# Energetic Particle Acceleration in Compressible Magnetohydrodynamic
Turbulence

Jian-Fu Zhang11affiliation: Department of Physics, Xiangtan University,
Xiangtan, Hunan 411105, China and Fu-Yuan Xiang11affiliation: Department of
Physics, Xiangtan University, Xiangtan, Hunan 411105, China<EMAIL_ADDRESS>
###### Abstract

Magnetohydrodynamic (MHD) turbulence is an important agent of energetic
particle acceleration. Focusing on the compressible properties of magnetic
turbulence, we adopt test particle method to study the particle acceleration
from Alfvén, slow and fast modes in four turbulence regimes that may appear in
a realistic astrophysical environment. Our studies show that (1) the second-
order Fermi mechanism drives the acceleration of particles in the cascade
processes of three modes by particle-turbulence interactions, regardless of
whether the shock wave appears; (2) not only can the power spectra of maximum
acceleration rates reveal the inertial range of compressible turbulence, but
also recover the scaling and energy ratio relationship between the modes; (3)
fast mode dominates the acceleration of particles, especially in the case of
super-Alfvénic and supersonic turbulence, slow mode dominates the acceleration
for sub-Alfvénic turbulence in the very high energy range, and the
acceleration of Alfvén mode is significant at the early stage of the
acceleration; (4) particle acceleration from three modes results in a power-
law distribution in the certain range of evolution time. From the perspective
of particle-wave mode interaction, this paper promotes the understanding for
both the properties of turbulence and the behavior of particle acceleration,
which will help insight into astrophysical processes involved in MHD
turbulence.

ISM: general — ISM: magnetic fields — magnetohydrodynamics (MHD) —
acceleration — turbulence

##  1 Introduction

The visibility of cosmic astrophysics is due to the presence of a non-thermal
particle population that dominates emission spanning more than 20 orders of
magnitude of the electromagnetic spectrum (e.g., Longair et al. 2011).
Correctly explaining the essence of astrophysical activity phenomenon is to
understand how non-thermal high-energy particles are accelerated. The widely
accepted effective mechanisms for the acceleration of high-energy particles
are the diffusion shock wave (termed as the first-order Fermi process; Bell
1978) and the turbulent magnetic reconnection (Lazarian & Vishniac 1999,
hereafter LV99; Lazarian et al. 2015, 2020 for recent reviews), the latter of
which has been also considered to be a first-order Fermi acceleration process
(de Gouveia Dal Pino & Lazarian 2005; Kowal et al. 2012; Zhang et al. 2021).
At present, the second-order Fermi acceleration has been suggested for
acceleration of particles in a number of astrophysical settings, such as
black-hole X-ray binary (Zhang et al. 2018b), supernova remnant (Fan et al.
2010), solar physics (Yan et al. 2008; Petrosian & Bykov 2008), the
interstellar medium (Cho & Lazarian 2003) and diffuse intracluster medium in
galaxy clusters (Brunetti & Lazarian 2016). It is noticed that MHD turbulence
is an important agent of particle acceleration as pointed out by Alfvén (1942)
for stochastic particle acceleration.

The new paradigm of MHD turbulence, especially compressible properties
associated with Alfvén, slow and fast modes, has significantly promoted the
understanding of energetic particle-turbulence interactions, including
particle acceleration, scattering, diffusion and propagation processes (Yan &
Lazarian 2002, 2004; Chandran 2003; Cho & Lazarian 2006; Beresnyak & Li 2016;
Xu & Yan 2013, 2018; Lazarian & Xu 2021). For instance, stochastic particle
acceleration from slow mode (Chandran 2003) and fast one (Yan et al. 2008)
assuming efficient pitch-angle scattering was applied to elucidating solar
flares. Cho & Lazarian (2006) made use of the established scaling of slow and
fast modes to provide a systematic analytical research of particle
acceleration in strong and weak MHD turbulence. In the limits of both slow and
fast particle diffusion, they found that fast mode accelerates particles more
efficiently than slow mode, the efficiency of particle acceleration by slow
and fast modes depends on the spatial diffusion rate, and highly supersonic
turbulence is an efficient agent for the acceleration. Since Alfvén and slow
modes undergo an anisotropy cascade mainly in the direction perpendicular to
the underlying mean magnetic field, they are generally considered the
comparatively inefficiency in the acceleration of particles. This needs
further verification of numerical simulation.

Generally, test particle method is popular in studying the interaction between
cosmic ray particles and MHD turbulence. In this way, to trace particle
trajectories by integrating the Lorentz force, Xu & Yan (2013) pointed out
that cosmic ray diffusion coefficients are spatially correlated to the local
magnetic turbulence properties and confirmed that cosmic rays are
superdiffusive in MHD turbulence. Similarly, the recent study using the open
source PENCIL code demonstrated that the fast mode dominates the cosmic ray
propagation, whereas Alfvén and slow modes are much less efficient with
similar pitch angle scattering rates (Maiti et al. 2021). With the purpose of
studying the perpendicular and parallel diffusion of magnetic field, they both
ignored the effect of the induced electric field $\textbf{\em E}=-\textbf{\em
v}\times\textbf{\em B}$ arising from MHD turbulence. When investigating
particle acceleration in a turbulent reconnection layer or a purely turbulent
setting, kinetic equation including the induced electric field has been
numerically solved using the similar test particle method (Kowal et al. 2011,
2012; del Valle et al. 2016; Zhang et al. 2021).

Recently, a variety of observation-based new techniques has shed new light on
the understanding of the nature of MHD turbulence (e.g., Lazarian & Pogosyan
2012; Zhang et al. 2016, 2018, 2020). Especially with application of the
quadrupole ratio modulus for synchrotron polarization intensity, Wang et al.
(2020) successfully recovered the anisotropic properties of compressible MHD
turbulence. Statistics of the polarization intensity from Alfvén and slow
modes demonstrate the significant anisotropy, while the statistics of the
polarization intensity from fast mode show isotropic feature. This is in
agreement with earlier directly numerical results provided in Cho & Lazarian
(2002). These findings indeed show that synchrotron polarization radiation
signature is closely related to MHD turbulence properties. In almost all the
studies that we know currently, the specific acceleration properties of non-
thermal high-energy particles have still not been explored when applying
techniques to the measurement of the turbulence. The purpose of our current
work is to explore how high-energy particles are accelerated in compressible
MHD turbulence, and whether the acceleration of high-energy particles are
relevant to the properties of compressible MHD turbulence.

We organize the main structure of this paper as follows: theoretical
description of MHD turbulence and particle acceleration mechanism in Section
2, simulation methods in Section 3, simulation results in Section 4,
discussions in Section 5 and summary in Section 6.

##  2 Theoretical Description

###  2.1 MHD turbulence theory

It is widely accepted that Goldreich & Sridhar (1995, hereafter GS95)
initiated the development of modern MHD turbulence theory (see Beresnyak &
Lazarian 2019 for a recent review). In the case of strong turbulence, that is,
$M_{\rm A}=V_{\rm L}/V_{\rm A}\simeq 1$, where $V_{\rm L}$ is the injection
velocity at the scale $L_{\rm in}$ and $V_{\rm A}=B_{0}/\sqrt{4\pi\rho}$ is
the Alfvénic velocity with the mean magnetic field strength $B_{0}$ and the
density $\rho$, this work predicted the scale-dependent anisotropy
($l_{\|}\propto l_{\perp}^{2/3}$) of incompressible MHD turbulence through the
assumption of critical equilibrium, i.e., $v_{l}l_{\perp}^{-1}=V_{\rm
A}l_{\|}^{-1}$. Here, $v_{l}$ indicates the fluctuation velocity of turbulence
at the scale $l$, while $l_{\|}$ and $l_{\perp}$ denote parallel and
perpendicular scales of the eddies, respectively.

In the framework of a collection of anisotropic eddies for MHD turbulence,
GS95 was generalized to both $M_{\rm A}<1$ and $M_{\rm A}>1$ (LV99 and
Lazarian 2006). In the case of $M_{\rm A}<1$, i.e., the MHD turbulence being
driven with sub-Alfvénic velocities at injection scale $L_{\rm in}$, it was
found that a weak turbulence spans from $L_{\rm in}$ to the transition scale
$l_{\rm tr}=L_{\rm in}M_{\rm A}^{2}$, while the strong sub-Alfvénic turbulence
is in the range from the dissipation scale $l_{\rm dis}$ to the transition
scale $l_{\rm tr}$, where the turbulence velocity $v_{l}$ is equal to $V_{\rm
A}$. In this strong turbulence range, we have the relation

$l_{\|}\approx L_{\rm in}^{1/3}l_{\perp}^{2/3}M_{\rm A}^{-4/3}$ (1)

between the parallel scale $l_{\|}$ of the eddy extended along the magnetic
field and its transversal scale $l_{\bot}$. When setting $M_{A}=1$, we obtain
the original GS95 relation of $l_{\|}\propto l_{\perp}^{2/3}$.

The opposite case, $M_{\rm A}>1$, corresponds to super-Alfvénic turbulence,
i.e., $V_{\rm L}>V_{\rm A}$, where $V_{\rm L}$ is the turbulence injection
velocity. For a limiting case of ${M}_{{\rm{A}}}\gg 1$, since the weakly
turbulent magnetic field undergoes a marginal influence of MHD turbulence, the
turbulence at the scale close to $L_{\rm in}$ scale has an essentially
hydrodynamic Kolmogorov performance, i.e., $v_{l}=V_{\rm L}(l/L_{\rm
in})^{1/3}$. Interestingly, changes happen at the scale of $l_{\rm A}=L_{\rm
in}M_{\rm A}^{-3}$ to the hydrodynamic property of turbulence cascade (see
Section 2.2 in Lazarian 2006), where the turbulent velocity is equal to the
Alfvén velocity. In the inertial range from $l_{\rm A}$ to $l_{\rm dis}$, the
super-Alfvénic turbulence follows again the characteristics of the GS95
anisotropy (GS95 and Lazarian 2006).

Another important component of MHD turbulence theory is its compressibility
(Beresnyak 2019 for recent review; Beresnyak & Lazarian 2019 for textbook),
which was confirmed in numerical simulations (Lithwick & Goldreich 2001; Cho &
Lazarian 2002, 2003 ) using the Fourier (Cho & Lazarian 2002) or wavelet
(Kowal & Lazarian 2010) decomposition MHD turbulence into three modes, i.e.,
Alfvén, slow and fast modes. It was demonstrated that while density is
seriously modified by compressibility, the magnetic and velocity fluctuations
of Alfvén and slow modes are only slightly different from that of the
incompressible case (Cho & Lazarian 2002; Kowal & Lazarian 2010). Generally,
Alfvén modes are compatible with the GS95 anisotropic property ($l_{\|}\propto
l_{\perp}^{2/3}$), following Kolmogorov spectrum of $E\propto k^{-5/3}$, and
slow modes sheared by Alfvén modes are evolved passively, showing the GS95
anisotropic property, while fast modes have isotropic properties
($l_{\|}\propto l_{\perp}$), abiding by the scaling of $E\propto k^{-3/2}$
(Cho & Lazarian 2002). What is worth noting is that the properties of these
modes have been well confirmed by observational research, such as synchrotron
radiation statistics (Lazarian & Pogosyan 2012; Wang et al. 2020) and velocity
channel analysis of spectroscopic measurement (Kandel et al. 2016, 2017).

###  2.2 Particle acceleration mechanism

Particle acceleration mechanism can generally be classified as dynamic,
hydrodynamic and electromagnetic processes (Longair et al. 2011). In general,
magnetic fields themselves do no work and cannot be directly responsible for
particle acceleration, but time-varying magnetic fields lead to an inductive
electric field, $\textbf{\em E}=-\partial\textbf{\em B}/\partial t$, to
accelerate particles. The classic model of particle acceleration dates back to
Fermi (1949) in which particles can statistically gain energy through
collisions with interstellar clouds, resulting in a second-order acceleration,
$\Delta E/E\propto(V/c)^{2}\sim O(V^{2}/c^{2})$, where $V$ and $c$ are the
velocity of the cloud and the speed of light, respectively. By assuming a
characteristic escape time, $t_{\rm esc}$, he obtained a distribution of
particle energies, $N\propto E^{-m}$ with $m=1+(\alpha t_{\rm esc})^{-1}$,
where $\alpha$ and $t_{\rm esc}$ are model-dependent parameters. Thus, this
model does not predict a single power-law exponent, i.e., a change in the
power-law index. In the modern understanding of the second-order Fermi
acceleration (Longair et al. 2011), the particles gain energy by interacting
with various types of plasma wave or irregularities in the interstellar
magnetic field. As for this mechanism, it is generally thought that there are
few chances of a particle gaining significant energy due to the second-order
behavior of $O(V^{2}/c^{2})$.

To enhance acceleration efficiency, many researches on particle acceleration
have been motivated independently and applied to many diverse astrophysical
environments (Bell 1978; Blandford & Ostriker 1978; Drury et al. 1999). The
acceleration process was associated with acceleration in strong shock waves,
referred to as diffusive shock acceleration, in which the acceleration is
first-order, $\Delta E/E\propto(V/c)\sim O(V/c)$ in the shock velocity and
predicts a power-law spectrum with an index of 2. This model, as first-order
Fermi acceleration, is efficient for accelerating particles. However, this
model ignored detailed physics of the magnetic field and energy losses of the
particles. Considering the effect of magnetic fields, it is involved in a
complex non-linear process and causes a more of less universal power-law
energy spectrum (Berezhko & Völk 2007; Kang et al. 2006, 2009).

In analogy to classical diffusive shock acceleration, it was suggested that a
first-order Fermi process operates within the 3D current sheet of turbulent
reconnection (LV99, de Gouveia Dal Pino & Lazarian 2005; Drury 2012; Kowal et
al. 2011, 2012; Zhang et al. 2021), where particles trapped in the converging
magnetic fluxes of opposite polarity bounce back and forth in the large-scale
reconnection zone. Similarly, Drake et al. (2006) proposed that the particles
trapped in contracting 2D magnetic islands or plasmoids caused by tearing
reconnection also undergo a similar first-order Fermi process. In driving the
first-order Fermi process, it was demonstrated to be equivalent between plasma
kinetic scale reconnection and MHD turbulent reconnection (Kowal et al. 2011).

##  3 Simulation Methods

###  3.1 Simulation of MHD Turbulence

The equations describing MHD turbulence are given as follows

${\partial\rho}/{\partial t}+\nabla\cdot(\rho{\textbf{\em v}})=0,$ (2)
$\rho[\partial{\textbf{\em v}}/{\partial t}+({\textbf{\em
v}}\cdot\nabla){\textbf{\em v}}]+\nabla p-{\textbf{\em J}}\times{\textbf{\em
B}}/4\pi={\textbf{\em f}},$ (3) ${\partial{\textbf{\em B}}}/{\partial
t}-\nabla\times({\textbf{\em v}}\times{\textbf{\em B}})=0,$ (4)
$\nabla\cdot{\textbf{\em B}}=0,$ (5)

where $p=c_{\rm s}^{2}\rho$ is the gas pressure, $c_{\rm s}$ the sonic speed,
$t$ the evolution time of the fluid, ${\textbf{\em
J}}=\nabla\times{\textbf{\em B}}$ the current density, and f a random driving
force. In our simulation, we use a third-order-accurate hybrid, essentially
non-oscillatory code to solve the above control equations in a periodic box at
the length of $2\pi$. In wavenumber space, we drive the turbulence by a random
solenoidal driving force on large scale (small wavenumber $k\simeq 2.5$ ), and
set a non-zero mean magnetic field along the $x$-axis direction. The data
cubes with numerical resolution of $512^{3}$ we obtained, corresponding to
four different turbulence regimes, are listed in Table 1, where each group of
data is mainly depicted by the plasma parameter, the Alfvénic and sonic Mach
numbers.

Cho & Lazarian (2002) provided a method to decompose MHD modes into Alfvén,
slow and fast modes. The main procedure of this decomposition method is
characterized by the following displacement vectors

$\hat{\zeta}_{\rm
f}\varpropto(1+\frac{\beta}{2}+\sqrt{D})(k_{\perp}\hat{\textbf{\em
k}}_{\perp})+(-1+\frac{\beta}{2}+\sqrt{D})(k_{\parallel}\hat{\textbf{\em
k}}_{\parallel}),$ (6) $\hat{\zeta}_{\rm
s}\varpropto(1+\frac{\beta}{2}-\sqrt{D})(k_{\perp}\hat{\textbf{\em
k}}_{\perp})+(-1+\frac{\beta}{2}-\sqrt{D})(k_{\parallel}\hat{\textbf{\em
k}}_{\parallel}),$ (7) $\hat{\zeta}_{\rm A}\varpropto-\hat{\textbf{\em
k}}_{\perp}\times\hat{\textbf{\em k}}_{\parallel},$ (8)

where $D=(1+\frac{\beta}{2})^{2}-2\beta\cos^{2}\theta$ and
$\cos\theta=\hat{\textbf{\em k}}_{\parallel}\cdot\hat{\textbf{\em B}_{0}}$. By
projecting magnetic field and velocity into $\hat{\zeta}_{\rm f}$,
$\hat{\zeta}_{\rm s}$ and $\hat{\zeta}_{\rm A}$, we can obtain their
components for fast, slow and Alfvén modes. It is noticed that when each MHD
mode is separated by the Fourier transformation, this technique is closely
related to the mean magnetic field in the global frame. In the view of
depending on the wandering of large scale magnetic field and density
inhomogeneities, this technique is only applicable for the case of the sub-
Alfvénic turbulence with a strong mean magnetic field and small perturbation.
Afterward, Kowal & Lazarian (2010) proposed an improved procedure to extend
the earlier decomposition (Cho & Lazarian 2002) by introducing a discrete
wavelet transformation before the Fourier separation. This improved method
relies on the local magnetic field instead of the mean magnetic field.
Therefore, it has more significant advantages in decomposing turbulent
fluctuations of high amplitude in the case of the sup-Alfvénic turbulence.

Models | $B_{0}$ | $M_{\rm A}$ | $M_{\rm s}$ | $\beta$ | Description
---|---|---|---|---|---
run1 | 1.0 | 0.65 | 0.48 | 3.67 | Sub-Alfvénic & subsonic
run2 | 1.0 | 0.50 | 9.92 | 0.01 | Sub-Alfvénic & supersonic
run3 | 0.1 | 1.72 | 0.45 | 29.12 | super-Alfvénic & subsonic
run3 | 0.1 | 1.76 | 7.02 | 0.13 | super-Alfvénic & supersonic
Table 1: Data cubes with numerical resolution of $512^{3}$ for different
turbulence regimes. The mean magnetic field $B_{0}$ in units of code is set
along the $x$-axis. $M_{\rm A}$ is the Alfvénic Mach number, $M_{\rm s}$ the
sonic Mach number, and $\beta=2M_{\rm A}^{2}/M_{\rm s}^{2}$ the plasma
parameter.

###  3.2 Method of Test Particle

In the MHD turbulence, the motion of a charged particle satisfies the
following kinetic equation

$\frac{d}{dt}\left(\gamma m\textbf{\em u}\right)=q\left(\textbf{\em
E}+\textbf{\em u}\times\textbf{\em B}\right),$ (9)

where $\gamma\equiv 1/\sqrt{1-u^{2}/c^{2}}$ is the Lorentz factor of
relativistic particle, u the particle velocity, $m$ the particle mass and $q$
the particle electric charge. In this work, we consider the acceleration
process resulting from magnetic filed B and its induced electric field
$\textbf{\em E}=-\textbf{\em v}\times\textbf{\em B}$, igoring resistivity
effects of electric field. Consequently, Equation (9) can be rewritten as

$\frac{d}{dt}\left(\gamma m\textbf{\em u}\right)=q\left[\left(\textbf{\em
u}-\textbf{\em v}\right)\times\textbf{\em B}\right].$ (10)

We integrate this equation to trace particle’s trajectories using the 8th
order embedded Dormand-Prince method (Hairer et al. 2008) with adaptive time
step. Using cubic interpolation with the discontinuity detector, we can obtain
the local values of the plasma velocity v and magnetic field B at each step of
the integration. In practice, test particles, in the form of a thermal
distribution with a temperature of $T=10^{6}$ K, are injected into a frozen-
in-time 3D MHD domain at a given snapshot. We assume that the speed of light
is approximately equal to 20 times greater than the injection velocity and the
magnetic field strength is $10^{-3}\ \rm G$.

##  4 Simulation Results

On the basis of data cubes from simulations of MHD turbulence (see Table 1)
and plasma mode decomposition procedure described above, we in this section
provide numerical results for Alfvén, slow and fast modes using test particle
methods. These results include the evolution of particle kinetic energy over
time, the mean acceleration time of test particles, the power spectrum of
maximum acceleration rate and spectral energy distributions, ranging from four
turbulence regimes: sub-Alfvénic and subsonic, sub-Alfvénic and supersonic,
super-Alfvénic and subsonic, and super-Alfvénic and supersonic.

Figure 1: Kinetic energy distributions of 10, 000 protons for Alfvén, slow and
fast modes, arising from sub-Alfvénic & subsonic (upper panels), and sub-
Alfvénic & supersonic (lower panels) turbulence regimes. In each subpanel, the
color bar shows the change in the number of protons.

Figure 2: Kinetic energy distributions of 10, 000 protons for Alfvén, slow and
fast modes, arising from super-Alfvénic & subsonic (upper panels), and super-
Alfvénic & supersonic (lower panels) turbulence regimes. In each subpanel, the
color bar shows the change in the number of protons.

Figures 1 and 2 show kinetic energy distributions of 10, 000 protons injected
initially in a thermal distribution over the evolution time for Alfvén, slow
and fast modes, in which various turbulence regimes appearing in astrophysics
are explored. Compared to contour maps for three modes, we can see the
differences of particle distribution between them, as shown in the color bar
in four turbulence regimes. The injected particles first undergo a less
efficient acceleration process in the range of less than $t\simeq 5$ hours,
and then accelerate significantly with a relation of $E\propto t^{3.6}$. When
evolving to the late stage approximately more than $t\simeq 100$ hours, the
distribution of particle energy begins to change with the relation of
$E\propto t^{0.8}$. Despite the noticeable change of the number of accelerated
particle’s distribution, it statistically follows a similar distribution
relationship under various turbulence conditions from $E\propto t^{3.6}$ to
$E\propto t^{0.8}$.

The less efficient particle acceleration at the early stage should result from
the fact that the gyration radius of the test particles is smaller than the
eddy size, resulting in insufficient energy exchange from turbulence to
particles. Efficient particle acceleration stage, i.e., $E\propto t^{3.6}$,
corresponds to full interactions between particles and plasma waves in the
inertial range of turbulence. The transition to $E\propto t^{0.8}$ can be
understood as a weak turbulence interaction in the spatial region near the
injection scale $L_{\rm in}\simeq 200$ code units. Specifically, this
corresponds to the scale range of $L_{\rm in}M_{\rm A}^{2}<l<L_{\rm in}$ as is
in sub-Alfvénic turbulence, and $L_{\rm in}M_{\rm A}^{-3}<l<L_{\rm in}$ in
super-Alfvénic turbulence. In general, this result shows that particle
acceleration has universal properties under various turbulent conditions. This
implies that the mechanism which causes particles to accelerate is a second-
order Fermi acceleration process, comparing with a more effective first-order
Fermi acceleration in the turbulent reconnection layer with the relation from
$E\propto t^{4.5}$ to $E\propto t^{2.5}$ (Zhang et al. 2021). Due to the
setting of periodic boundary conditions, particles with a gyration radius
larger than the box scale will reenter and cause a continuous increase in
their kinetic energies, which is similar to a 1.5-order Fermi acceleration in
a large-scale turbulence environment around the turbulent reconnection layer
(Brunetti & Lazarian 2016; Zhang et al. 2021).

Figure 3: The mean acceleration time as a function of proton kinetic energy
for Aflvén, slow and fast modes in different turbulence regimes: sub-Alfvénic
and subsonic (left upper panel), sub-Alfvénic and supersonic (right upper),
super-Alfvénic and subsonic (left lower), and super-Alfvénic and supersonic
(right lower).

In order to further study the acceleration of particles, we adopt the
definition of $t_{\rm acc}=\sum_{i=1}^{N_{\rm E}}t_{i}/N_{\rm E}$ provided in
del Valle et al. (2016), where $t_{i}$ is the acceleration time of $i$-th
particle and $N_{\rm E}$ is the number of particles in the energy interval
from $E$ to $E+\bigtriangleup E$. The mean acceleration time as a function of
kinetic energy is plotted in Figure 3, from which we can see more detailed
differences between three modes in various turbulence regimes. As for sub-
Alfvénic, subsonic (left upper) and supersonic (right upper) regimes, the
acceleration of slow mode takes more time than that of Alfvén and fast modes
in low energy range, both of which accelerate similarly. The mean acceleration
time for slow mode decreases with increasing kinetic energy in the very low
energy range, but then reverses and increases with energy in the high energy
range, representing the transition of power-law relations from $t_{\rm
acc}\propto E^{0.9}$ to $t_{\rm acc}\propto E^{0.2}$ then to $t_{\rm
acc}\propto E^{0.8}$. Except for the low energy range, it maintains as similar
scaling for Alfvén and fast modes as that for slow mode. The lower panels
corresponding to super-Alfvénic, subsonic (left lower) and supersonic (right
lower) regimes present that fast mode takes less time than Alfvén and slow
modes. In particular, there is no remarkable negative correlation between
kinetic energy and mean acceleration time in the low energy range. In fact,
the power law relationship we find in Figure 3 has a one-to-one correspondence
to those in Figures 1 and 2, that is, $t_{\rm acc}\propto E^{0.9}$ matches
with a flat particle distribution in the range of less than about 5 hours,
$t_{\rm acc}\propto E^{0.2}$ (plotted in dotted line) with a significantly
increasing process of kinetic energy ($E\propto t^{3.6}$) and $t_{\rm
acc}\propto E^{0.8}$ with the moderately increasing process at the late stage
of evolution ($E\propto t^{0.8}$).

Figure 4: Power spectra of the maximum acceleration rate (left column) and the
magnetic field (right column) for Aflvén, slow and fast modes in different
turbulence regimes: sub-Alfvénic and subsonic (upper panel), sub-Alfvénic and
supersonic (upper middle), super-Alfvénic and subsonic (lower middle), and
super-Alfvénic and supersonic (lower).

The influence of three modes on the maximum acceleration rate is explored in
Figure 4 by analyzing power spectrum, compared to underlying turbulent
magnetic fields. The left column of this figure shows power spectra of the
maximum acceleration rate of particles, and the right column corresponds to
power spectra of the underlying magnetic fields in four turbulence regimes. As
shown, the maximum acceleration rates for fast mode present well the scaling
of $k^{-3/2}$ in four turbulence regimes explored, while Alfvén and slow modes
indicate approximately the scaling of $k^{-5/3}$. Compared to power spectra of
magnetic field in the same turbulence regime, we see that there is a more
limited inertial range by the maximum acceleration rate, the spectrum of which
first becomes steeper at small scales (large wavenumber), then flattens at
close dissipation scales. In short, the maximum acceleration rates for three
modes reveal the inertial range, the scaling of spectrum and energy exchange
dependence from underlying compressible MHD turbulence cascade. Thus, the
acceleration efficiency of particles relies on the properties of each mode.

Figure 5:  Spectral energy distributions of the accelerated protons for Alfvén
(solid lines), slow (dotted lines) and fast (dashed lines) modes, arising from
different turbulence regimes: sub-Alfvénic and subsonic (upper panel), sub-
Alfvénic and supersonic (upper middle), super-Alfvénic and subsonic (lower
middle), and super-Alfvénic and supersonic (lower), at different evolution
times in units of hour.

Furthermore, spectral energy distributions of accelerated particles for three
modes are plotted in Figure 5 at several snapshots $t=0$, 0.5, 5.0, 10, 100,
500 and 1000 hours, in which the thick solid lines indicate spectral
distributions for Alfvén mode, the thick dotted lines for slow mode, and the
thick dashed lines for fast mode. As time evolves, the particle spectra of the
thermal distribution broaden and move to high energy wavebands, and form a
changing power-law distribution, e.g., with of $N(E)\propto E^{-1.5}$ at the
time $t\simeq 5$ hours. In the extremely high energy range, the non-thermal
spectra present a thermal-like spectral distribution characteristics. In the
case of sub-Alfvénic turbulence (upper panel for subsonic, and upper middle
panel for supersonic), Alfvén and fast modes dominate the acceleration of
particles in the range of less than $t\simeq 10$ hours, and then it turns to
slow mode dominance. In the case of super-Alfvénic and subsonic turbulence
(lower middle panel), we observe the transition of the plasma modes that
dominates the acceleration process, that is, at the early stages of evolution
($t\lesssim 1$ hour), the dominant acceleration process is Alfvén and slow
modes, then fast mode ($1\lesssim t\lesssim 50$ hours), and finally Alfvén and
slow modes ($t\gtrsim 50$ hours). In the case of super-Alfvénic and supersonic
turbulence (lower panel), we find that fast mode dominates the acceleration of
particles throughout the entire process.

In brief, the main features regarding the particle acceleration of three modes
are that (1) fast mode plays a major role in accelerating particles in various
turbulence regimes we explored; (2) slow mode obviously accelerates particles
in the very high energy part in the case of sub-Alfvén turbulence; (3) Alfvén
mode mainly contributes in the early stage of evolution.

##  5 Discussions

In the current work we have considered proton as probe particle and injected
the number of 10, 000 protons into a frozen-in-time 3D MHD domain that has
reached an evolutionary steady state. Due to the fact that the macroscopic MHD
dynamical time is much longer than the evolution time of the test particles,
it is a reasonable treatment for considering a single snapshot to accelerate
the particles. Similar to most PIC acceleration simulations in the
collisionless plasma settings, electron-positron pairs can also be regarded as
probe particles in the collisional MHD turbulence environment. However,
simulating electron-positron pairs evolution in the case of MHD turbulence
will be extremely time-consuming because of their more small mean free path,
and explored in future work

We studied the effect of mean magnetic fields on the acceleration process
using the structure function analysis of the maximum acceleration rate. We
found that the maximum acceleration rate shows an approximate $SF\propto
R^{1.4}$ for Alfvén, slow and fast modes along three coordinate axes (not
shown in the paper), from the scaling of which we cannot almost distinguish
between them. However, we observe that significant changes happen to the
amplitude of structure function, which is associated with the possible maximum
energy that particle can be reached. In addition, we also studied the
relationship between the particle’s maximum acceleration rate and the local
magnetic field, but no spatial anisotropy of the maximum acceleration rate was
found for three modes. We feel that the lack of anisotropic features may be
attributed to the limitation of numerical resolution. As pointed out in Lee et
al. (2016), the calculation of the structure function, in particular, the
recovering of power law scaling, requires a higher numerical resolution; it
needs further high resolution numerical simulation in the future.

We would like to say that the mechanism that induces the particles to
accelerate is the same, that is, it is a second-order Fermi process for three
modes. It is very interesting that although a shock acceleration process is
expected for supersonic turbulence, we did not find significant differences
compared with subsonic case (see Figure 1). We think that the particle
interacting with a single shock leads to the first-order Fermi acceleration,
but the first-order effect is canceled out after particles encountering many
shock compressions or expansions, and only the second-order Fermi process
results in an increase of particle energy.

From the perspective of numerical practice, the execution of this work
consists of three main steps: (1) MHD turbulence simulations to generate data
cubes including the information of both magnetic fields and velocities; (2)
implementation of wavelet transformation to decompose MHD modes into Alfvén,
slow and fast modes, and (3) test particle simulations to study the properties
of particle acceleration. The cost of simulation time restricts us to use a
numerical resolution of 512 along each dimension of coordinates. Especially in
the process of test particle simulation, the time-consuming is strongly
dependent on the number of injected particles and the terminal time of
simulation evolution. The 10,000 test particles we injected have ensured the
reliability of the numerical results, which is an order of magnitude higher
than the earlier simulation of 1,000 particles (Xu & Yan 2013). As shown in
Figure 4, the inertial range of the MHD turbulence cascade revealed from the
particle’s maximum acceleration rate is narrower than that displayed by the
underlying magnetic field (or velocity). Nevertheless, the execution of higher
numerical simulations is necessary to confirm our research results.

This work did not include the effect of radiative losses, such as synchrotron
emission and inverse Compton scattering, on the particle acceleration.
Although the radiative loss is almost negligible for the proton considered as
a probe particle in this paper, the loss effect should be included anyway in
the some astrophysical environments dominated by hadrons. The radiative loss
taken into account, the spectral energy distributions predicted by the current
work might vary in the very high-energy regime. Related testing and
verification are welcome.

##  6 Summary

Considering four turbulence regimes that could appear in the realistic
astrophysical settings, we have performed the study of particle acceleration
in compressible MHD turbulence using the test particle method. The main
results that we found are summarized as follows:

  1. 1.

The acceleration of particles caused by Alfvén, slow and fast modes is a
second-order Fermi process in four turbulence regimes explored in this paper.
The transition of particle distributions, from $N\propto t^{3.6}$ to $N\propto
t^{0.8}$ in the acceleration process, is related to the injection scale of
turbulence, or, to put it another way, it is related to the strong or weak
turbulence regimes.

  2. 2.

The power spectrum of the particle’s maximum acceleration efficiency directly
reflects the inertial range of MHD turbulence cascade, and recovers the
scaling relations of Alfvén, slow and fast modes. In contrast to the inertial
scale range revealed by the underlying magnetic fields, the power spectrum of
maximum acceleration rate provides a narrower inertial range. Our studies
indeed demonstrated that fast mode shows an ideal scaling of $\propto
k^{-3/2}$ as expected, and Alfvén and slow modes can also recover
approximately the scaling of $\propto k^{-5/3}$.

  3. 3.

Spectral energy distributions of the accelerated particles for three modes
first present a power law form in the low energy range, and then transfers
into thermal-like spectrum in the high energy range.

  4. 4.

Fast mode dominates the acceleration of particles during the most evolution
time, especially in the subsonic turbulence. In the case of sub-Alfvén
turbulence, slow mode dominates the particle acceleration in the very high
energy range. Different from the earlier theoretical predictions, the
contribution of Alfvén mode cannot be ignored, which even plays a dominant
role in the particle acceleration at the very early stage of evolution.

We thank the anonymous referee for valuable suggestions that significantly
improved this paper. We would like to thank Grzegorz Kowal for the support of
numerical techniques regarding wavelet decomposition and test particle method,
Jungyeon Cho for generating data cube of MHD turbulence, and Alex Lazarian for
valuable discussions about the particle acceleration in the compressible MHD
turbulence. J.F.Z. thanks the supports from the National Natural Science
Foundation of China (grant Nos. 11973035 and 11703020) and the Hunan Province
Innovation Platform and Talent Plan–HuXiang Youth Talent Project (No.
2020RC3045). F.Y.X. acknowledges the support from the Joint Research Funds in
Astronomy U2031114 under cooperative agreement between the National Natural
Science Foundation of China and the Chinese Academy of Sciences.

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# Curvature properties of Riemannian manifolds with skew-circulant structures

Iva Dokuzova

###### Abstract.

We consider a $4$-dimensional Riemannian manifold $M$ endowed with a right
skew-circulant tensor structure $S$, which is an isometry with respect to the
metric $g$ and the fourth power of $S$ is minus identity. We determine a class
of manifolds $(M,g,S)$, whose curvature tensors are invariant under $S$. For
such manifolds we obtain properties of the Ricci tensor. Also we get
expressions of the sectional curvatures of some special $2$-planes in a
tangent space of $(M,g,S)$.

Mathematics Subject Classification (2020): 53B20, 53C25, 53C15, 53C55.

Keywords: Riemannian manifold; sectional curvature; skew-circulant matrix;
Ricci curvature.

##  1\. Introduction

Almost Hermitian manifolds were classified with respect to the covariant
derivative of the almost complex structure $J$ by A. Gray and L. Hervella
([8]). The Hermitian manifolds form a class of manifolds with an integrable
structure $J$. Part of these manifolds are the well-known and extensively
explored Kähler manifolds. The locally conformal Kähler manifolds are
Hermitian manifolds whose metric is conformal to a Kähler metric in a
neighborhood at each point. These manifolds are studied by many geometers (for
example [1, 10, 11, 12, 14, 19]).

In [7], there are introduced three classes of almost Hermitian manifolds,
determined by Gray’s curvature identities, and it is proved that every Kähler
manifold satisfies these identities. Many investigations of the geometry of
the almost Hermitian manifolds are devoted to the study of the curvature
tensors, the Ricci tensors, the scalar curvatures and the sectional curvatures
of some characteristic 2-planes of the tangent spaces of the manifolds. We
will mention some papers that consider classes of manifolds, whose curvature
tensors are invariant under the almost complex structure $J$ ([4, 6, 9, 15,
16, 18, 20]).

The present paper is a continuation of our research made in [5] and [17]. We
study a 4-dimensional Riemannian manifold $M$ with a metric $g$ and equipped
with a tensor $S$ of type $(1,1)$, compatible with $g$. The tensor structure
$S$ satisfies $S^{4}=-\textrm{id}$ and its local components form a right skew-
circulant matrix. We note that such a manifold $(M,g,S)$ is associated with a
locally conformal Kähler manifold $(M,g,J=S^{2})$ ([17]). Our purpose is to
find geometrical properties of manifolds $(M,g,S)$ that satisfy analogues of
Gray’s curvature identities. Thus, we study a class of manifolds whose
curvature tensors are invariant under $S$.

The paper is organized as follows. In Section 2, we give some necessary facts
about a 4-dimensional Riemannian manifold $(M,g,S)$ obtained in [5]. Also, we
determine a class of manifolds $(M,g,S)$ whose curvature tensors are invariant
under $S$. The following results in Section 3 and Section 4 refer to such
manifolds. In Section 3, we express the Ricci tensor of $(M,g,S)$ by the
metric $g$ and by the additional structure $S$. We establish that $(M,g,S)$ is
an almost Einstein manifold and get a condition under which it is an Einstein
manifold. We find the Ricci curvatures in the direction of the basis vectors
$x,Sx,S^{2}x,S^{3}x$ in a tangent space $T_{p}M$. In Section 4, we obtain
relations between the sectional curvatures of the $2$-planes determined by the
$S$-basis $\\{S^{3}x,S^{2}x,Sx,x\\}$. Also, we get expressions of these
sectional curvatures by the angles between vectors.

##  2\. Preliminaries

Let $M$ be a $4$-dimensional differentiable manifold with a metric $g$. Let
$S$ be a tensor of type $(1,1)$ on $M$, with local coordinates given by the
right skew-circulant matrix

(1) $(S_{i}^{j})=\begin{pmatrix}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\
\end{pmatrix}.$

Then

(2) $S^{4}=-\textrm{id}.$

We suppose that the structure $S$ of the manifold $M$ acts as an isometry with
respect to $g$, i.e.,

(3) $g(Sx,Sy)=g(x,y).$

Anywhere in this work $x,y,z,u,w$ will stand for arbitrary elements of the
algebra of the smooth vector fields on $M$ or vectors in the tangent space
$T_{p}M$ at a point $p\in M$. The Einstein summation convention is used, the
range of the summation indices being always $\\{1,2,3,4\\}$.

The equalities (1) and (3) imply the following right skew-circulant matrix of
components of $g$:

(4) $(g_{ij})=\begin{pmatrix}A&B&0&-B\\\ B&A&B&0\\\ 0&B&A&B\\\ -B&0&B&A\\\
\end{pmatrix}.$

In (4), functions $A$ and $B$ of four variables $x^{1}$, $x^{2}$, $x^{3}$ and
$x^{4}$ are smooth on $M$. It is supposed that $A>\sqrt{2}B>0$ in order $g$ to
be positive definite. The manifold $(M,g,S)$ is introduced in [5].

On $(M,g,S)$ we consider an associated metric $\tilde{g}$ with $g$, determined
by

(5) $\tilde{g}(x,y)=g(x,Sy)+g(Sx,y).$

In [5], it is verified that $\tilde{g}$ is a necessary indefinite. The matrix
of its components is

(6) $(\tilde{g}_{ij})=\begin{pmatrix}2B&A&0&-A\\\ A&2B&A&0\\\ 0&A&2B&A\\\
-A&0&A&2B\\\ \end{pmatrix}.$

Let $\nabla$ be the Riemannian connection of $g$. The curvature tensor $R$ of
$\nabla$ is determined by
$R(x,y)z=\nabla_{x}\nabla_{y}z-\nabla_{y}\nabla_{x}z-\nabla_{[x,y]}z.$ The
tensor of type $(0,4)$ associated with $R$ is defined by
$R(x,y,z,u)=g(R(x,y)z,u).$

In [5], it is determined a class of manifolds $(M,g,S)$ which satisfy the
identity

(7) $R(x,y,Sz,Su)=R(x,y,z,u).$

The above property defines a more general class of manifolds than this one
with $\nabla S=0$.

On the other hand, equality (7) and the symmetries of $R$ imply

(8) $R(Sx,Sy,Sz,Su)=R(x,y,z,u).$

Therefore, the class of manifolds $(M,g,S)$ with the condition (8) is more
general than the class $(M,g,S)$ with the condition (7).

######  Remark 2.1.

Every manifold $(M,g,S)$ is associated with an almost Hermitian manifold
$(M,g,J)$, where $J=S^{2}$. Moreover, the manifold $(M,g,J)$ is locally
conformal Kähler ([17]). For almost Hermitian manifolds, Grey’s classification
is valid ([7]). It is made with respect to the curvature tensor $R$. Due to
this classification the class $\mathcal{L}_{3}$ consists of manifolds which
satisfy the identity $R(Jx,Jy,Jz,Ju)=R(x,y,z,u)$.

It is easy to see that if $(M,g,S)$ has the property (8), then the associated
manifold $(M,g,J)$ belongs to $\mathcal{L}_{3}$.

##  3\. Almost Einstein manifolds

The Ricci tensor $\rho$ with respect to $g$ is given by the well-known formula
$\rho(y,z)=g^{ij}R(e_{i},y,z,e_{j}).$ The scalar curvature $\tau$ with respect
to $g$ and its associated quantity $\tau^{*}$ are determined by
$\tau=g^{ij}\rho(e_{i},e_{j}),\ \tau^{*}=\tilde{g}^{ij}\rho(e_{i},e_{j}).$
Here $\\{e_{i}\\}$ is a local basis of $T_{p}M$, $g^{ij}$ and $\tilde{g}^{ij}$
are the components of the inverse matrices of $g$ and $\tilde{g}$,
respectively.

By $R_{ijkh}$ and $\rho_{ij}$ we will denote the components of the curvature
tensor $R$ and the components of the Ricci tensor $\rho$ with respect to
$\\{e_{i}\\}$. Hence, we establish the following propositions.

######  Proposition 3.1.

The property (8) of the curvature tensor $R$ of the manifold $(M,g,S)$ is
equivalent to the conditions

$\displaystyle R_{1313}=$ $\displaystyle R_{2424},\
R_{1212}=R_{1414}=R_{2323}=R_{3434},$ (9) $\displaystyle R_{1213}=$
$\displaystyle R_{2414}=R_{2423}=R_{2313}=R_{1334},\
R_{1324}=2R_{1234}=2R_{1423},$ $\displaystyle R_{1223}=$ $\displaystyle
R_{1214}=R_{1434}=R_{2334},\ R_{1224}=R_{1413}=R_{2434}.$

###### Proof.

The local form of (8) is

(10) $R_{abcd}S_{i}^{a}S_{j}^{b}S_{k}^{c}S_{h}^{d}=R_{ijkh}.$

Then, using (1) and applying the Bianchi first identity to the components of
$R$, we obtain (3.1).

Vice versa, from (1) and (3.1) it follows (10), so (8) holds true. ∎

######  Proposition 3.2.

If a manifold $(M,g,S)$ has the property (8), then the components of the Ricci
tensor $\rho$ satisfy

(11)
$\rho_{11}=\rho_{22}=\rho_{33}=\rho_{44},\quad\rho_{12}=\rho_{23}=\rho_{34}=-\rho_{14},\quad\rho_{13}=\rho_{24}=0.$

###### Proof.

Due to Proposition 3.1, the components of $R$ satisfy (3.1). For brevity, we
denote

(12) $\begin{split}R_{1}=R_{1313},\ R_{2}=R_{1212},\ R_{3}=R_{1223},\
R_{4}=R_{1234},\ R_{5}=R_{1213},\ R_{6}=R_{1224}.\end{split}$

The inverse matrix of $(g_{ij})$ is

(13) $(g^{ij})=\frac{1}{A^{2}-2B^{2}}\begin{pmatrix}A&-B&0&B\\\ -B&A&-B&0\\\
0&-B&A&-B\\\ B&0&-B&A\\\ \end{pmatrix}.$

Now, having in mind (3.1), (12) and (13), we calculate the components of the
Ricci tensor $\rho$ as follows:

$\begin{split}\rho_{11}&=\rho_{22}=\rho_{33}=\rho_{44}=\frac{1}{A^{2}-2B^{2}}\big{(}2B(R_{5}+R_{6})-A(2R_{2}+R_{1})\big{)},\\\
\rho_{12}&=\rho_{23}=\rho_{34}=-\rho_{14}=\frac{1}{A^{2}-2B^{2}}\big{(}B(2R_{3}-R_{2}+3R_{4})-A(R_{5}+R_{6})\big{)},\\\
\rho_{13}&=\rho_{24}=0.\end{split}$

So the equalities (11) are valid. ∎

A Riemannian manifold is said to be Einstein if its Ricci tensor $\rho$ is a
constant multiple of the metric tensor $g$, i.e.

(14) $\rho(x,y)=\alpha g(x,y).$

In [5], a Riemannian manifold $(M,g,S)$ is called almost Einstein if its Ricci
tensor $\rho$, the metrics $g$ and $\tilde{g}$ satisfy

(15) $\rho(x,y)=\alpha g(x,y)+\beta\tilde{g}(x,y),$

where $\alpha$ and $\beta$ are smooth functions on $M$.

######  Theorem 3.3.

The manifold $(M,g,S)$ with property (8) is almost Einstein.

###### Proof.

The inverse matrix of $(\tilde{g}_{ij})$ is

(16) $(\tilde{g}^{ij})=\frac{1}{2(A^{2}-2B^{2})}\begin{pmatrix}-2B&A&0&-A\\\
A&-2B&A&0\\\ 0&A&-2B&A\\\ -A&0&A&-2B\\\ \end{pmatrix}.$

According to Proposition 3.2, for $(M,g,S)$ the equalities (11) are valid.
Consequently, using (11), (13) and (16) we calculate the values of the scalar
curvature $\tau$ and the associated quantity $\tau^{*}$ as follows:

$\tau=\frac{4}{A^{2}-2B^{2}}\big{(}A\rho_{11}-2B\rho_{12}\big{)},\quad\tau^{*}=\frac{4}{A^{2}-2B^{2}}\big{(}A\rho_{12}-B\rho_{11}\big{)}.$

Immediately from (11) and the latter equalities we have

(17)
$\rho_{11}=\frac{\tau}{4}A+\frac{2\tau^{*}}{4}B,\quad\rho_{12}=\frac{\tau}{4}B+\frac{\tau^{*}}{4}A,\quad\rho_{13}=\frac{\tau}{4}0+\frac{\tau^{*}}{4}0,$

and bearing in mind (4) and (6) we get

$\rho_{11}=\frac{\tau}{4}g_{11}+\frac{\tau^{*}}{4}\tilde{g}_{11},\quad\rho_{12}=\frac{\tau}{4}g_{12}+\frac{\tau^{*}}{4}\tilde{g}_{12},\quad\rho_{13}=\frac{\tau}{4}g_{13}+\frac{\tau^{*}}{4}\tilde{g}_{13}.$

Then, taking into account (4), (6), (11) and (17), we obtain

$\rho_{ij}=\frac{\tau}{4}g_{ij}+\frac{\tau^{*}}{4}\tilde{g}_{ij},$

i.e.

(18) $\rho(x,y)=\frac{\tau}{4}g(x,y)+\frac{\tau^{*}}{4}\tilde{g}(x,y).$

Therefore, comparing (18) with (15), we state that $(M,g,S)$ is an almost
Einstein manifold. ∎

Let $(M,g,S)$ satisfy the conditions of Theorem 3.3. If we suppose that
$(M,g,S)$ is an Einstein manifold, then its Ricci tensor $\rho$ has the form
(14). Hence (18) yields the following

######  Corollary 3.4.

If the manifold $(M,g,S)$ with property (8) is Einstein then

$\tau^{*}=0.$

###  3.1. Ricci curvatures

Now, we recall that the Ricci curvature in the direction of a nonzero vector
$x$ is the value

(19) $r(x)=\frac{\rho(x,x)}{g(x,x)}.$

Let $x$ be an arbitrary nonzero vector in a tangent space $T_{p}M$ of
$(M,g,S)$. The vectors $x$, $Sx$, $S^{2}x$ and $S^{3}x$ determine six angles
which belong to $(0,\pi)$. For these angles, the next statements are verified
in [5]. Namely, if $\varphi=\angle(x,Sx)$ then

(20)
$\begin{split}\angle(x,Sx)=&\angle(Sx,S^{2}x)=\angle(S^{2}x,S^{3}x)=\varphi,\quad\angle(x,S^{3}x)=\pi-\varphi,\\\
\angle(x,S^{2}x)=&\angle(Sx,S^{3}x)=\frac{\pi}{2},\end{split}$ (21)
$\frac{\pi}{4}<\varphi<\frac{3\pi}{4}.$

A basis of type $\\{S^{3}x,S^{2}x,Sx,x\\}$ of $T_{p}M$ is called an $S$-basis.
In this case we say that the vector $x$ induces an $S$-basis of $T_{p}M$. The
existence of an orthogonal $S$-basis of $T_{p}M$ is based on (20) and (21).

######  Theorem 3.5.

Let $(M,g,S)$ be a manifold with property (8). If a vector $x$ induces an
$S$-basis, then the Ricci curvatures in the direction of the basis vectors are

(22)
$r(x)=r(Sx)=r(S^{2}x)=r(S^{3}x)=\frac{\tau}{4}+\frac{\tau^{*}}{2}\cos\varphi,$

where $\angle(x,Sx)=\varphi$.

###### Proof.

In the course of the proof of Theorem 3.3, we find that $\rho$ is given by
(18). Therefore, using (3) and (5), we obtain

(23)
$\begin{split}\rho(x,x)=\rho(Sx,Sx)=\rho(S^{2}x,S^{2}x)&=\rho(S^{3}x,S^{3}x)=\frac{\tau}{4}g(x,x)+\frac{\tau^{*}}{4}\tilde{g}(x,x).\end{split}$

Let a vector $x$ induce an $S$-basis. The equalities (3) and (5) imply

$\begin{split}g(x,Sx)=g(x,x)\cos\varphi,\qquad\tilde{g}(x,x)=2g(x,x)\cos\varphi,\end{split}$

and using (19) and (23) we find (22). ∎

######  Corollary 3.6.

Let $(M,g,S)$ with (8) be an Einstein manifold. If a vector $x$ induces an
$S$-basis, then the Ricci curvatures in the direction of the basis vectors are

$r(x)=r(Sx)=r(S^{2}x)=r(S^{3}x)=\frac{\tau}{4}.$

###### Proof.

The above equalities follow directly by substituting $\tau^{*}=0$ into (22). ∎

###  3.2. An example

Examples of Einstein and almost Einstein manifolds $(M,g,S)$ with condition
(7) are constructed on Lie groups in [5]. Another example of an Einstein
manifold $(M,g,S)$ is constructed on a Lie group in [17]. Since this example
is not specified as a manifold with identity (8), we will now focus only on
this property of it.

Let $G$ be a $4$-dimensional real connected Lie group. Let $\mathfrak{g}$ be
the corresponding Lie algebra with a basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$ of
left invariant vector fields. We introduce a skew-circulant structure $S$ and
a metric $g$ as follows:

$\displaystyle Se_{1}=-e_{4},\quad Se_{2}=e_{1},\quad Se_{3}=e_{2},\quad
Se_{4}=e_{3},$ $\displaystyle g(e_{i},e_{j})=\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta.

The used basis $\\{e_{i}\\}$ is an orthonormal $S$-basis. The equalities (2)
and (3) are valid and $(g,S)$ is a structure of the considered type. We denote
the corresponding manifold by $(G,g,S)$.

The real four-dimensional indecomposable Lie algebras are classified by
Mubarakzyanov ([13]). More accessible in the libraries is the paper [2] and
the references therein. We consider the class $\\{\mathfrak{g}_{4,5}\\}$,
which represents an indecomposable Lie algebra, depending on two real
parameters $a$ and $b$.

According to the definition of the class $\\{\mathfrak{g}_{4,5}\\}$, we have
that the nonzero brackets are as follows ([2]):

$[e_{1},e_{4}]=e_{1},\quad[e_{2},e_{4}]=ae_{2},\quad[e_{3},e_{4}]=be_{3},\quad-1\leq
b\leq a\leq 1,\ ab\neq 0.$

The components of $\nabla$ are

$\begin{array}[]{lll}\nabla_{e_{1}}e_{1}=-e_{4},&\nabla_{e_{1}}e_{4}=e_{1},&\nabla_{e_{2}}e_{2}=-ae_{4},\\\
\nabla_{e_{2}}e_{4}=ae_{2},&\nabla_{e_{3}}e_{4}=be_{3},&\nabla_{e_{3}}e_{3}=-be_{4}.\end{array}$

In [17], for a manifold $(G,g,S)$ with a Lie algebra $\mathfrak{g}$ from the
class $\\{\mathfrak{g}_{4,5}\\}$, we get that if $a=b=1$ then

i) the nonzero components of the curvature tensor $R$ and of the Ricci tensor
$\rho$ with respect to $\\{e_{i}\\}$, are as follows:

(24) $\displaystyle R_{1212}=R_{1414}=R_{2323}=R_{3434}=R_{1313}=R_{2424}=1,$
$\rho_{11}=\rho_{22}=\rho_{33}=\rho_{44}=-3.$

ii) $(G,g,S)$ is a non-flat Einstein manifold with a negative scalar curvature
$\tau=-12$.

Bearing in mind the above results we state the following

######  Proposition 3.7.

Let $(G,g,S)$ be a manifold with a Lie algebra $\mathfrak{g}$ from the class
$\\{\mathfrak{g}_{4,5}\\}$. If $a=b=1$ are valid, then $(G,g,S)$ satisfies the
property (8).

###### Proof.

Directly from (24) we get that the components of the curvature tensor satisfy
(3.1). ∎

So we obtain an example of an Einstein manifold with the property (8). Also,
this manifold does not satisfy (7).

##  4\. Some sectional curvatures

If $\\{x,y\\}$ is a non-degenerate $2$-plane spanned by vectors $x,y$ in
$T_{p}M$, then its sectional curvature is

(25) $k(x,y)=\frac{R(x,y,x,y)}{g(x,x)g(y,y)-g^{2}(x,y)}.$

Let a vector $x$ induce an $S$-basis in $T_{p}M$ of $(M,g,S)$. There are
determined six 2-planes $\\{x,Sx\\}$, $\\{x,S^{2}x\\}$, $\\{x,S^{3}x\\}$,
$\\{Sx,S^{2}x\\}$, $\\{Sx,S^{3}x\\}$ and $\\{S^{2}x,S^{3}x\\}$ of $T_{p}M$.
For the angles between the pairs of vectors equalities (20) are valid.
Moreover, the angle $\varphi=\angle(x,Sx)$ satisfies (21).

In the next theorems we establish relations among the sectional curvatures of
the 2-planes generated by an $S$-basis.

######  Theorem 4.1.

Let $(M,g,S)$ be a manifold with property (8). If a vector $x$ induces an
$S$-basis, then for the sectional curvatures of the basic $2$-planes we have

(26) $k(x,Sx)=k(Sx,S^{2}x)=k(S^{2}x,S^{3}x)=k(S^{3}x,x),$ (27)
$k(x,S^{2}x)=k(Sx,S^{3}x).$

###### Proof.

From (8) we obtain

(28) $\begin{split}R(x,y,z,u)=&R(Sx,Sy,Sz,Su)\\\
=&R(S^{2}x,S^{2}y,S^{2}z,S^{2}u)=R(S^{3}x,S^{3}y,S^{3}z,S^{3}u).\end{split}$

In (28) we substitute

1) $Sx$ for $y$, $x$ for $z$, $Sx$ for $u$, and we get

(29) $\begin{split}R(x,Sx,x,Sx)=&R(Sx,S^{2}x,Sx,S^{2}x)\\\
=&R(S^{2}x,S^{3}x,S^{2}x,S^{3}x)=R(S^{3}x,x,S^{3}x,x).\end{split}$

2) $S^{2}x$ for $y$, $x$ for $z$, $S^{2}x$ for $u$, and we find

(30) $R(x,S^{2}x,x,S^{2}x)=R(Sx,S^{3}x,Sx,S^{3}x).$

The equality (26) follows from (3), (25) and (29). In a similar way, from (3),
(25) and (30) we obtain (27). ∎

Due to Theorem 4.1 there are only two different sectional curvatures
determined by an $S$-basis. Thus, we consider only the sectional curvatures of
the type $k(x,Sx)$ and $k(x,S^{2}x)$.

######  Theorem 4.2.

Let $(M,g,S)$ be a manifold with property (8). If vectors $x$ and $u$ induce
$S$-bases and $\\{S^{3}x,S^{2}x,Sx,x\\}$ is an orthonormal basis, then the
sectional curvatures satisfy

(31)
$\begin{split}2(1-&\cos^{2}\varphi)k(u,Su)+k(u,S^{2}u)=k(x,S^{2}x)+2(1-\cos^{2}\varphi)k(x,Sx)\\\
&+\cos^{2}\varphi\big{(}4R(x,Sx,Sx,S^{2}x)+6R(x,Sx,S^{2}x,S^{3}x)\big{)}\\\
&+4\cos\varphi\big{(}R(x,Sx,x,S^{2}x)+R(x,Sx,Sx,S^{3}x)\big{)}.\end{split}$

where $\angle(u,Su)=\varphi$.

###### Proof.

In (28) we substitute

1) $Sx$ for $y$, $S^{2}x$ for $z$ and $x$ for $u$, and we get

(32) $\begin{split}R(x,Sx,S^{2}x,x)=&R(Sx,S^{2}x,S^{3}x,Sx)\\\
=&-R(S^{2}x,S^{3}x,x,S^{2}x)=R(S^{3}x,x,Sx,S^{3}x),\end{split}$

2) $Sx$ for $y$, $Sx$ for $z$ and $S^{2}x$ for $u$, and we have

(33) $\begin{split}R(x,Sx,Sx,S^{2}x)=&R(Sx,S^{2}x,S^{2}x,S^{3}x)\\\
=&-R(S^{2}x,S^{3}x,S^{3}x,x)=-R(S^{3}x,x,x,Sx),\end{split}$

3) $Sx$ for $y$, $S^{2}x$ for $z$ and $S^{3}x$ for $u$, and we find

(34) $\begin{split}R(x,Sx,S^{2}x,S^{3}x)=&-R(Sx,S^{2}x,S^{3}x,x)\\\
=&R(S^{2}x,S^{3}x,x,Sx)=-R(S^{3}x,x,Sx,S^{2}x),\end{split}$

4) $Sx$ for $y$, $Sx$ for $z$ and $S^{3}x$ for $u$, and we get

(35) $\begin{split}R(x,Sx,Sx,S^{3}x)=&-R(Sx,S^{2}x,S^{2}x,x)\\\
=&-R(S^{2}x,S^{3}x,S^{3}x,Sx)=-R(S^{3}x,x,x,S^{2}x),\end{split}$

5) $S^{2}x$ for $y$, $Sx$ for $z$ and $S^{3}x$ for $u$, and we have

(36) $\begin{split}R(x,S^{2}x,Sx,S^{3}x)=&-R(Sx,S^{3}x,S^{2}x,x)\\\
=&R(S^{2}x,x,S^{3}x,Sx)=-R(S^{3}x,Sx,x,S^{2}x).\end{split}$

Let $u=\alpha S^{3}x+\beta S^{2}x+\gamma Sx+\delta x$, where
$\alpha,\beta,\gamma,\delta\in\mathbb{R}$. From (1) we get $Su=\beta
S^{3}x+\gamma S^{2}x+\delta Sx-\alpha x$, $S^{2}u=\gamma S^{3}x+\delta
S^{2}x-\alpha Sx-\beta x$ and $S^{3}u=\delta S^{3}x-\alpha S^{2}x-\beta
Sx-\gamma x$. Then, by using the linear properties of the curvature tensor $R$
and having in mind (28), (29), (30), (32) – (36), we obtain

$\displaystyle R(u,Su,u,Su)$
$\displaystyle=(\alpha^{2}+\gamma^{2})(\beta^{2}+\delta^{2})R_{1}$
$\displaystyle+\Big{(}(\alpha^{2}+\gamma^{2})^{2}+(\beta^{2}+\delta^{2})^{2}+2\delta\beta(\alpha^{2}-\gamma^{2})+2\alpha\gamma(\delta^{2}-\beta^{2})\Big{)}R_{2}$
$\displaystyle+2(\alpha^{2}+\gamma^{2})(\beta^{2}+\delta^{2})R_{3}$
$\displaystyle+6\Big{(}\delta\beta(\gamma^{2}-\alpha^{2})+\alpha\gamma(\beta^{2}-\delta^{2})\Big{)}R_{4}$
$\displaystyle+2\Big{(}(\beta\gamma-\alpha\delta)(\beta^{2}+\delta^{2})+(\alpha\beta+\delta\gamma)(\alpha^{2}+\gamma^{2})\Big{)}R_{5}$
$\displaystyle+2\Big{(}(\alpha\beta+\gamma\delta)(\delta^{2}+\beta^{2})+(\beta\gamma-\delta\alpha)(\alpha^{2}+\gamma^{2})\Big{)}R_{6},$
$\displaystyle R(u,S^{2}u,u,S^{2}u)$
$\displaystyle=\Big{(}(\alpha^{2}+\gamma^{2})^{2}+(\beta^{2}+\delta^{2})^{2}\Big{)}R_{1}$
$\displaystyle+2(\alpha^{2}+\gamma^{2})(\beta^{2}+\delta^{2})R_{2}$
$\displaystyle+8\Big{(}\delta\beta(\gamma^{2}-\alpha^{2})+\alpha\gamma(\beta^{2}-\delta^{2})\Big{)}R_{3}$
$\displaystyle+6(\alpha^{2}+\gamma^{2})(\beta^{2}+\delta^{2})R_{4}$
$\displaystyle+4\Big{(}(\alpha\beta+\gamma\delta)(\delta^{2}+\beta^{2})+(\beta\gamma-\delta\alpha)(\alpha^{2}+\gamma^{2}))R_{5}$
$\displaystyle+4\Big{(}(\beta\gamma-\alpha\delta)(\beta^{2}+\delta^{2})+(\alpha\beta+\delta\gamma)(\alpha^{2}+\gamma^{2})\Big{)}R_{6},$

where

(37)
$\begin{array}[]{lll}R_{1}=R(x,S^{2}x,x,S^{2}x),&R_{2}=R(x,Sx,x,Sx),&R_{3}=R(x,Sx,Sx,S^{2}x),\\\
R_{4}=R(x,Sx,S^{2}x,S^{3}x),&R_{5}=R(x,Sx,x,S^{2}x),&R_{6}=R(x,Sx,Sx,S^{3}x).\end{array}$

Now we get the sum

(38)
$\begin{split}2R(u,Su,u,Su)&+R(u,S^{2}u,u,S^{2}u)=(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2})^{2}R_{1}\\\
&+2\big{(}(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2})^{2}-(\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta)^{2}\big{)}R_{2}\\\
&+(\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta)^{2}(4R_{3}+6R_{4})\\\
&+4(\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta)(R_{5}+R_{6}).\end{split}$

Since the basis $\\{S^{3}x,S^{2}x,Sx,x\\}$ is orthonormal, we have

$g(u,u)=g(Su,Su)=\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2},$
$g(u,Su)=\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta,\quad
g(u,S^{2}u)=0.$

We assume that $u$ is a unit vector and from the latter equalities we get

(39)
$\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}=1,\quad\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta=\cos\varphi,$

where $\varphi=\angle(u,Su)$. Using (38) and (39) we calculate

(40)
$\begin{split}2R(u,Su,u,Su)+&R(u,S^{2}u,u,S^{2}u)=R_{1}+2(1-\cos^{2}\varphi)R_{2}\\\
+&\cos^{2}\varphi(4R_{3}+6R_{4})+4\cos\varphi(R_{5}+R_{6}).\end{split}$

Bearing in mind that $x$ induces an orthonormal $S$-basis and $u$ is a unit
vector, with the help of (3), (20) and (25), we find

(41) $\begin{split}k(x,Sx)=R(x,Sx,x,Sx),\quad
k(x,S^{2}x)=R(x,S^{2}x,x,S^{2}x),\\\
k(u,Su)=\frac{R(u,Su,u,Su)}{1-\cos^{2}\varphi},\quad
k(u,S^{2}u)=R(u,S^{2}u,u,S^{2}u).\end{split}$

Taking into account (41), from (40) we obtain

(42)
$\begin{split}2(1-\cos^{2}\varphi)k(u,Su)+&k(u,S^{2}u)=k(x,S^{2}x)+2(1-\cos^{2}\varphi)k(x,Sx)\\\
+&\cos^{2}\varphi(4R_{3}+6R_{4})+4\cos\varphi(R_{5}+R_{6}).\end{split}$

The equality (31) follows from (37) and (42). ∎

Next, we express the sectional curvatures of the 2-planes $\\{u,Su\\}$ and
$\\{u,S^{2}u\\}$ by the sectional curvatures of the 2-planes of the same type,
whose angles between basis vectors are constants.

######  Theorem 4.3.

Let $(M,g,S)$ be a manifold with property (8). If vectors $x$, $u$, $v$ and
$w$ induce $S$-bases and $\\{S^{3}x,S^{2}x,Sx,x\\}$ is an orthonormal basis,
then the sectional curvatures satisfy

(43)
$\begin{split}2(1-\cos^{2}\varphi)k(u,Su)+k(u,S^{2}u)&=(1-4\cos^{2}\varphi)\big{(}2k(x,Sx)+k(x,S^{2}x)\big{)}\\\
&+(2\cos^{2}\varphi+\cos\varphi)\big{(}\frac{3}{2}k(v,Sv)+k(v,S^{2}v)\big{)}\\\
&+(2\cos^{2}\varphi-\cos\varphi)\big{(}\frac{3}{2}k(w,Sw)+k(w,S^{2}w)\big{)},\end{split}$

where $\angle(u,Su)=\varphi$, $\angle(v,Sv)=\frac{\pi}{3}$,
$\angle(w,Sw)=\frac{2\pi}{3}$.

###### Proof.

In (42) first we substitute $u=v$ and then $u=w$. Successively we get

$\displaystyle\frac{3}{2}\big{(}k(v,Sv)-k(x,Sx)\big{)}+k(v,S^{2}v)-k(x,S^{2}x)=\frac{1}{2}(2R_{3}+3R_{4})+2(R_{5}+R_{6}),$
$\displaystyle\frac{3}{2}\big{(}k(w,Sw)-k(x,Sx)\big{)}+k(w,S^{2}w)-k(x,S^{2}x)=\frac{1}{2}(2R_{3}+3R_{4})-2(R_{5}+R_{6}).$

We solve the above system and express the tensors $2R_{3}+3R_{4}$ and
$R_{5}+R_{6}$ by the curvatures. We substitute these solutions into (42), and
find (43). ∎

The following two statements refer to the manifolds $(M,g,S)$ with property
(7), which are a special case of the manifolds considered so far.

######  Proposition 4.4.

Let $(M,g,S)$ be a manifold with property (7). If a vector $x$ induces an
$S$-basis, then for the sectional curvatures of the basic $2$-planes we have

(44) $\begin{split}k(x,Sx)&=k(Sx,S^{2}x)=k(S^{2}x,S^{3}x)=k(S^{3}x,x),\\\
k(x,S^{2}x)&=S(Sx,S^{3}x)=2(1-\cos^{2}\varphi)k(x,Sx),\end{split}$

where $\angle(x,Sx)=\varphi.$

###### Proof.

If $(M,g,S)$ has the property (7), then it satisfies (8). Thus having in mind
(7), (29) and (30), we obtain

$\begin{split}R(x,S^{2}x,x,S^{2}x)=&R(Sx,S^{3}x,Sx,S^{3}x)=2R(x,Sx,x,Sx)\\\
=2R(x,S^{3}x,x,S^{3}x)&2R(Sx,S^{2}x,Sx,S^{2}x)=2R(S^{2}x,S^{3}x,S^{2}x,S^{3}x).\end{split}$

Then (3) and (25) imply (44). ∎

######  Theorem 4.5.

Let $(M,g,S)$ be a manifold with property (7). If vectors $x$, $u$ and $v$
induce $S$-bases and $\\{S^{3}x,S^{2}x,Sx,x\\}$ is an orthonormal basis, then
the following equality is valid

(45)
$\begin{split}k(u,Su)=\frac{1}{1-\cos^{2}\varphi}(1+2\cos^{2}\varphi-3\cos\varphi)k(x,Sx)+\frac{3\cos\varphi}{2(1-\cos^{2}\varphi)}k(v,Sv),\end{split}$

where $\angle(u,Su)=\varphi$, $\angle(v,Sv)=\frac{\pi}{3}$.

###### Proof.

We apply (7) to (29), (30), (32) – (36), and for the components (37) we get

(46) $R_{1}=2R_{2}=2R_{3}=2R_{4},\quad R_{5}=R_{6}.$

Using the presentation of $R(u,Su,u,Su)$ in Theorem 4.2 and having in mind
(46) we calculate

(47)
$\begin{split}R(u,Su,u,Su)&=\big{(}(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2})^{2}+2(\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta)^{2}\big{)}R(x,Sx,x,Sx)\\\
&+2(\alpha\beta+\beta\gamma+\delta\gamma-\alpha\delta)R(x,Sx,x,S^{2}x).\end{split}$

We assume that $u$ is a unit vector and from (39) and (47) it follows

(48) $\begin{split}R(u,Su,u,Su)=(1+2\cos^{2}\varphi)R(x,Sx,x,Sx)+2\cos\varphi
R(x,Sx,x,S^{2}x).\end{split}$

Then (41) imply

(49)
$\begin{split}k(u,Su)=\frac{1+2\cos^{2}\varphi}{1-\cos^{2}\varphi}k(x,Sx)+\frac{2\cos\varphi}{1-\cos^{2}\varphi}R(x,Sx,x,S^{2}x).\end{split}$

Now we substitute $v$ for $u$ in (49) and express $R(x,Sx,x,S^{2}x)$ by
$k(v,Sv)$ and $k(x,Sx)$. We apply this expression to (49) and obtain (45). ∎

##  5\. Conclusion

In this paper we obtain curvature properties of a 4-dimensional Riemannian
manifold $(M,g,S)$ with (8) associated with a locally conformal Hermitian
manifold $(M,g,J=S^{2})$. Furthermore, the curvature properties of the
associated manifold $(M,g,J)$ deserve to be studied, also in the case when the
manifold belongs to the class $\mathcal{L}_{3}$, according to Grey’s
classification. The problem of constructing examples of 4-dimensional almost
Einstein manifolds $(M,g,S)$ with (8) also remains.

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Iva Dokuzova
Department of Algebra and Geometry
University of Plovdiv Paisii Hilendarski
24 Tzar Asen, 4000 Plovdiv, Bulgaria
<EMAIL_ADDRESS>