T]}}{\int\limits_{0}^{\infty}\text{Exp}{[-U(d)/k_{B}T]\mathrm{\partial}d}}$
(6)

where $U(d)=E_{c}(d)+F_{\mathrm{GB}}d$ is the total energy of the suspended
nanoplate, where $F_{\mathrm{GB}}$ is the sum of the gravity and buoyancy
forces Ge et al. (2020b); Varela et al. (2011). Here, we consider the gold
nanoplate with area $A$=20 $\mu\mathrm{m}\times$ 20 $\mu\mathrm{m}$ (see the
experiment samples in Zhao et al. (2019)). The normalized probability with
respect to the separation is shown in Figs. 5(a) and 5(b), where we set
$L_{s}$=300 and 150 nm, respectively. The higher of the probability at the
equilibrium distance, the narrower the suspension distribution. The peak
probability $\rho(d_{e},T)$ for $L_{s}$=150 nm is almost 2 times larger than
that of $L_{s}$=300 nm, indicating a stronger stiffness at the quantum
trapping. Overall, the probability $\rho(d_{e},T)$ increases with increasing
the applied voltage. By contrast, the probability $\rho(d_{e},T)$ decreases
slightly with increasing the temperature from 300 to 350 and 400 K. The
distribution functions could be overlapping with each other at an intervening
50 K. The average of separations would be Varela et al. (2011):

$\bar{d}=\frac{\int\limits_{0}^{\infty}d\cdot\text{Exp}{[-U(d)/k_{B}T]\mathrm{\partial}d}}{\int\limits_{0}^{\infty}\text{Exp}{[-U(d)/k_{B}T]\mathrm{\partial}d}}$
(7)

The $\bar{d}$ is obtained by averaged over multiple measurements Zhao et al.
(2019). Here, the offset $\triangle=\bar{d}-d_{e}$ given in Fig. 5 is smaller
than 1 nm, due to the symmetry of the probability function near the
equilibrium distance. The precise measurement of the thermal Casimir effect
relies on the suspended separation of the nanoplate. Fortunately, the high $Q$
spectra in optical cavities offer an opportunity to monitor the separation
accurately. As depicted in the insets of Figs. 5(a) and 5(b), the high
$Q$-factor is maintained over a wide range of separations. Compared with
previous literature Rodriguez et al. (2010); Esteso et al. (2016), our work
presents an accurate way to detect thermal Casimir effect via spectroscopic
measurements of optical cavities.

##  IV Conclusions

The optical FP cavities tuned by the Casimir forces is investigated in this
work. The system consists of a gold nanoplate confronted to a Teflon-coated
MOS substrate in a liquid environment. The suspension of the gold nanoplate is
dependent on the balance among the Casimir, gravity and buoyancy forces. One
way to modulate the suspension distance of the gold nanoplate is achieved by
electrical gating. The frequency shifting of the reflection spectra can be
tens of nanometers via gating voltages. Furthermore, the control of the
optical resonances via the temperature is also demonstrated. The temperature
modulations can be manifested greatly at sub-micrometer separations. In
addition, the Brownian motion is discussed in different configurations. An new
scheme to measure thermal Casimir effect is suggested by the spectroscopic
measurements of the optical FP cavities.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11804288, and No. 61974127), Natural Science Foundation of Henan
Province (Grant No. 232300420120), and the Innovation Scientists and
Technicians Troop Construction Projects of Henan Province.

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# CURVATURE PROPERTIES OF GÖDEL METRIC

Ryszard Deszcz, Marian Hotloś, Jan Jełowicki, Haradhan Kundu and Absos Ali
Shaikh
Ryszard Deszcz and Jan Jełowicki,
Department of Mathematics,
Wrocław University of Environmental and Life Sciences
Grunwaldzka 53, 50-357 Wrocław , Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>
Marian Hotloś
Institute of Mathematics and Computer Science
Wrocław University of Technology
Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland<EMAIL_ADDRESS>
Haradhan Kundu and Absos Ali Shaikh
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The main aim of this article is to investigate the geometric structures
admitting by the Gödel spacetime which produces a new class of semi-Riemannian
manifolds (see Theorem 4.1 and Theorem 4.4). We also consider some extension
of Gödel metric (see Example 4.1).

00footnotetext:  $\mathbf{2010}$ Mathematics Subject Classification: 53B20,
53B30, 53B50, 53C50, 53C80, 83C57.
Key words and phrases: Gödel spacetime, Weyl conformal curvature tensor,
conharmonic curvature tensor, Tachibana tensor, pseudosymmetric manifold,
pseudosymmetry type curvature condition, quasi-Einstein manifold.

##  1\. Introduction

In 1949 Gödel [1] obtained an exact solution of Einstein field equation with a
non-zero cosmological constant corresponding to a universe in rotation and
with an incoherent matter distribution. In that paper he described a metric
nowadays called Gödel metric as exact and stationary solution of Einstein
field equation, which describes a rotating, homogeneous but non-isotropic
spacetime. Possessing a series of strange properties, it remains still today
quite interesting mathematically and significant physically. For example, it
contains rotating matter but have not singularity, and also it is cyclic Ricci
parallel [2]. It is known that the Weyl conformal tensor of the Gödel solution
has Petrov type $D$, and Gödel solution is, up to local isometry, the only
perfect fluid solution of Einstein field equation admitting five dimensional
Lie algebra of Killing vectors. Gödel spacetime is geodesically complete, its
timelike curves are closed [3]. Also Gödel spacetime is not globally
hyperbolic but diffeomorphic to $\mathbb{R}^{4}$ and is simply connected.
Gödel metric is the Cartesian product of a factor $\mathbb{R}$ with a three
dimensional Lorentzian manifold with signature $(-+++)$.

Gödel metric and its properties have been studied by various authors to
describe the Gödel universe. Kundt [4] studied its geodesics in 1956, and
Hawking and Ellis [5] emphasized on coordinates showing its rotational
symmetry to draw a nice picture of its dynamics in their book in 1973.
Malament [6] calculated the minimal energy of a closed timelike curve of Gödel
spacetime. In 2001 Radojević [7] presented modification of Gödel metric in
order to find out some other perfect fluid solutions. Induced matter theory
and embedding of Gödel universe in five-dimensional Ricci flat space was
studied by Fonseca-Neto et. al. [8] in 2005. Riemann extension of Gödel metric
was considered by Dryuma [2] in 2005, and Dautcourt et. al. [9] studied light
cone of Gödel universe. Lanczos spin tensor of Gödel geometry was studied by
Garcia-Olivo et. al. [10] in 2006. Gödel metric in various dimensions was
studied by Gürses et. al. [11]. Generalized Gödel metric is given by Plaue et.
al. [12] in 2008. The Gödel metric [1] is given by:

(1.1)
$ds^{2}=g_{ij}dx^{i}dx^{j}=a^{2}\left(-(dx^{1})^{2}+\frac{1}{2}e^{2x^{1}}(dx^{2})^{2}-(dx^{3})^{2}+(dx^{4})^{2}+2e^{x^{1}}dx^{2}dx^{4}\right),$

where $-\infty<x^{i}<\infty$, $i,j\in\\{1,2,3,4\\}$ and
$a^{2}=\frac{1}{2\omega^{2}}$, $\omega$ is a non-zero real constant, which
turns out to be the angular velocity, as measured by any non-spinning observer
located at any one of the dust grains.

The object of the paper is to present the curvature properties of Gödel
metric. Section 2 deals with semi-Riemannian manifolds with cyclic parallel
and Codazzi type Ricci tensor and we provide a metric whose Ricci tensor is of
Codazzi type but not cyclic parallel (see Example 2.1). However, Gödel
spacetime is a manifold with cyclic parallel Ricci tensor but the Ricci tensor
is not of Codazzi type. Section 3 is concerned with rudiments of
pseudosymmetry type manifolds, and in the last section we investigate the
geometric structures admitting by Gödel metric. Among others, it is shown that
Gödel spacetime is neither pseudosymmetric nor Ricci pseudosymmetric but it is
quasi-Einstein and a special type of Ricci generalized pseudosymmetric (i.e.,
$R\cdot R=Q(S,R)$). Although, it is not conformally pseudosymmetric but its
Weyl conformal curvature tensor is pseudosymmetric (i.e., $C\cdot
C=\frac{\kappa}{6}Q(g,C)$) (see Theorem 4.1). Hence Gödel spacetime induces a
new class of semi-Riemannian manifolds which are quasi-Einstein with
pseudosymmetric Weyl conformal curvature tensor satisfying $R\cdot R=Q(S,R)$.
Finally, we consider some extension of Gödel metric (see Example 4.1).

##  2\. Manifolds with cyclic parallel and Codazzi type Ricci tensor

Let $(M,g)$, $\dim M=n\geq 3$, be a connected paracompact manifold of class
$C^{\infty}$ with the metric $g$ of signature $(s,n-s)$, $0\leq s\leq n$. The
manifold $(M,g)$ will be called a semi(pseudo)-Riemannian manifold. Clearly,
if $s=0$ or $s=n$ then $(M,g)$ is a Riemannian manifold. If $s=1$ or $s=n-1$
then $(M,g)$ is a Lorentzian manifold. Further, let $\nabla$, $R$, $S$ and
$\kappa$ be the Levi-Civita connection, the curvature tensor, the Ricci tensor
and the scalar curvature of the semi-Riemannian manifold $(M,g)$,
respectively.

The semi-Riemannian manifold $(M,g)$ is called locally symmetric if $\nabla
R=0$ (locally $R^{h}_{ijk,\,l}=0$), which is equivalent to the fact that for
each point $x\in M$ the local geodesic symmetry is an isometry. For
$2$-dimensional manifolds being of locally symmetric and being of constant
curvature are equivalent. But for $n\geq 3$, the locally symmetric manifolds
are a generalization of the manifolds of constant curvature. A full
classification of locally symmetric manifolds is given by Cartan [13] for the
Riemannian case, and Cahen and Parker ([14], [15]) for the non-Riemannian
case.

The semi-Riemannian manifold $(M,g)$ is said to be Ricci symmetric if $\nabla
S=0$ (locally $S_{ij,\,k}=0$). Every locally symmetric semi-Riemannian
manifold is Ricci symmetric but not conversely. However, the converse
statement is true when $n=3$. For a compendium of natural symmetries of semi-
Riemannian manifolds, we refer to [16] and [17]. We mention that Gray in [18],
among other things, investigated various extensions of the class of Ricci
symmetric manifolds. We denote by $\mathcal{A}$ the class of semi-Riemannian
manifolds whose Ricci tensor $S$ is cyclic parallel, i.e.,

(2.1) $(\nabla_{X}S)(Y,Z)+(\nabla_{Y}S)(Z,X)+(\nabla_{Z}S)(X,Y)=0$

for all vector fields $X$, $Y$, $Z\in\chi(M)$, $\chi(M)$ being the Lie algebra
of all smooth vector fields on $M$. The local expression of (2.1) is
$S_{ij,\,k}+S_{jk,\,i}+S_{ki,\,j}=0$. A semi-Riemannian manifold satisfying
(2.1) is said to be a manifold with cyclic parallel Ricci tensor. We mention
that D’Atri and Nickerson [19] proposed to study some class of Riemannian
manifolds whose curvature tensor satisfies certain conditions of which the
first one is equivalent to (2.1).

Evidently, every Ricci symmetric semi-Riemannian manifold is a manifold with
cyclic parallel Ricci tensor but not conversely. However, the converse
statement is true if the Ricci tensor is a Codazzi tensor. We recall that an
$(0,2)$-symmetric tensor $B$ is said to be a Codazzi tensor if it satisfies
the Codazzi equation, i.e. $(\nabla_{X}B)(Y,Z)=(\nabla_{Y}B)(X,Z)$. The local
expression of the last equation is $B_{ij,\,k}=B_{kj,\,i}$, where $B_{ij}$ are
the local components of the tensor $B$. A Codazzi tensor is trivial if it is a
constant multiple of the metric tensor [20]. We denote by $\mathcal{B}$ the
class of semi-Riemannian manifolds with Ricci tensor $S$ as Codazzi tensor,
i.e. $S$ satisfies $(\nabla_{X}S)(Y,Z)=(\nabla_{Y}S)(X,Z)$, for all vector
fields $X$, $Y$, $Z\in\chi(M)$. Every Ricci symmetric semi-Riemannian manifold
is of class $\mathcal{B}$ but not conversely. Codazzi tensors are of great
interest in the geometric literature and have been studied by several authors,
as Berger and Ebin [21], Bourguignon [22], Derdzinski ([23], [24]), Derdzinski
and Shen [20], Ferus [25], Simon [26]; a compendium of results is reported in
Besse’s book [27].

We note that every semi-Riemannian manifold of constant curvature and hence
Einstein semi-Riemannian manifold is of class $\mathcal{A}$ as well as of
$\mathcal{B}$. We note that the scalar curvature $\kappa$ of every semi-
Riemannian manifold of the class $\mathcal{A}$ or $\mathcal{B}$ is constant.
We note that Gödel spacetime is of class $\mathcal{A}$ but not of class
$\mathcal{B}$.

It is known that Cartan hypersurfaces are Riemannian manifolds, with non-
parallel Ricci tensor, satisfying the generalized Einstein metric condition
(2.1) ([28], Theorem 4.1). As it was noted in [29](p. 109), the Cartan
hypersurfaces do not satisfy

(2.2)
$\nabla_{Z}\left(S(X,Y)-\frac{\kappa}{2(n-1)}g(X,Y)\right)=\nabla_{Y}\left(S(X,Z)-\frac{\kappa}{2(n-1)}g(X,Z)\right).$

We mention that (2.2) is presented in the Table 1, pp. 432-433 of [27]. We
also refer to [27] for results on Riemannian manifolds satisfying (2.2). This
was noted that Codazzi tensors occur naturally in the study of harmonic
Riemannian manifolds. The Ricci tensor is a Codazzi tensor if and only if
$div\,R=0$ i.e., if and only if the manifold has harmonic curvature tensor
[27].

We note that in a $3$-dimensional Riemannian manifold $(M,g)$, the following
conditions:
(a) $(M,g)$ is locally symmetric, (b) $(M,g)$ is Ricci symmetric and (c)
$(M,g)$ is a conformally flat manifold with cyclic parallel Ricci tensor, are
equivalent [30]. Also for a Riemannian manifold $(M,g)$, $n\geq 4$, the
following conditions: (a) $(M,g)$ is Ricci symmetric, (b) $(M,g)$ is a
manifold with cyclic parallel Ricci tensor and harmonic conformal curvature
tensor, and (c) $(M,g)$ is a manifold with cyclic parallel and Codazzi type
Ricci tensor, are equivalent [30].

Example 2.1 (i) Let $M=\\{(x^{1},x^{2},x^{3},x^{4}):x^{i}>0,i=1,2,3,4\\}$ be
the subset of $\mathbb{R}^{4}$ endowed with the metric $g$ defined by
$ds^{2}=\varepsilon(dx^{1})^{2}+x^{1}\left((dx^{2})^{2}+(dx^{3})^{2}+(dx^{4})^{2}\right)$,
$\varepsilon=\pm 1$. It is easy to check that $(M,g)$ is a conformally flat
quasi-Einstein Riemannian manifold,
$\mbox{rank}\,(S-\frac{1}{4(x^{1})^{2}}\,g)=1$, whose scalar curvature
$\kappa$ is equal to zero, and the Ricci tensor $S$ is of Codazzi type but not
cyclic parallel. Moreover, we have $R\cdot R=Q(S,R)$ and $R\cdot R=L\,Q(g,R)$,
$L=-\frac{1}{4\varepsilon(x^{1})^{2}}$.
(ii) Let $M=\\{(x^{1},x^{2},\ldots,x^{5}):x^{i}>0,i=1,2,\ldots,5\\}$ be the
subset of $\mathbb{R}^{5}$ endowed with the metric $g$ defined by
$ds^{2}=\varepsilon(dx^{1})^{2}+x^{1}\left((dx^{2})^{2}+\ldots+(dx^{5})^{2}\right)$,
$\varepsilon=\pm 1$. It is easy to check that $(M,g)$ is a conformally flat
quasi-Einstein Riemannian manifold,
$\mbox{rank}\,(S-\frac{1}{2(x^{1})^{2}}\,g)=1$, whose scalar curvature
$\kappa$ is non-zero, $\kappa=\frac{1}{\varepsilon(x^{1})^{2}}$, and the Ricci
tensor $S$ is not of Codazzi type and not cyclic parallel. Moreover, we have
$R\cdot R=Q(S,R)$ and $R\cdot R=L\,Q(g,R)$,
$L=-\frac{1}{4\varepsilon(x^{1})^{2}}$.

##  3\. Pseudosymmetry type curvature conditions

We define on a semi-Riemannian manifold $(M,g)$, $n\geq 3$, the endomorphisms
$X\wedge_{A}Y$, $\mathcal{R}(X,Y)$, $\mathcal{C}(X,Y)$, $\mathcal{K}(X,Y)$ and
$conh(\mathcal{R})$ by ([31], [32], [33], [34], [35])

$\displaystyle(X\wedge_{A}Y)Z$ $\displaystyle=$ $\displaystyle
A(Y,Z)X-A(X,Z)Y,$ $\displaystyle\mathcal{R}(X,Y)Z$ $\displaystyle=$
$\displaystyle[\nabla_{X},\nabla_{Y}]Z-\nabla_{[X,Y]}Z,$
$\displaystyle\mathcal{C}(X,Y)$ $\displaystyle=$
$\displaystyle\mathcal{R}(X,Y)-\frac{1}{n-2}(X\wedge_{g}\mathcal{L}Y+\mathcal{L}X\wedge_{g}Y-\frac{\kappa}{n-1}X\wedge_{g}Y),$
$\displaystyle\mathcal{K}(X,Y)$ $\displaystyle=$
$\displaystyle\mathcal{R}(X,Y)-\frac{\kappa}{n(n-1)}X\wedge_{g}Y,$
$\displaystyle{conh(\mathcal{R})}(X,Y)$ $\displaystyle=$
$\displaystyle{\mathcal{R}}(X,Y)-\frac{1}{n-2}(X\wedge_{g}{\mathcal{L}}Y+{\mathcal{L}}X\wedge_{g}Y),$

respectively, where $A$ is an $(0,2)$-tensor on $M$, $X,Y,Z\in\chi(M)$. The
Ricci operator $\mathcal{L}$ is defined by $g(X,\mathcal{L}Y)=S(X,Y)$, where
$S$ is the Ricci tensor and $\kappa$ the scalar curvature of $(M,g)$,
respectively. The tensor $S^{2}$ is defined by $S^{2}(X,Y)=S(X,\mathcal{L}Y)$.
Further, we define the Gaussian curvature tensor $G$, the Riemann-Christoffel
curvature tensor $R$, the Weyl conformal curvature tensor $C$, concircular
curvature tensor $K$ and conharmonic curvature tensor $conh(R)$ of $(M,g)$, by
([31], [32], [33], [34])

$\displaystyle G(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g((X_{1}\wedge_{g}X_{2})X_{3},X_{4}),$ $\displaystyle
R(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g(\mathcal{R}(X_{1},X_{2})X_{3},X_{4}),$ $\displaystyle
C(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g(\mathcal{C}(X_{1},X_{2})X_{3},X_{4}),$ $\displaystyle
K(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g(\mathcal{K}(X_{1},X_{2})X_{3},X_{4}),$ $\displaystyle
conh(R)(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g(conh(\mathcal{R})(X_{1},X_{2})X_{3},X_{4}),$

respectively. For $(0,2)$-tensors $A$ and $B$ we define their Kulkarni-Nomizu
product $A\wedge B$ by (see e.g. [32], [33], [35])

$\displaystyle(A\wedge B)(X_{1},X_{2};X,Y)$ $\displaystyle=$ $\displaystyle
A(X_{1},Y)B(X_{2},X)+A(X_{2},X)B(X_{1},Y)$ $\displaystyle-$ $\displaystyle
A(X_{1},X)B(X_{2},Y)-A(X_{2},Y)B(X_{1},X).$

We note that the Weyl conformal curvature tensor $C$ can be presented in the
following form

(3.1) $C=R-\frac{1}{n-2}g\wedge S+\frac{\kappa}{(n-2)(n-1)}G.$

For an $(0,k)$-tensor $T$, $k\geq 1$ and a symmetric $(0,2)$-tensor $A$ we
define the $(0,k)$-tensor $A\cdot T$ and the $(0,k+2)$-tensors $R\cdot T$,
$C\cdot T$ and $Q(A,T)$ by

$\displaystyle(A\cdot
T)(X_{1},\cdots,X_{k})=-T(\mathcal{A}X_{1},X_{2},\cdots,X_{k})-\cdots-T(X_{1},X_{2},\cdots,\mathcal{A}X_{k}),$
$\displaystyle(R\cdot T)(X_{1},\cdots,X_{k};X,Y)=(\mathcal{R}(X,Y)\cdot
T)(X_{1},\cdots,X_{k})$
$\displaystyle=-T(\mathcal{R}(X,Y)X_{1},X_{2},\cdots,X_{k})-\cdots-T(X_{1},\cdots,X_{k-1},\mathcal{R}(X,Y)X_{k}),$
$\displaystyle(C\cdot T)(X_{1},\cdots,X_{k};X,Y)=(\mathcal{C}(X,Y)\cdot
T)(X_{1},\cdots,X_{k})$
$\displaystyle=-T(\mathcal{C}(X,Y)X_{1},X_{2},\cdots,X_{k})-\cdots-T(X_{1},\cdots,X_{k-1},\mathcal{C}(X,Y)X_{k}),$
$\displaystyle Q(A,T)(X_{1},\cdots,X_{k};X,Y)=((X\wedge_{A}Y)\cdot
T)(X_{1},\cdots,X_{k})$
$\displaystyle=-T((X\wedge_{A}Y)X_{1},X_{2},\cdots,X_{k})-\cdots-T(X_{1},\cdots,X_{k-1},(X\wedge_{A}Y)X_{k}),$

where $\mathcal{A}$ is the endomorphism of $\chi(M)$ defined by
$g(\mathcal{A}X,Y)=A(X,Y)$. Putting in the above formulas $T=R,T=S$, $T=C$ or
$T=K$, $A=g$ or $A=S$, we obtain the tensors: $R\cdot R$, $R\cdot S$, $R\cdot
C$, $R\cdot K$, $C\cdot R$, $C\cdot S$, $C\cdot C$, $C\cdot K$, $Q(g,R)$,
$Q(g,S)$, $Q(g,C)$, $Q(g,K)$, $Q(S,R)$, $Q(S,C)$, $Q(g,K)$, $S\cdot R$,
$S\cdot C$, $S\cdot K$, $conh(R)\cdot conh(R)$, $conh(R)\cdot R$, $R\cdot
conh(R)$, $conh(R)\cdot S$, etc. The tensor $Q(A,T)$ is called the Tachibana
tensor of the tensors $A$ and $T$, or the Tachibana tensor for short ([36]).
We like to point out that in some papers, $Q(g,R)$ is called the Tachibana
tensor (see e.g. [36], [37], [38], [39]). We also have

######  Proposition 3.1.

(cf. [40]) For any semi-Riemannian manifold $(M,g)$, $n\geq 4$, we have

$\displaystyle conh(R)\cdot S$ $\displaystyle=$ $\displaystyle C\cdot
S-\frac{\kappa}{(n-2)(n-1)}\,Q(g,S),$ $\displaystyle R\cdot conh(R)$
$\displaystyle=$ $\displaystyle R\cdot C,$ $\displaystyle conh(R)\cdot R$
$\displaystyle=$ $\displaystyle C\cdot R-\frac{\kappa}{(n-2)(n-1)}\,Q(g,R),$
(3.2) $\displaystyle conh(R)\cdot conh(R)$ $\displaystyle=$ $\displaystyle
C\cdot C-\frac{\kappa}{(n-2)(n-1)}\,Q(g,C).$

A semi-Riemannian manifold $(M,g)$, $n\geq 3$, satisfying the condition

(3.3) $R\cdot R=0$

is called semisymmetric ([41]). We mention that non-conformally flat and non-
locally symmetric semi-Riemannian manifolds having parallel Weyl conformal
curvature tensor are semisymmetric ([42], Theorem 9), their scalar curvature
is equal to zero ([42], Theorem 7) and the Ricci tensor is a Codazzi tensor
([42], eq. (6)). We refer to [43]-[46] for the recent results on semi-
Riemannian manifolds with parallel Weyl conformal curvature tensor and, in
particular, for classification results. Semi-Riemannian warped products having
parallel Weyl conformal curvature tensor were investigated in [47]. We also
mention that, recently, conformally semisymmetric manifolds and special
semisymmetric Weyl conformal tensors are studied in [48]. Another important
subclass of semisymmetric semi-Riemannian manifolds form manifolds satisfying

(3.4) $\nabla\nabla R=0.$

We refer to [49] and [50] and references therein for results on manifolds
satisfying (3.4).

A semi-Riemannian manifold $(M,g)$, $n\geq 3$, is said to be pseudosymmetric
[51] if the tensor $R\cdot R$ and the Tachibana tensor $Q(g,R)$ are linearly
dependent at every point of $M$. This is equivalent to

(3.5) $R\cdot R=L_{R}\,Q(g,R)$

on $U_{R}=\\{x\in M:R-\frac{\kappa}{n(n-1)}G\neq 0\ \mbox{at}\ x\\}$, where
$L_{R}$ is some function on this set. We refer to [52], [53], [16], [37] and
[17] for surveys on such manifolds. In particular, a geometrical
interpretation of pseudosymmetric manifolds, in the Riemannian case, is given
in [37].

We note that [51] is the first publication, in which a semi-Riemannian
manifold satisfying (3.5) was named the pseudosymmetric manifold. In [51]
pseudosymmetric warped products with $1$-dimensional base manifold and
$(n-1)$-dimensional fibre, $n\geq 4$, which is not a semi-Riemannian space of
constant curvature, were investigated. In [54] it was shown that hypersurfaces
in spaces of constant curvature, with exactly two distinct principal
curvatures at every point, are pseudosymmetric. Thus in particular, Cartan’s
and Schouten’s investigations of quasi-umbilical hypersurfaces in spaces of
constant curvature are closely related to pseudosymmetric manifolds (cf.
[16]). It is clear that every semisymmetric manifold is pseudosymmetric.
However, the converse statement is not true. For instance, the Schwarzschild
spacetime, the Kottler spacetime and the Reissner-Nordström spacetime satisfy
(3.5) with non-zero function $L_{R}$ [55] (see also [56], [57]). We also
mention that Friedmann-Lemaître-Robertson-Walker spacetimes are
pseudosymmetric (cf. [16]). It is well-known that the Schwarzschild spacetime
was discovered in 1916 by Schwarzschild, during his study on solutions of
Einstein’s equations. It seems that the Schwarzschild spacetime is the first
example of a non-semisymmetric, pseudosymmetric warped product. Finally, we
note that (3.5) is equivalent to (e.g. see [53])

(3.6) $(R-L_{R}\,G)\cdot(R-L_{R}\,G)=0.$

We also note that in [58] Chaki introduced another kind of pseudosymmetry.
However, both notions of pseudosymmetry are not equivalent. Throughout the
paper we will confine the pseudosymmetry related to (3.5).

A semi-Riemannian manifold $(M,g)$, $n\geq 3$, is said to be Ricci-
pseudosymmetric ([59], [60]) if the tensor $R\cdot S$ and the Tachibana tensor
$Q(g,S)$ are linearly dependent at every point of $M$. Thus the manifold
$(M,g)$ is Ricci-pseudosymmetric if and only if

(3.7) $R\cdot S=L_{S}\,Q(g,S)$

holds on $U_{S}=\\{x\in M:S-\frac{r}{n}\,g\neq 0\ \mbox{at}\ x\\}$, where
$L_{S}$ is some function on this set. We note that $U_{S}\subset U_{R}$. It is
easy to check that (3.7) is equivalent to

(3.8) $(R-L_{S}\,G)\cdot(S-L_{S}\,g)=0.$

We refer to [52], [32], [16], [29] and [37] for surveys and comments on such
manifolds. A geometrical interpretation of Ricci-pseudosymmetric manifolds, in
the Riemannian case, is given in [38]. It is clear that every pseudosymmetric
semi-Riemannian manifold is Ricci-pseudosymmetric. However, the converse
statement is not true. For instance, the Cartan hypersurfaces of dimension
$6$, $12$ or $24$ are non-quasi-Einstein and non-pseudosymmetric Ricci-
pseudosymmetric manifolds ([61], see also [32], [29]). The $3$-dimensional
Cartan hypersurface is a quasi-Einstein pseudosymmetric manifold [54]. We
mention that recently quasi-Einstein Ricci-pseudosymmetric hypersurfaces in
semi-Riemannian spaces of constant curvature were investigated in [62].

A semi-Riemannian manifold $(M,g)$, $n\geq 4$, is said to be conformally
pseudosymmetric ([53], [63]) if the tensor $R\cdot C$ and the Tachibana tensor
$Q(g,C)$ are linearly dependent at every point of $M$. Again a semi-Riemannian
manifold $(M,g)$, $n\geq 4$, is said to be a manifold with pseudosymmetric
Weyl conformal curvature tensor ([63], [53]) if the tensor $C\cdot C$ and the
Tachibana tensor $Q(g,C)$ are linearly dependent at every point of $M$. This
is equivalent to

(3.9) $C\cdot C=L_{C}\,Q(g,C)$

on $U_{C}=\\{x\in M:C\neq 0\ \mbox{at}\ x\\}$, where $L_{C}$ is some function
on this set. We note that $U_{C}\subset U_{R}$. It is easy to check that (3.9)
is equivalent to

(3.10) $(C-L_{C}\,G)\cdot(C-L_{C}\,G)=0.$

Using (3.1), we also can check that (3.6) and (3.10) are equivalent on every
Einstein manifold. As it was stated in [63], any warped product
$M_{1}\times_{F}M_{2}$, with $\dim\,M_{1}=\dim M_{2}=2$, satisfies (3.9). Thus
in particular, the Schwarzschild spacetime, the Kottler spacetime and the
Reissner-Nordström spacetime satisfy (3.9).

A semi-Riemannian manifold $(M,g)$, $n\geq 3$, is said to be Ricci-generalized
pseudosymmetric ([64], [65]) if at every point of $M$ the tensor $R\cdot R$
and the Tachibana tensor $Q(S,R)$ are linearly dependent. Hence $(M,g)$ is
Ricci-generalized pseudosymmetric if and only if

(3.11) $R\cdot R=L\,Q(S,R)$

holds on $U=\\{x\in M:Q(S,R)\neq 0\ \mbox{at}\ x\\}$, where $L$ is some
function on this set. An important subclass of Ricci-generalized
pseudosymmetric manifolds is formed by the manifolds realizing the condition
([64], [66])

(3.12) $R\cdot R=Q(S,R).$

At the end of this section we also present some other curvature conditions
closely related to the above presented conditions. Namely, it was stated in
[67], on every hypersurface $M$ immersed isometrically in a semi-Riemannian
space of constant curvature $N$, $\dim\,N=n+1$, $n\geq 4$, we have

(3.13) $R\cdot R-Q(S,R)=-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}\,Q(g,C),$

where $\widetilde{\kappa}$ is the scalar curvature of the ambient space. It is
clear, that if the ambient space is a semi-Euclidean space then (3.13) reduces
to (3.12). We also note that any warped product $M_{1}\times_{F}M_{2}$, with
$\dim\,M_{1}=1$, $\dim M_{2}=3$, satisfies [68]

(3.14) $R\cdot R-Q(S,R)=L\,Q(g,C),$

for some function $L$ on $U_{C}$. Thus generalized Robertson-Walker spacetimes
fulfills (3.14). In particular, Friedmann-Lemaître-Robertson-Walker spacetimes
satisfy (3.12). We mention that the Vaidya spacetime also satisfies (3.14)
([35], Example 5.2).

The conditions: (3.5), (3.7), (3.9), (3.11) and (3.14) or other conditions of
this kind are called conditions of pseudosymmetry type. We refer to [52],
[31], [53], [32] and [16] for surveys on semi-Riemannian manifolds satisfying
such conditions. In particular, we refer to [69] for recent results on quasi-
Einstein manifolds satisfying curvature conditions of this kind.

It is easy to check that on every Ricci-pseudosymmetric manifold the following
condition is satisfied ([70], Lemma 3.3; [71], Proposition 3.1(iv))

(3.15)
$R({\mathcal{L}}X,Y,Z,W)+R({\mathcal{L}}Z,Y,W,X)+R({\mathcal{L}}W,Y,X,Z)=0.$

Semi-Riemannian manifolds $(M,g)$, $n\geq 3$, satisfying (3.15) are called
Riemann compatible ([72], [73]). We also note that (3.15) remains invariant
under geodesic mappings. In [36] (Proposition 2.1) it was proved that (3.15)
holds on every manifold satisfying (3.14). Thus in particular, manifolds
satisfying (3.12) are also Riemann compatible ([64], Lemma 2.2 (i)). We refer
to [74], [75] and [76] for further results on Riemann compatible manifolds.

##  4\. Gödel metric admitting geometric structures

Let on $\mathbb{R}^{4}$ be given the Gödel metric $g$ defined by (1.1) and let
$h,i,j,k,l.m\in\\{1,2,3,4\\}$. From (1.1) the non-zero components
$\Gamma^{h}_{ij}$ of the Christoffel symbols of second kind of $g$ are given
by [1]:

$\Gamma^{4}_{12}=\frac{e^{x^{1}}}{2},\,\,\,\Gamma^{2}_{14}=-e^{-x^{1}},\,\,\,\Gamma^{4}_{14}=1,\,\,\,\Gamma^{1}_{22}=\frac{e^{2x^{1}}}{2},\,\,\,\Gamma^{1}_{24}=\frac{e^{x^{1}}}{2}.$

Further, the non-zero components $R_{hijk}$ and $S_{ij}$ of the Riemann-
Christoffel curvature tensor $R$ and the Ricci tensor $S$, respectively, and
the scalar curvature $\kappa$ are given by [1]:

$R_{1212}=\frac{3}{4}a^{2}e^{2x^{1}},\,\,\,R_{1214}=\frac{1}{2}a^{2}e^{x^{1}},\,\,\,R_{1414}=\frac{a^{2}}{2},\,\,\,R_{2424}=\frac{1}{4}a^{2}e^{2x^{1}},$
$S_{22}=e^{2x^{1}},\,\,\,S_{24}=e^{x^{1}},\,\,\,S_{44}=1\mbox{ and
}\kappa=\frac{1}{a^{2}}.$

Again the non-zero components $R_{hijk,l}$ and $S_{ij,l}$ of the covariant
derivatives of the Riemann-Christoffel curvature tensor $\nabla R$ and the
Ricci tensor $\nabla S$, respectively, are given by:

$R_{1212,1}=a^{2}e^{2x^{1}},\,\,\,R_{1214,1}=\frac{1}{2}a^{2}e^{x^{1}},\,\,\,R_{1224,2}=\frac{1}{4}a^{2}e^{3x^{1}},$
$S_{12,2}=-\frac{e^{2x^{1}}}{2},\,\,\,S_{14,2}=-\frac{e^{x^{1}}}{2},\,\,\,S_{22,1}=e^{2x^{1}},\,\,\,S_{24,1}=\frac{e^{x^{1}}}{2}.$

The non-zero components $C_{hijk}$ of the Weyl conformal curvature tensor $C$
are given below:

$C_{1212}=\frac{1}{3}a^{2}e^{2x^{1}},\,\,\,C_{1214}=\frac{1}{6}a^{2}e^{x^{1}},\,\,\,C_{1414}=\frac{a^{2}}{6},\,\,\,C_{2424}=\frac{1}{12}a^{2}e^{2x^{1}},$
$C_{1313}=-\frac{1}{6}a^{2},\,\,\,C_{2323}=-\frac{5}{12}a^{2}e^{2x^{1}},\,\,\,C_{2334}=\frac{1}{3}a^{2}e^{x^{1}},\,\,\,C_{3434}=-\frac{1}{3}a^{2}.$

The non-zero components $(R\cdot R)_{hijklm}$ of the tensor $R\cdot R$ are
given below:

$(R\cdot R)_{122412}=\frac{1}{4}a^{2}e^{3x^{1}},\,\,\,(R\cdot
R)_{122414}=\frac{1}{4}a^{2}e^{2x^{1}},$ $(R\cdot
R)_{121224}=-\frac{1}{2}a^{2}e^{3x^{1}},\,\,\,(R\cdot
R)_{121424}=-\frac{1}{4}a^{2}e^{2x^{1}}.$

The non-zero components $Q(S,R)_{hijklm}$ of the Tachibana tensor $Q(S,R)$ are
given below:

$Q(S,R)_{122412}=\frac{1}{4}a^{2}e^{3x^{1}},\,\,\,Q(S,R)_{122414}=\frac{1}{4}a^{2}e^{2x^{1}},$
$Q(S,R)_{121224}=-\frac{1}{2}a^{2}e^{3x^{1}},\,\,\,Q(S,R)_{121424}=-\frac{1}{4}a^{2}e^{2x^{1}}.$

The non-zero components $(C\cdot C)_{hijklm}$ of the tensor $C\cdot C$ are
given below:

$\displaystyle(C\cdot C)_{122412}=\frac{1}{24}a^{2}e^{3x^{1}},\,\,\,(C\cdot
C)_{132312}=\frac{1}{12}a^{2}e^{2x^{1}},\,\,\,(C\cdot
C)_{133412}=-\frac{1}{12}a^{2}e^{x^{1}},$ $\displaystyle(C\cdot
C)_{122313}=-\frac{1}{8}a^{2}e^{2x^{1}},\,\,\,(C\cdot
C)_{123413}=\frac{1}{12}a^{2}e^{x^{1}},\,\,\,(C\cdot
C)_{142313}=-\frac{1}{12}a^{2}e^{x^{1}},$ $\displaystyle(C\cdot
C)_{143413}=\frac{1}{12}a^{2},\,\,\,(C\cdot
C)_{122414}=\frac{1}{24}a^{2}e^{2x^{1}},\,\,\,(C\cdot
C)_{132314}=\frac{1}{12}a^{2}e^{x^{1}},$ $\displaystyle(C\cdot
C)_{133414}=-\frac{1}{12}a^{2},\,\,\,(C\cdot
C)_{121323}=\frac{1}{24}a^{2}e^{2x^{1}},\,\,\,(C\cdot
C)_{232423}=-\frac{1}{24}a^{2}e^{3x^{1}},$ $\displaystyle(C\cdot
C)_{243423}=\frac{1}{24}a^{2}e^{2x^{1}},\,\,\,(C\cdot
C)_{121224}=-\frac{1}{12}a^{2}e^{3x^{1}},\,\,\,(C\cdot
C)_{121424}=-\frac{1}{24}a^{2}e^{2x^{1}},$ $\displaystyle(C\cdot
C)_{232324}=\frac{1}{12}a^{2}e^{3x^{1}},\,\,\,(C\cdot
C)_{233424}=-\frac{1}{24}a^{2}e^{2x^{1}}.$

The non-zero components $Q(g,C)_{hijklm}$ of the Tachibana tensor $Q(g,C)$ are
given below:

$\displaystyle
Q(g,C)_{122412}=\frac{1}{4}a^{4}e^{3x^{1}},\,\,\,Q(g,C)_{132312}=\frac{1}{2}a^{4}e^{2x^{1}},\,\,\,Q(g,C)_{133412}=-\frac{1}{2}a^{4}e^{x^{1}},$
$\displaystyle
Q(g,C)_{122313}=-\frac{3}{4}a^{4}e^{2x^{1}},\,\,\,Q(g,C)_{123413}=\frac{1}{2}a^{4}e^{x^{1}},\,\,\,Q(g,C)_{142313}=-\frac{1}{2}a^{4}e^{x^{1}},$
$\displaystyle
Q(g,C)_{143413}=\frac{1}{2}a^{4},\,\,\,Q(g,C)_{122414}=\frac{1}{4}a^{4}e^{2x^{1}},\,\,\,Q(g,C)_{132314}=\frac{1}{2}a^{4}e^{x^{1}},$
$\displaystyle
Q(g,C)_{133414}=-\frac{1}{2}a^{4},\,\,\,Q(g,C)_{121323}=\frac{1}{4}a^{4}e^{2x^{1}},\,\,\,Q(g,C)_{232423}=-\frac{1}{4}a^{4}e^{3x^{1}},$
$\displaystyle
Q(g,C)_{243423}=\frac{1}{4}a^{4}e^{2x^{1}},\,\,\,Q(g,C)_{121224}=-\frac{1}{2}a^{4}e^{3x^{1}},\,\,\,Q(g,C)_{121424}=-\frac{1}{4}a^{4}e^{2x^{1}},$
$\displaystyle
Q(g,C)_{232324}=\frac{1}{2}a^{4}e^{3x^{1}},\,\,\,Q(g,C)_{233424}=-\frac{1}{4}a^{4}e^{2x^{1}}.$

Thus we see that $\mathbb{R}^{4}$ equipped with the Gödel metric $g$ has the
following curvature properties:
(i) The Ricci tensor is cyclic parallel [2], the rank of the Ricci tensor $S$
is $1$ [1], precisely,

(4.1)
$S=\kappa\,\omega\otimes\omega,\,\,\,\kappa=\frac{1}{a^{2}},\,\,\,\omega=(\omega_{1},\omega_{2},\omega_{3},\omega_{4})=(0,ae^{x^{1}},0,a),$

and the vector field $X$ corresponding to $1$-form $\omega$ is given by
$X=(0,0,0,\frac{1}{a})$,
(ii) $R\cdot R=Q(S,R)$,
(iii) $C\cdot C=\frac{\kappa}{6}\,Q(g,C)$,
(iv) $3R\cdot K-2\,Q(S,K)=Q(S,C)$.
Gödel metric also realizes the following pseudosymmetric type conditions:
(v)

$\displaystyle(2a^{2}L_{1}+\frac{2}{3}L_{2})(R\cdot C+C\cdot R)$
$\displaystyle=$
$\displaystyle(-\frac{2}{3}L_{1}+\frac{1}{9a^{2}}L_{2})\Big{(}Q(g,R)-3Q(S,R)\Big{)}$
$\displaystyle+$ $\displaystyle L_{1}Q(g,C)+L_{2}Q(S,C),$

where $L_{1}$ and $L_{2}$ are some functions. This condition implies that
$R\cdot C$, $C\cdot R$, $Q(g,R)$, $Q(S,R)$, $Q(g,C)$ and $Q(S,C)$ are linearly
dependent.
(vi)

$\displaystyle(L_{1}+L_{2})(C\cdot K+K\cdot C)$ $\displaystyle=$
$\displaystyle(\frac{1}{12a^{2}}L_{1}+\frac{7}{12a^{2}}L_{2})Q(g,C)+L_{1}Q(S,C)$
$\displaystyle-$ $\displaystyle\frac{1}{2a^{2}}L_{2}Q(g,K)+L_{2}Q(S,K),$

where $L_{1}$ and $L_{2}$ are some functions. This condition implies that
$C\cdot K$, $K\cdot C$, $Q(g,C)$, $Q(S,C)$, $Q(g,K)$ and $Q(S,K)$ are linearly
dependent.
(vii)

$\displaystyle(-\frac{2}{5}L_{1}+\frac{12a^{2}}{5}L_{2})(K\cdot
conh(R)+conh(R)\cdot K)$ $\displaystyle=$
$\displaystyle(\frac{1}{30a^{2}}L_{1}-\frac{6}{5}L_{2})Q(g,K)+L_{1}Q(S,K)+L_{2}Q(g,conh(R))$
$\displaystyle+$
$\displaystyle(-\frac{7}{5}L_{1}+\frac{12a^{2}}{5}L_{2})Q(S,conh(R)),$

where $L_{1}$ and $L_{2}$ are some functions. This condition implies that
$K\cdot conh(R)$, $conh(R)\cdot K$, $Q(g,K)$, $Q(S,K)$, $Q(g,conh(R))$ and
$Q(S,conh(R))$ are linearly dependent.
(viii)

$\displaystyle 2a^{2}L_{1}(R\cdot conh(R)+conh(R)\cdot R)$ $\displaystyle=$
$\displaystyle-L_{1}\Big{(}Q(g,R)-Q(g,conh(R))\Big{)}$ $\displaystyle+$
$\displaystyle L_{2}Q(S,R)+(2a^{2}L_{1}-L_{2})Q(S,conh(R)),$

where $L_{1}$ and $L_{2}$ are some functions. This condition implies that
$R\cdot conh(R)$, $conh(R)\cdot R$, $Q(g,R)$, $Q(S,R)$, $Q(g,conh(R))$ and
$Q(S,conh(R))$ are linearly dependent.
We note that the condition $\mbox{rank}\,S=1$ holds at a point of a semi-
Riemannian manifold $(M,g)$, $n\geq 3$, if and only if $S\wedge S=0$ at this
point. Further, it is easy to check that (iv) is an immediate consequence of
(ii) and $S\wedge S=0$ and the definitions of the tensors $C$ and $K$.

We also note that the condition (3.12) holds at every point $x$ of a semi-
Riemannian manifold $(M,g)$, $n\geq 3$, at which the condition

$\omega(X_{1})\mathcal{R}(X_{2},X_{3})+\omega(X_{2})\mathcal{R}(X_{3},X_{1})+\omega(X_{3})\mathcal{R}(X_{1},X_{2})=0$

is satisfied, where $\omega$ is a non-zero covector at $x$ ([66], Theorem 3.1,
[64], p. 110). However, in case of Gödel metric (3.12) holds, but does not
satisfy the above condition. Since the Gödel metric is a product metric of a
$3$-dimensional metric and an $1$-dimensional metric, the property (3.12) also
follows from Corollary 4.1 of [65]. As it was stated in Section 3, any semi-
Riemannian manifold satisfying (3.12) is Riemann compatible. Thus the Gödel
metric satisfies also (3.15). Now (3.15), by (4.1), turns into $\
\omega_{r}g^{rs}(\omega_{h}R_{sijk}+\omega_{j}R_{sikh}+\omega_{k}R_{sihj})=0$,
where $R_{sijk}$ and $g^{rs}$ are the local components of the the Riemann-
Christoffel curvature tensor $R$ and the tensor $g^{-1}$ of the Gödel metric
$g$. We note that the $1$-form $\omega$, with respect to Definition 3.1 of
[77], is called $R$-compatible and hence Weyl compatible. Thus we have

######  Theorem 4.1.

The Gödel spacetime $(M,g)$ is a cyclic Ricci parallel and Riemann compatible
manifold satisfying: $\mbox{rank}\,S\,=\,1$, $R\cdot R=Q(S,R)$, $conh(R)\cdot
C=conh(R)\cdot conh(R)=0$, $C\cdot C=C\cdot conh(R)=\frac{\kappa}{6}\,Q(g,C)$
and the $1$-form $\omega$, defined by (4.1), is $R$-compatible as well as Weyl
compatible.

The above presented results lead to the following generalizations.

Let $(\overline{M}\times\widetilde{N},g=\overline{g}\times\widetilde{g})$ be
the product manifold of an $(n-1)$-dimensional semi-Riemannian manifold
$(\overline{M},\overline{g})$, $n\geq 4$, and an $1$-dimensional manifold
$(\widetilde{N},\widetilde{g})$. Moreover, let $(\overline{M},\overline{g})$
be a conformally flat manifold, provided that $n\geq 5$. The local components
$C_{hijk}$, $h,i,j,k\in\\{1,2,\ldots,n\\}$, of the Weyl conformal curvature
tensor $C$ of $(\overline{M}\times\widetilde{N},g)$ which may not vanish
identically are the following (cf. [78], eqs. (49)-(51))

(4.2)
$C_{abcd}=\frac{1}{(n-3)(n-2)}\,(\overline{g}_{ad}A_{bc}-\overline{g}_{ac}A_{bd}+\overline{g}_{bc}A_{ad}-\overline{g}_{bd}A_{ac}),$
(4.3) $C_{nbcn}=-\frac{1}{n-2}\,\widetilde{g}_{nn}A_{bc},$

where
$A_{ab}=\overline{S}_{ab}-\frac{\overline{\kappa}}{n-1}\,\overline{g}_{ab}$,
and $\overline{g}_{ab}$ and $\overline{S}_{ab}$ denote the local components of
the metric tensor $\overline{g}$ and the Ricci tensor $\overline{S}$ of
$(\overline{M},\overline{g})$, respectively, $a,b,c,d\in\\{1,2,\ldots,n-1\\}$,
and $\overline{\kappa}$ is the scalar curvature of
$(\overline{M},\overline{g})$. Further, we denote by $U_{C}$ the set of all
points of $(\overline{M}\times\widetilde{N},g)$ at which the Weyl conformal
curvature tensor $C$ of $(\overline{M}\times\widetilde{N},g)$ is non-zero. We
note that the tensor $C$ is non-zero at a point of $U_{C}$ if and only if
$\overline{S}\neq\frac{\overline{\kappa}}{n-1}\,\overline{g}$ at this point.

As an immediate consequence of Proposition 2 of [78] we get the following
equivalence: (3.9) holds on the set $U_{C}$ of the defined above manifold
$(\overline{M}\times\widetilde{N},g)$ for some function $L_{C}$ on this set,
if and only if at every point of $U_{C}$ we have

(4.4) $g^{bc}A_{ab}A_{cd}=(n-3)(n-2)L_{C}\,A_{ad}+\lambda\,\overline{g}_{ad},$
(4.5)
$B_{ad}B_{bc}-B_{ac}B_{bd}=\left((n-2)^{2}L_{C}^{2}-\frac{\lambda}{n-1}\right)\,(\overline{g}_{ad}\overline{g}_{bc}-\overline{g}_{ac}\overline{g}_{bd}),$

where $B=A+(n-2)L_{C}\,\overline{g}$, and $\lambda$ is a constant.
Furthermore, in view of Lemma 3.1 of [79], at every point of $U_{C}$ (4.5) is
equivalent to $\mbox{rank}\,B=1$, i.e.

(4.6)
$\mbox{rank}\left(\overline{S}-\left(\frac{\overline{\kappa}}{n-1}-(n-2)L_{C}\right)\,\overline{g}\right)=1.$

Further, we denote by $S$ and $\kappa$ the Ricci tensor and the scalar
curvature of
$(\overline{M}\times\widetilde{N},g=\overline{g}\times\widetilde{g})$,
respectively. It is obviuos that $S=\overline{S}$ and
$\kappa=\overline{\kappa}$. Therefore (4.6) yields

(4.7)
$\mbox{rank}\left(S-\left(\frac{\kappa}{n-1}-(n-2)L_{C}\right)\,g\right)=1.$

From the above presented considerations and Proposition 3.1 it follows

######  Theorem 4.2.

Let $(\overline{M}\times\widetilde{N},g=\overline{g}\times\widetilde{g})$ be
the product manifold of an $(n-1)$-dimensional semi-Riemannian manifold
$(\overline{M},\overline{g})$, $n\geq 4$, and an $1$-dimensional manifold
$(\widetilde{N},\widetilde{g})$. Moreover, let $(\overline{M},\overline{g})$
be a conformally flat manifold, provided that $n\geq 5$. If on $\overline{M}$
we have $\mbox{rank}\,(\overline{S}-\rho\,\overline{g})=1$, for some function
$\rho$, then $\mbox{rank}\,(S-\rho\,g)=1$ and (3.9), i.e. $C\cdot
C=L_{C}\,Q(g,C)$, with $L_{C}=\frac{1}{n-2}(\frac{\kappa}{n-1}-\rho)$, hold on
$\overline{M}\times\widetilde{N}$. In particular, if the rank of the Ricci
tensor of $(\overline{M},\overline{g})$ is one, then the rank of the Ricci
tensor of $\overline{M}\times\widetilde{N}$ is also one and (3.9), with
$L_{C}=\frac{\kappa}{(n-2)(n-1)}$, or equivalently, $conh(R)\cdot conh(R)=0$
holds on this manifold.

We present now an application of the last theorem.

From Theorem 4.1 of [67] it follows that a hypersurface $M$ immersed
isometrically in a semi-Riemannian space of constant curvature $N$, $\dim
N\geq 5$, is a quasi-umbilical hypersurface if and only if it is a conformally
flat manifold. Furthermore, using the Gauss equation of $M$ in $N$, we can
easily prove that if $M$ is quasi-umbilical hypersurface then it is also a
quasi-Einstein manifold. These facts, together with Theorem 4.2, leads to the
following

######  Theorem 4.3.

Let $(\overline{M},\overline{g})$ be a manifold which is isometric with a
quasi-umbilical hypersurface immersed isometrically in a semi-Riemannian space
of constant curvature $N$, $\dim N\geq 5$. Let $(\widetilde{N},\widetilde{g})$
be an $1$-dimensional manifold. Then the manifold
$(\overline{M}\times\widetilde{N},g=\overline{g}\times\widetilde{g})$ is a
quasi-Einstein manifold with pseudosymmetric Weyl conformal curvature tensor.

In this way we obtain a family of quasi-Einstein manifolds with
pseudosymmetric Weyl conformal curvature tensor. We mention that quasi-
Einstein hypersurfaces with pseudosymmetric Weyl conformal curvature tensor
immersed isometrically in semi-Riemannian spaces of constant curvature were
investigated in [33].

Theorem 4.3 together with Theorem 4.2 and Example 4.1 of [68], yields

######  Theorem 4.4.

Let $(\overline{M},\overline{g})$ be a manifold which is isometric with a
quasi-umbilical hypersurface immersed isometrically in a semi-Euclidean space
$N$, $\dim N\geq 5$. Let $(\widetilde{N},\widetilde{g})$ be an $1$-dimensional
manifold. Then the manifold
$(\overline{M}\times\widetilde{N},g=\overline{g}\times\widetilde{g})$ is a
quasi-Einstein manifold with pseudosymmetric Weyl conformal curvature tensor
satisfying $R\cdot R=Q(S,R)$.

Finally, we consider some extension of the Gödel metric.
Example 4.1. (i) We define the metric $g$ on
$M\,=\,\\{(t,r,\phi,z):t>0,r>0\\}\subset{\mathbb{R}}^{4}$ by (cf. [80],
Section 1)

(4.8) $\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(dt+H(r)\,d\phi)^{2}-D^{2}(r)\,d\phi^{2}-dr^{2}-dz^{2},$

where $H$ and $D$ are certain functions on $M$. In the special case, if
$H(r)\,=\,\frac{2\sqrt{2}}{m}\,sinh^{2}(\frac{mr}{2})$ and
$D(r)\,=\,\frac{2}{m}\,sinh(\frac{mr}{2})\,cosh(\frac{mr}{2})$ then $g$ is the
Gödel metric (e.g. see [80], eq. (1.6)).
(ii) Since the metric $g$ defined by (4.8) is the product metric of a
$3$-dimensional metric and a $1$-dimensional metric (3.12) holds on $M$. We
can check that the Riemann-Christoffel curvature tensor $R$ of $(M,g)$ is
expressed by a linear combination of the Kulkarni-Nomizu products formed by
$S$ and $S^{2}$, i.e. by the tensors $S\wedge S$, $S\wedge S^{2}$ and
$S^{2}\wedge S^{2}$,

$\displaystyle R$ $\displaystyle=$ $\displaystyle\phi_{1}\,S\wedge
S+\phi_{2}\,S\wedge S^{2}+\phi_{3}\,S^{2}\wedge S^{2},$
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\frac{D^{2}}{\tau}(2D^{2}H^{\prime\prime
2}-4DD^{\prime}H^{\prime}H^{\prime\prime}-3H^{\prime
4}+8DD^{\prime\prime}H^{\prime 2}+2D^{\prime 2}H^{\prime
2}-8D^{2}D^{\prime\prime 2}),$ $\displaystyle\phi_{2}$ $\displaystyle=$
$\displaystyle\frac{2D^{4}}{\tau}(H^{\prime 2}-4DD^{\prime\prime}),$
$\displaystyle\phi_{3}$ $\displaystyle=$ $\displaystyle-\frac{4D^{6}}{\tau},\
\ H^{\prime}\ =\ \frac{dH}{dr},\ \ H^{\prime\prime}\ =\
\frac{dH^{\prime}}{dr},$ $\displaystyle\tau$ $\displaystyle=$
$\displaystyle(H^{\prime 2}-2DD^{\prime\prime})(D^{2}H^{\prime\prime
2}-2DD^{\prime}H^{\prime}H^{\prime\prime}-H^{\prime
4}+2DD^{\prime\prime}H^{\prime 2}+D^{\prime 2}H^{\prime 2}),$

provided that the function $\tau$ is non-zero at every point of $M$.
(iii) If $H(r)\,=\,ar^{2}$, $a\,=\,const.\neq 0$ and $D(r)\,=\,r$ then (4.8)
turns into ([80], eq. (3.20))

(4.9) $\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(dt+ar^{2}\,d\phi)^{2}-r^{2}\,d\phi^{2}-dr^{2}-dz^{2}.$

The spacetime $(M,g)$ with the metric $g$ defined by (4.9) is called the Som-
Raychaudhuri solution of the Einstein field equations ([81]). For the metric
(4.9) the function $\tau$ is non-zero at every point of $M$.
(iv) We refer to [32] and [82] for surveys on semi-Riemannian manifolds
$(M,g)$, $n\geq 4$, having Riemann-Christoffel curvature tensor $R$ expressed
by a linear combination of the Kulkarni-Nomizu products formed by $g$ and $S$,
i.e. by the tensors $g\wedge g$, $g\wedge S$ and $S\wedge S$. In particular,
we mention that in the class of the Reissner-Nordström-de Sitter spacetimes
there are spacetimes having that property ([35], Example 5.3).

It may be mentioned that we have calculated the local components of various
tensors using Wolfram Mathematica, as well as SymPy and Maxima packages for
symbolic calculation.

## Conclusion:

By considering the dust particles as galaxies, the Gödel spacetime can be
taken as a cosmological model of rotating universe. Although Gödel spacetime
is not a realistic model of the universe in which we live but it realized many
peculiar properties. For example, the existence of closed timelike curves
implies a form of time travel in an alternative universe described by the
Gödel spacetime. Also Gödel spacetime is quasi-Einstein, Ricci tensor is
cyclic parallel but not Codazzi type, which may be physically interpreted as
the content of the spacetime is of rotating matter without singularity. It is
neither pseudosymmetric nor Ricci pseudosymmetric but a special type of Ricci
generalized pseudosymmetric, and it is not conformally pseudosymmetric but its
Weyl conformal curvature tensor is pseudosymmetric (i.e., $C\cdot
C=\frac{\kappa}{6}Q(g,C)$) and also the spacetime is Riemann compatible as
well as Weyl compatible (Theorem 4.1). Hence Gödel spacetime forced us to
obtain a new class of semi-Riemannian manifolds which is quasi-Einstein with
pseudosymmetric Weyl conformal curvature tensor and is a special type of Ricci
generalized pseudosymmetric manifolds (Theorem 4.4).

## Acknowledgments

The first and third named authors are supported by a grant of the Wrocław
University of Environmental and Life Sciences, Poland [WIKSiG/441/212/S]. The
fourth and fifth named authors gratefully acknowledge the financial support of
CSIR, New Delhi, India [File no: 09/025(0194)/2010-EMR-I, Project F. No.
25(0171)/09/EMR-II].

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