# Unleashing infinite momentum bandgap using resonant material systems

X. Wang1 These authors contributed equally. P. Garg2 These authors contributed
equally. M. S. Mirmoosa3 A. G. Lamprianidis2 C. Rockstuhl1,2 V. S. Asadchy4
1Institute of Nanotechnology, Karlsruhe Institute of Technology, Karlsruhe,
Germany
2Institute of Theoretical Solid State Physics, Karlsruhe Institute of
Technology, Karlsruhe, Germany
3Department of Physics and Mathematics, University of Eastern Finland, P.O.
Box 111, FI-80101 Joensuu, Finland
4Department of Electronics and Nanoengineering, Aalto University, Espoo,
Finland

###### Abstract

The realization of photonic time crystals is a major opportunity but also
comes with significant challenges. The most pressing one, potentially, is the
requirement for a substantial modulation strength in the material properties
to create a noticeable momentum bandgap. Reaching that noticeable bandgap in
optics is highly demanding with current, and possibly also future, material
platforms since their modulation strength is small by tendency. Here we
demonstrate that by introducing temporal variations in a resonant material,
the momentum bandgap can be drastically expanded, potentially approaching
infinity with modulation strengths in reach with known low-loss materials and
realistic laser pump powers. The resonance can emerge from an intrinsic
material resonance or a suitably spatially structured material supporting a
structural resonance. Our concept is validated for resonant bulk media and
optical metasurfaces and paves the way toward the first experimental
realizations of photonic time crystals.

##  I Introduction

Time-varying electromagnetics has emerged as a captivating field of research,
capturing significant attention from the scientific community Galiffi  _et
al._ (2022); Engheta (2020). Within this domain, a recent and intriguing
concept that has garnered interest is that of photonic time crystals Zurita-
Sánchez  _et al._ (2009); Lustig  _et al._ (2018). Distinct from conventional
spatial photonic crystals, photonic time crystals are artificial materials
whose electromagnetic properties are uniform in space but periodically
modulated in time. This temporal periodicity generates a momentum bandgap in
which light exponentially grows over time. Such behavior results in exotic
light-matter interaction in the optical regime, including amplification of
spontaneous emission of an excited atom Lyubarov  _et al._ (2022), subluminal
Cherenkov radiation Dikopoltsev  _et al._ (2022), superluminal momentum-gap
solitons Pan  _et al._ (2023), and others. The momentum bandgap stems from the
Bragg-like interference of multiple reflected waves generated at the periodic
temporal interfaces, that is, time moments at which the crystal switches its
refractive index Galiffi  _et al._ (2022). To develop a detectable momentum
bandgap, the relative change of the refractive index of the crystal $\Delta
n/n$ must be comparable to unity Zurita-Sánchez  _et al._ (2009); Lustig  _et
al._ (2018); Hayran  _et al._ (2022).

While photonic time crystals have been experimentally confirmed at microwave
frequencies Reyes-Ayona and Halevi (2015); Park  _et al._ (2022); Wang  _et
al._ (2023), designing photonic time crystals at optical frequencies remains a
prime challenge. Indeed, the material temporal modulation must be extremely
fast, typically twice the oscillation period of the light that probes the
response. Such a rapid modulation is unreachable with acousto-optic, thermo-
optic, or even electro-optic mechanisms which operate up to 100 GHz Williamson
_et al._ (2020). All-optical modulation of the refractive index $n$ (e.g., via
third-order optical nonlinearities) is currently considered the most prominent
approach to realizing photonic time crystals at optical frequencies. However,
nonlinear effects in low-loss materials are very weak, yielding a relative
change in the refractive index to saturate at less than 1%. It also implies
substantial power densities that can only be achieved through femtosecond
laser pulses with low duty cycles (substantially lower than those required for
material modulation) Borchers  _et al._ (2012). Recently, transparent
conductive oxides were suggested as alternative material candidates to
synthesize photonic time crystals. In particular, operating at the epsilon-
near-zero regime, indium tin oxide (ITO) and aluminium-doped zinc oxide (AZO)
were demonstrated to exhibit relative changes in the refractive index of the
order of 100% Alam  _et al._ (2016); Bohn  _et al._ (2021); Zhou  _et al._
(2020); Caspani  _et al._ (2016). However, they still demand extremely high
pumping power densities, i.e., in the order of tens of TW/${\rm cm}^{3}$
Hayran  _et al._ (2022). When combined with significant material dissipation,
this could lead to rapid thermal damage to the material. Furthermore, the
dissipation can lead to the excitation of a parasitic “dynamic-grating”
nonlinear effect that may obstruct the observation of the photonic-time-
crystal regime Khurgin  _et al._ (2021). Therefore, only time refraction
(single-cycle modulation) has been experimentally realized in transparent
conducting oxides to date Lustig  _et al._ (2023). Even though the distinction
between a photonic time crystal and a parametric amplifier remains
challenging, these experiments suggest that photonic time crystals are in
reach Khurgin (2023a).

To avoid all these obstacles, we introduce a distinct approach to designing
photonic time crystals. We capitalize on artificial composites that support
high-quality resonances rather than seeking new materials with improved
nonlinear characteristics. Interestingly, a similar generic idea was recently
suggested in Ref. Khurgin (2023b), however, no practical design or
implementation was demonstrated. The approach based on resonant artificial
composites allows us to create photonic time crystals with pronounced momentum
bandgaps with significantly reduced required modulation strength ($\Delta n/n$
ratio) in reach with known low-loss materials and realistic laser pump powers.
We show that at a conceptual level, the resonance can be intrinsic to the
material. But we also go beyond and suggest exploiting a resonance sustained
by a spatially structured material to significantly lower the requirement on
the modulation strength for reaching a notable momentum bandgap. We validate
our concept of a resonant photonic time crystal for bulk materials and
realistic optical metasurfaces. We show that the momentum bandgap size can be
expanded limitlessly under even low modulation strengths (assuming
sufficiently small material dissipation). Our results potentially provide the
first material platform to realize a photonic time crystal at optical
frequencies.

##  II Results

###  II.1 Photonic time crystals made of intrinsically resonant materials

The earlier designs for photonic time crystals relied on varying the density
of free-charge carriers in a bulk material, such as the number of electrons
per unit volume Alam  _et al._ (2016); Bohn  _et al._ (2021); Zhou  _et al._
(2020); Caspani  _et al._ (2016). In this Section, we first demonstrate that
this approach is ineffective in achieving a large momentum bandgap. Next, we
will show that temporally modulating the material’s resonance frequency
significantly increases the bandgap size while having the same modulation
strength.

The permittivity of a bulk material with a Lorentzian dispersion whose charge
concentration is modulated in time can be written as
$\epsilon(\omega,t)=1+\omega_{\rm p}^{2}(t)/(\omega_{\rm
r0}^{2}-\omega^{2}+j\gamma\omega)$ (Mirmoosa  _et al._ , 2022, Sec. 4.3).
Here, $\omega_{\rm p}(t)$, $\omega_{\rm r0}$, and $\gamma$ are the plasma
frequency, resonance frequency, and the damping factor of the charger carriers
in the material, respectively. A time convention as $e^{+j\omega t}$ is
considered, with $j$ being the imaginary unit. This form of the complex
permittivity function is valid if the damping coefficient and the natural
frequency of the material can be assumed to be constant in time Mirmoosa  _et
al._ (2022). Assuming a time-harmonic modulation of the charge concentration
with the magnitude $m$ and modulation frequency $\omega_{\rm m}$, i.e.,
$\omega_{\rm p}^{2}(t)=\omega_{\rm p0}^{2}[1+m\cos(\omega_{\rm m}t)]$, the
permittivity can be written as
$\epsilon(\omega,t)=1+\chi(\omega)[1+m\cos(\omega_{\rm m}t)]$ with
$\chi(\omega)$ the stationary dispersive complex susceptibility of the
material just considered.

When solving for the eigenmodes to Maxwell’s equations in such a time-varying
medium, the temporal modulation induces a split of the degenerate eigenmodes
at $\omega=\omega_{\rm m}/2$, resulting in a momentum bandgap. Assuming a weak
modulation strength ($m\ll 1$) Asadchy  _et al._ (2022), the bandgap covers
the momentum space from $k=k_{-}$ to $k=k_{+}$, where

$k_{\pm}={\omega_{\rm{m}}\over 2c_{0}}\sqrt{1+\chi(\omega_{\rm{m}}/2)[1\pm
m/2]}.$ (1)

The detailed derivation of (1) can be found in Sec. 1 of the Supplementary
Material sup . Here, $c_{0}$ is the speed of light in a vacuum, and $\gamma=0$
was assumed. We plot the dispersion relation in the first Brillouin zone of
such a photonic time crystal in Fig. 1(b), taking as an example $m=0.2$ (other
material parameters are specified in the figure caption). Inside the momentum
bandgap, the eigenfrequency has two complex solutions with positive and
negative imaginary parts corresponding to the exponentially decaying and
growing modes in time (dashed orange lines).

Figure 1: (a) Dispersion relations of a stationary (not modulated) bulk
material when its plasma frequency is $\omega_{\rm p1}^{2}=\omega_{\rm
p0}^{2}(1-m/2)$ (red line) and $\omega_{\rm p2}^{2}=\omega_{\rm
p0}^{2}(1+m/2)$ (green line). (b) Band structure of the time-modulated bulk
material when its plasma frequency is modulated as $\omega_{\rm
p}^{2}(t)=\omega_{\rm p0}^{2}[1+m\cos(\omega_{\rm m}t)]$. The gray region
emphasizes the bandgap in (b) and the variation of eigenwavenumber at
$\omega=\omega_{\rm m}/2$ in (a). (c) Dispersion relations of a stationary
(not modulated) bulk material when its resonance frequency is $\omega_{\rm
r1}^{2}=\omega_{\rm r0}^{2}(1+m/2)$ (red line) and $\omega_{\rm
r2}^{2}=\omega_{\rm r0}^{2}(1-m/2)$ (green line). (d) Band structure of the
time-modulated bulk material when its resonance frequency is modulated as
$\omega_{\rm r}^{2}(t)=\omega_{\rm r0}^{2}[1+m\cos(\omega_{\rm m}t)]$. In all
the subfigures, the wavenumber is normalized by $k_{r0}=\omega_{\rm
r0}\sqrt{\epsilon_{0}\mu_{0}}$, and the material parameters are chosen as
$m=0.2$, $\omega_{\rm m}/2=0.95\omega_{\rm r0}$, $\gamma=0$, and $\omega_{\rm
p0}=3.5\omega_{\rm r0}$.

The weak-modulation approximation is very powerful. Not only does it allow us
to find a closed-form solution for the edges of the momentum bandgap, but it
also highlights a crucial observation. As is seen from Eq. (1), the bandgap
edges $k_{-}$ and $k_{+}$ in the modulated material precisely correspond to
the eigenwavenumbers $k_{1}$ and $k_{2}$ of two non-modulated (stationary)
materials with scaled susceptibilities $\chi(\omega)(1\pm m/2)$ evaluated at
the frequency $\omega=\omega_{\rm{m}}/2$. Indeed, inside a stationary
material, the eigenwavenumber is given by
$k(\omega)=\omega\sqrt{1+\chi(\omega)}/c_{0}$. This correspondence is rather
general and applies to all the photonic time crystal topologies with $m\ll 1$
explored in this work, including the case when the material is lossy.
Interestingly, even when the modulation strength becomes large, the
correspondence has only a small error (less than $2\%$ and $5\%$ for $m=0.5$
and $m=0.9$, respectively).

Figure 1(a) depicts the dispersion relations of the two stationary materials
with scaled susceptibilities. Since $\chi$ is linearly proportional to
$\omega_{\rm p}^{2}$, the two materials can be alternatively described with
scaled plasma frequencies $\omega_{\rm p1}^{2}=\omega_{\rm p0}^{2}(1-m/2)$ and
$\omega_{\rm p2}^{2}=\omega_{\rm p0}^{2}(1+m/2)$. Thus, by plotting the
dispersion relation of the material with modulation turned off, one can
determine the momentum bandgap size for a given modulation. We emphasize this
property in Figs. 1(a) and (b) using a vertically shaded region. More
importantly, this correspondence provides a simple visual insight into how a
given material dispersion affects the bandgap size and position. It is clear
from Fig. 1(a) that the temporal modulation of the plasma frequency results in
a relatively narrow bandgap. The narrow bandgap is caused by the tiny
variation of the dispersion curves of a stationary material when the resonance
frequency is fixed ($k_{1}\approx k_{2}$). The relative size of the obtained
bandgap, calculated as the gap-to-midgap ratio $\Delta
k=2|(k_{2}-k_{1})/(k_{1}+k_{2})|$, reaches only $10.3\%$.

On the contrary, when the resonance frequency $\omega_{\rm r}$ is time-
modulated while the plasma frequency $\omega_{\rm p}$ and damping factor
$\gamma$ are fixed, the momentum bandgap can be qualitatively different. Now,
we assume that the intrinsic resonance frequency is modulated harmonically so
that $\omega_{\rm r}^{2}(t)=\omega_{\rm r0}^{2}[1+m\cos(\omega_{\rm m}t)]$.
Following the above-described correspondence between the stationary and
modulated material scenarios for $m\ll 1$, we plot in Fig. 1(c) the dispersion
relation of a stationary bulk material for two values of the resonance
frequency: $\omega_{\rm r1}^{2}=\omega_{\rm r0}^{2}(1+m/2)$ and $\omega_{\rm
r2}^{2}=\omega_{\rm r0}^{2}(1-m/2)$. All other material properties remain the
same and are mentioned in the figure caption. It is evident that for the same
value of $m=0.2$, the two dispersion curves now differ significantly at
$\omega=\omega_{\rm m}/2$. When choosing $\omega_{\rm r2}<\omega_{\rm
m}/2<\omega_{\rm r1}$, the horizontal dashed line at $\omega=\omega_{\rm m}/2$
intersects only with one of the dispersion curves, at $k=k_{1}$ (Fig. 1(c)).
This indicates that the momentum bandgap in the corresponding time-modulated
material has only one edge and extends infinitely towards higher momenta, as
shown by the gray-shaded region in the figure. Indeed, by rigorously computing
the band structure for this scenario, we observe a semi-infinite momentum
bandgap whose edge at $k_{-}$ is located right at $k_{1}$ for the
corresponding stationary material (see Fig. 1(d)). The detailed derivations
for the band structure extraction can be found in Sec. 2 of the Supplementary
Material sup . The imaginary part of the eigenfrequency inside the bandgap is
significantly increased compared to modulating the plasma frequency. Moreover,
the shape of the imaginary part curve (orange dashed line) is not typical. It
points out that waves with different momentum $k$ can have different and, in
principle, engineered amplification rates. We observed that by changing
$\omega_{\rm m}$ within the range $\omega_{\rm r2}<\omega_{\rm
m}/2<\omega_{\rm r1}$, it is possible to control the curve shape.

Since, in realistic scenarios, near the resonance, material losses tend to
increase, it is crucial to explore how a nonzero damping factor $\gamma$
influences the band structure. The effect of losses is studied in Sec. 3 of
the Supplementary Material sup . In the presence of loss, the imaginary parts
of eigenfrequencies exhibit a positive offset, resulting in reduced temporal
amplification rates. In contrast, the bandgap size remains. Even when
considering losses, the amplification rate remains significantly higher
compared to modulating the plasma frequency in the lossless case.

The temporal modulation of the intrinsic resonance frequency of a bulk
material can be achieved, for example, through strong dynamic electric biasing
Serra and Silveirinha (2023) but, in practice, could be very challenging.
Figure 2(a) illustrates this scenario where an external electric field
modulates the effective spring constants $\kappa$ of each nucleus-electron
oscillator. However, instead of modulating materials by modulating the
intrinsic resonance frequency of their natural atoms, we propose exploiting a
metamaterial concept and utilizing the resonances in metamaterials generated
through spatially structured meta-atoms. A conceptual illustration of our
proposal is depicted in Fig. 2(b). The metamaterial consists of meta-atoms
(with spherical shapes as an example) whose material properties (permittivity
$\epsilon_{\rm r}$ or permeability $\mu_{\rm r}$), radius, or mutual
separation distance can be modulated in time. Each of these modulation
mechanisms leads to the modulation of the effective resonance frequency
$\omega_{\rm r}^{\rm eff}(t)$ of the metamaterial. Probably, the most
practical scenario is when the plasma frequency of the meta-atom’s material is
modulated, which can be achieved with the current optical modulation
techniques Alam  _et al._ (2016); Bohn  _et al._ (2021); Zhou  _et al._
(2020); Caspani  _et al._ (2016).

In what follows, instead of relying on bulk metamaterial geometries, we
exploit the recently proposed idea of constructing photonic time crystals
using metasurfaces Wang  _et al._ (2023). First, we explore the concept by
designing resonant metasurfaces using a generic LC-model applicable to all
operating frequencies. Then, we provide a feasible optical example of the
photonic time crystal – a two-dimensional array of spherical particles –
supporting a large momentum bandgap at realistic pump energies.

Figure 2: (a) Illustration of the temporal modulation of the intrinsic
resonance frequency of a bulk material. The modulation can be described by a
time-varying effective spring constant $\kappa(t)$. (b) Illustration of the
temporal modulation of the resonance frequency of a metamaterial consisting of
meta-atoms. The modulation can be achieved in various ways, including
modulating the permittivity $\epsilon_{\rm r}$ or permeability $\mu_{\rm r}$
of the material from which the meta-atoms are made, the size of the meta-atoms
$R$, or the metamaterial periodicity $d$.

Figure 3: (a) Generic illustration of a time-varying resonant $LC$
metasurface. Each meta-atom is described by a time-modulated surface
capacitance $C(t)$ and a constant surface inductance $L$. The red arrow
indicates a surface eigenmode whose amplitude is growing in time due to its
wavenumber being inside the momentum bangdap of the metasurface. (b) Upper:
Dispersion relations of a stationary metasurface for the two scaled values of
the surface capacitance. Lower: Band structure of the time-modulated resonant
metasurface with $C(t)=C_{0}[1+m\cos(\omega_{\rm m}t)]$. The gray region
emphasizes the bandgap in the lower panel and the variation of eigenwavenumber
at $\omega=\omega_{\rm m}/2$ in the upper panel. (c) and (d) magnetic field
snapshots for momenta $\beta=5k_{\rm r0}$ and $\beta=10k_{\rm r0}$ above the
metasurface at the time moment when modulation switches on ($t=0$) and after
some time passed ($t=30T_{\rm m}$). (e) Time evolution of the magnetic field
above the metasurface ($z=0$) calculated with full-wave simulations and
analytically from the band structure. The theoretical amplitude is calculated
as $H_{y}(t)=H_{y0}\exp{[\Im(\omega)t]}$ where $H_{y0}=1$ A/m is the initial
field at $t=0$ and $\Im(\omega)=0.025\omega_{\rm m}$. (f) Imaginary part of
the eigenfrequency for metasurfaces with different quality factors. The real
part of eigenfrequency is fixed as $\Re(\omega)=\omega_{\rm m}/2$. The black
dashed line separates propagating waves (p.w.) and surface waves (s.w.). (g)
The magnetic field evolution for a dipole source positioned above the time-
varying $LC$ metasurface. Due to the infinite bandgap, all momenta $k_{||}$
are amplified by the metasurface. In all the subfigures, $m=0.2$ was chosen.
In Figs. 3(c)(d)(g), the excitation source is turned off after modulation
starts.

###  II.2 Photonic time crystals based on resonant metasurfaces

We consider the photonic time crystal in a metasurface geometry. The
metasurface is assumed to support surface waves bound to the $xy$-plane, see
Fig. 3(a). The meta-atoms are deeply sub-wavelength and are all equivalent,
resulting in a spatial homogeneity of the metasurface at $z=0$. Similarly to
how bulk photonic time crystals exhibit momentum bandgaps $\Delta k$ for bulk
propagating eigenmodes, the metasurface-based counterparts support momentum
bandgaps $\Delta k_{||}$ for surface and/or propagating eigenmodes Wang  _et
al._ (2023) ($k_{||}$ denotes the parallel component of the wavenumber). While
here we model the metasurface as an impenetrable impedance boundary, the same
qualitative results hold true for penetrable metasurfaces (see the next
Section). A resonant impenetrable metasurface can be described by effective
reactive parameters, the surface capacitance $C$ and the surface inductance
$L$ connected in parallel (Fig. 3(a)). The advantage of this generic
$LC$-model is its ability to describe the interaction of light with a time-
modulated metasurface without restricting ourselves to any specific geometry
and operational frequency band. At optical frequencies, as we show in the
following section, such a resonant metasurface can be made from a two-
dimensional array of spherical particles with a time-modulated dielectric
constant. At microwave frequencies, the implementation could be based on a
mushroom-type high-impedance surface with varactors embedded between adjacent
patches Sievenpiper  _et al._ (1999); Wang  _et al._ (2023).

In our model, the surface capacitance is modulated in time harmonically as
$C(t)=C_{0}[1+m\cos(\omega_{\rm m}t)]$, while the surface inductance $L$
remains constant. Here, $C_{0}$ is the time-averaged value of the capacitance.
Such a configuration effectively provides the temporal modulation of the
resonance frequency. Without loss of generality, we consider a transverse-
magnetic (TM) polarization of the surface waves. Similarly to the derivation
in the previous Section, we solve for the eigenmodes of the time-modulated
metasurface (see Sec. 4 of the Supplementary Material sup ). Using the weak-
modulation approximation ($m\ll 1$), the edges of the momentum bandgap at
$\omega=\omega_{\rm m}/2$ can be calculated as

$k_{||}^{\pm}={\omega_{\rm{m}}\over
2c_{0}}\sqrt{1-\frac{Y_{0}^{2}}{Y_{\pm}^{2}}},$ (2)

where $Y_{0}=\sqrt{\varepsilon_{0}/\mu_{0}}$ is the free-space wave admittance
and $Y_{\pm}=j\frac{\omega_{\rm
m}}{2}C_{0}(1\pm\frac{m}{2})+\frac{1}{j\frac{\omega_{\rm m}}{2}L}$. In
agreement with our findings in the previous Section, the expressions for the
edges of the momentum bandgap $k_{||}^{+}$ and $k_{||}^{-}$ precisely
correspond to the eigenwavenumbers $k_{||,1}$ and $k_{||,2}$ of two non-
modulated impedance surfaces with scaled surface capacitances
$C_{1}=C_{0}(1+\frac{m}{2})$ and $C_{2}=C_{0}(1-\frac{m}{2})$ evaluated at the
frequency $\omega=\omega_{\rm m}/2$. This is clearly seen from the dispersion
relation of an impedance boundary
$k_{||}(\omega)=\frac{\omega}{c_{0}}\sqrt{1-Y_{0}^{2}/Y(\omega)}$ (Tretyakov,
2003, p. 234), where $Y(\omega)=j\omega C+1/j\omega L$ is the admittance of
the parallel connection of the capacitance $C$ and inductance $L$.

It should be noted that Eq. (2) holds true only for frequencies $\omega_{\rm
m}/2<1/\sqrt{LC_{0}(1+m/2)}=\omega_{\rm r1}$ (see Sec. 4 of Supplementary
Material sup  for more details). At the critical frequency of $\omega_{\rm
m}/2=\omega_{\rm r1}$, the denominator in Eq. (2) approaches zero, which
results in the semi-infinite bandgap ($Y_{+}\rightarrow j0$ and, therefore,
$k_{||}^{+}\rightarrow+\infty$). By working in the regime where $\omega_{\rm
r1}<\omega_{\rm m}/2<\omega_{\rm r2}$, we can ensure that the momentum bandgap
is semi-infinite (here, $\omega_{\rm r2}=1/\sqrt{LC_{0}(1-m/2)}$). For larger
modulation frequencies, the bandgap becomes closed. In Fig. 3(b), we plot a
band diagram for a time-modulated impedance surface for the case of $m=0.2$
and $\omega_{\rm m}/2=\omega_{\rm r0}=1/\sqrt{LC_{0}}$ (lower panel) together
with the corresponding dispersion relations for stationary surfaces (upper
panel). As expected, the bandgap size can be determined by the two dispersion
curves of stationary surfaces with $C_{1}=1.1C_{0}$ and $C_{2}=0.9C_{0}$, in
particular, by the points where they cross with the horizontal dashed line at
$\omega_{\rm r0}$. As is seen, the bandgap size extends from $k_{||}\approx
4k_{\rm r0}$ ($k_{\rm r0}=\omega_{\rm r0}/c_{0}$) to infinity.

Next, we verify with full-wave simulations (see Methods) the wave evolution
inside the momentum bandgap. A stationary capacitive surface is excited from
the left by a surface wave with momentum $k_{||}=5k_{\rm r0}$ (see Fig. 3(c)).
From the time moment $t=0$, the surface capacitance is harmonically modulated
in time, resulting in the exponential growth of the surface mode and the
higher-order frequency harmonics (some of them being inside the light cone are
propagating). This growth is seen in the magnetic-field snapshot taken at
$t=30T_{\rm m}(T_{\rm m}=2\pi/\omega_{\rm m})$ depicted in the lower panel of
Fig. 3(c). Due to the temporal modulation, the momentum of the eigenwaves
remains unchanged. However, the modulation generates backward and forward
harmonics with equal amplitudes, resulting in a standing wave along the
horizontal direction Wang  _et al._ (2023). The complete field evolution
animation is available in movie S1. For comparison, we also analyze the case
when the boundary is excited by a surface wave with wavenumber
$k_{||}=10k_{\rm r0}$ (Fig. 3(d)). This wave experiences an even faster
exponential growth since the imaginary part of the frequency $\Im(\omega)$
inside the bandgap is larger in this case (see the lower panel of Fig. 3(b)).
In Fig. 3(e), we plot the simulated magnetic-field evolution in time (blue
curve) for the latter case. The exponential growth of the field amplitude in
time coincides with the rate predicted by the theory.

It is important to analyze how the momentum bandgap of the modulated
metasurface depends on its quality factor. For that, we reduce the inductance
and increase the capacitance at the same rate to preserve the resonance
frequency of the stationary metasurface. The quality factor of the stationary
metasurface can be qualitatively described as a quality factor of an $RLC$
circuit (where $R=\sqrt{\mu_{0}/\varepsilon_{0}}$ is the free-space
characteristic impedance which is connected in parallel to the metasurface
equivalent circuit), that is, $Q=\omega_{\rm
r0}C_{0}\sqrt{\mu_{0}/\varepsilon_{0}}$ (Pozar, 2011, Sec. 6.1). For each real
value of $k_{||}$, we fix $\Re(\omega)=\omega_{\rm m}/2$ and calculate for the
imaginary part of the frequency $\Im(\omega)$ that indicates the presence or
absence of the bandgap. Figure 3(f) plots $\Im(\omega)$ for three metasurfaces
with different quality factors. One can see that metasurfaces with a higher
Q-factor provide wider momentum bandgaps for surface waves (s.w.) with larger
amplification rates assuming the same modulation function. In comparison, the
metasurface for which the results are shown in Figs. 3(b)–(e) has $Q=2.44$.
Moreover, for sufficiently large Q-factors ($Q\geq 9.75$), a second momentum
bandgap opens inside the light cone, i.e., for propagating waves (p.w.). Such
a regime of the open bandgap for propagating modes is inherent to metasurfaces
with high quality factors and cannot occur in non-resonant metasurfaces like
those in Ref. Wang  _et al._ (2023). The size of the second bandgap grows with
the quality factor of the metasurface since resonances with longer lifetimes
suffer from smaller radiation loss and need a weaker modulation to maintain
the same amplification rate. It should be noted that the amplification of
surface and oblique propagating waves in the metasurface is independent of the
phase of the incident light in sharp contrast to conventional parametric
degenerate amplifiers Wang  _et al._ (2023).

Figure 4: (a) A representative design of a photonic time crystal based on an
optical time-varying resonant metasurface made of dielectric spheres. The
purple arrow indicates a surface eigenmode whose amplitude grows in time due
to its wavenumber being inside the momentum bandgap of the metasurface. (b)
Band structure of a time-invariant metasurface. The color denotes the lowest
singular value $S_{\mathrm{min}}$ of the matrix in Eq. (5). The orange dotted
lines represent the light lines. (c) The two lowest-order Mie coefficients of
an isolated time-invariant sphere. The vertical scale is the same as in (b).
(d) Band structure of a time-varying metasurface with modulation frequency
$\omega_{\mathrm{m1}}$ (non-resonant case). (e) Zoomed band structure in the
green highlighted region of (d). (f) The corresponding imaginary part of the
frequency for fixed $\Re(\omega)=\omega_{\mathrm{m1}}/2$. (g)–(i) Original and
zoomed band structures of a time-varying metasurface with modulation frequency
$\omega_{\mathrm{m2}}$ (resonant case).

When the quality factor takes sufficiently large values, the two bandgaps
merge, and the metasurface can amplify incident waves with all possible
momenta $k_{||}$ (see the curve for $Q=122$ in Fig. 3(f)). To demonstrate this
infinite momentum bandgap, we place a dipole emitter above the metasurface
(Fig. 3(g)). The dipole radiation includes a wide spectrum of momenta, as
shown in the upper panel of the figure. Once the temporal modulation of the
metasurface is on, waves with all different momenta are amplified and radiated
in the specular and retro-directions with respect to the source, see the lower
panel in Fig. 3(g). This leads to interesting possibilities such as amplified
emission and lasing of light from a radiation source Lyubarov  _et al._
(2022). In contrast to the idea suggested in Lyubarov  _et al._ (2022), due to
the infinite bandgap, here it is possible to amplify emission with a large
and, in principle, tunable spectrum of wavenumbers. This provides
opportunities for beam shaping of the amplified signal and for creating
perfect lenses (evanescent content of the source radiation can be
reconstructed effectively Pendry (2000) thanks to the amplification of all
$k_{||}$).

###  II.3 Optical implementation

To provide a feasible optical realization of the resonant photonic time
crystal, we consider a penetrable metasurface consisting of dielectric spheres
with a time-varying permittivity (see Fig. 4(a)). For simplicity, we consider
the metasurface being surrounded by air. Here, each sphere effectively behaves
as an $LC$ resonator as it supports Mie resonances Rahimzadegan  _et al._
(2020). The spheres are arranged in an infinite square lattice with the period
$a$. The radius of each sphere is $R$. For simplicity, we ignore material
dispersion. As we verified numerically, it does not significantly modify the
bandgap width. Therefore, the permittivity of each sphere reads
$\varepsilon(t)=1+\chi_{0}[1+m\mathrm{cos}(\omega_{\mathrm{m}}t)]$ (Mirmoosa
_et al._ , 2022, Sec. 4.3). Varying the permittivity in time effectively
modulates the Mie resonance frequencies of the spheres (see Fig. 2(b)). Such a
metasurface corresponds to a (2+1)-dimensional spatiotemporal crystal. In the
following, we rely on the T-matrix method to study the optical response from
such a metasurface Garg  _et al._ (2022) (see Sec. 5 of the Supplementary
Material sup  for more details). Furthermore, the calculation method for band
structures using the T-matrix method has been described in the Materials and
Methods Section. Hereafter, the radius of the spheres $R$ is fixed at $810\
\mathrm{nm}$, and the lattice period $a$ is set to $3R$. The material
susceptibility is chosen $\chi_{0}=10.68$, corresponding to silicon which is
nearly lossless and dispersionless in the considered infrared frequency
regime.

First, using Eq. (5), we calculate the band structure of a time-invariant
metasurface substituting $m=0$ and $\omega_{\mathrm{m}}=0$ (Fig. 4(b)). One
can see flat bands that occur near the $M$ and $X$ points of the spatial
Brillouin zone due to the Mie resonances of the single sphere. For comparison,
we plot the absolute values of the dipolar Mie coefficients of a time-
invariant single sphere in Fig. 4(c) Rahimzadegan  _et al._ (2020). From Fig.
4(c), we observe the occurrence of a magnetic dipolar and an electric dipolar
resonance that explain the existence of flat bands in Fig. 4(b). As will be
shown below, exploiting these resonances dramatically reduces the modulation
strength $m$ for attaining wide momentum bandgaps. Note that the band
structure shown hereafter accommodates the eigenmodes for both TM and
transverse electric (TE) polarizations, however, in what follows, we optimize
the metasurfaces for TE-waves.

Figure 5: (a) Optical time-varying metasurface designed to exhibit a momentum
bandgap for propagating waves. The purple arrow indicates a propagating
eigenmode whose amplitude is growing in time. The geometry of the metasurface
is the same as in Fig. 4(a). (b) Band structure of a time-varying metasurface
with modulation frequency $\omega_{\mathrm{m3}}$. (c) Zoomed band structure in
the green highlighted region in (b). (d) The corresponding imaginary part of
the frequency for fixed $\Re(\omega)=\omega_{\mathrm{m3}}/2$.

Next, we plot the band structures of time-varying metasurfaces. We start by
choosing the modulation frequency to be $\omega_{\mathrm{m1}}=300$ THz as
shown in Fig. 4(b). Since $\frac{\omega_{\mathrm{m1}}}{2}$ is away from the
flat bands, this configuration corresponds to a scenario of modulating a non-
resonant metasurface. Moreover, we observe that the dispersion around
$\omega=\frac{\omega_{\mathrm{m1}}}{2}$ is linear [see Fig. 4(b)]. This is
because the metasurface is spatially homogenizable near and below
$\omega=\frac{\omega_{\mathrm{m1}}}{2}$ Venkitakrishnan  _et al._ (2021).
Therefore, the time-varying metasurface with ${\omega_{\mathrm{m1}}}$ mimics
the properties of a conventional (space-uniform) photonic time crystal with
the modulated plasma frequency. In Fig. 4(d), we show the band structure of
this nonresonant metasurface. We choose $m=0.01$ that corresponds to the
relative change of the refractive index $\Delta n/n\approx
m\chi_{0}/(1+\chi_{0})=1\%$, which is the maximum attainable value for the
nonlinear effects in low-loss materials with realistic pump powers Hayran  _et
al._ (2022). We observe the folding of the band structure of the time-
invariant case Joannopoulos  _et al._ (2008); Zurita-Sánchez  _et al._ (2009).
At the band crossing in the $\Gamma-M$ region, the momentum bandgap appears,
as is seen in the zoomed region (Fig. 4(e)). For plotting the band structures,
we limit ourselves only to the dominant dipolar moments in the vector
spherical expansion and to the three dominant frequency harmonics (see Sec. 5
in Supplementary Material sup ). While this assumption does not modify the
response of the metasurface inside the momentum bandgap, it allows to
eliminate side-bands in the band structure. The calculated relative gap width
is very narrow, $\Delta k_{||}=0.0183\%$. We also plot the imaginary part of
the frequency $\Im(\omega)$ for a fixed
$\Re(\omega)=\frac{\omega_{\mathrm{m1}}}{2}$ (see Fig. 4(f)). As expected, in
the middle of the bandgap, $\Im(\omega)$ reaches the largest absolute value.
Since the bandgap lies below the light cone [see Fig. 4(b)], only surface-wave
incident excitations couple to the modes inside it.

In sharp contrast to the previous case, when half of the modulation frequency
is at the position of the magnetic Mie resonance of the spheres (see the
horizontal line $0.5{\omega_{\mathrm{m2}}}=317$ THz in Fig. 4(b)), the bandgap
greatly expands. Keeping for a fair comparison the same modulation strength
$m=0.01$, the relative bandgap size of this resonant metasurface reaches the
value of $\Delta k_{||}=6.59\%$, as is calculated from the band structure in
Figs. 4(g)–(i). Remarkably, the resonance of the metasurface leads to the
widening of the bandgap by a factor of about $350$ as compared to the non-
resonant case. It should be noted that the correspondence between a time-
varying system and two stationary systems with scaled parameters also holds
for this metasurface (see Sec. 7 in the Supplementary Material sup ). It is
essential to note from Figs. 4(g) that the momentum bandgap is not complete
since there exist bands at the $\Re(\omega)>\omega_{\mathrm{m2}}/2$ within the
bandgap. Nevertheless, in contrast to spatial photonic crystals with their
energy bandgaps, the modes inside an incomplete momentum bandgap are always
dominant due to their amplifying nature Lustig  _et al._ (2018). The
comparison of Figs. 4(f) and (i) reveals that the structural resonance not
just enhances the bandgap size, but the amplification rate also greatly
increases: $\Im{(\omega)}$ is an order of magnitude larger for the resonant
case.

While in the above-considered scenarios, the bandgaps occurred below the light
cone, we designed as next a metasurface to generate bandgaps for propagating
waves (see illustration in Fig. 5(a)). From Fig. 4(b) for a stationary
metasurface, we observe the existence of a flat band within the light cone at
the spectral location $\omega=\omega_{\mathrm{m3}}/2=351$ THz around the
$\Gamma$-point. Note that because the modes in the flat band lie within the
light cone, they correspond to a leaky zone in the band structure. In
literature, such modes are termed as guided resonances Fan and Joannopoulos
(2002). By modulating the metasurface periodically at the modulation frequency
$\omega_{\mathrm{m3}}$, we obtain the band structure shown in Fig. 5(b)
($m=0.06$ in this case). The zoomed region of the band structure with the
momentum bandgap is plotted in Figs. 5(c)–(d). Remarkably, despite the small
modulation strength, the momentum bandgap is very large and spans over a wide
range of incident angles: up to $54^{\circ}$ in the incidence plane parallel
to $\Gamma-M$ and $33^{\circ}$ parallel to $\Gamma-X$. Due to the almost
constant $\Im{(\omega)}$ inside the bandgap (Fig. 5(d)), the waves that couple
to the eigenmodes in the bandgap get amplified at nearly the same rates
irrespective of the value of $\mathbf{k_{||}}$. Importantly, as mentioned in
the previous Section, the bandgap for propagating waves is unique to resonant
metasurfaces and does not occur for continuous material slabs. We explain the
fact that the bandgap in Fig. 5(b) does not span all possible
$\mathbf{k}_{||}$ due to relatively low quality factor of magnetic Mie
resonances in the spheres as predicted in the LC circuit analysis and limited
value of $m$. We foresee that the infinite-bandgap regime as in Figs. 3(f)–(g)
can be achieved in the optical metasurface by tuning $\omega_{\rm m}/2$ to the
frequency of the higher-order Mie resonances of the spheres.

##  III Discussion

The pivotal challenge in realizing photonic time crystals in the optical
domain lies in generating a substantial and detectable momentum bandgap under
current modulation limitations. In this work, we introduce a novel and
pragmatic platform based on structured materials to overcome this problem,
achieving a significant momentum bandgap under realistic pump conditions. In
particular, our findings underscore that by exploiting structural resonances
in material, we can achieve a great enhancement of the momentum bandgap size
(350 times wider compared with the same material without spatial structuring)
with the modulation strength as small as 1%. In principle, a strong resonance
has the potential to extend the momentum bandgap infinitely. A distinctive
feature of our approach is that the achieved momentum bandgap can cover the
entire $k$-space, encompassing both free-space propagating modes and surface
modes. This fundamentally provides new physics compared to the implementations
based on bulk media Lustig  _et al._ (2018); Hayran  _et al._ (2022) that only
support propagating modes and non-resonant metasurfaces Wang  _et al._ (2023)
that operate exclusively with surface modes. From this point of view, our
approach holds novel opportunities for designing more complex photonic time
and space-time crystals as well as for amplifying the spontaneous emission of
light from emitters located near the structure. Moreover, the designed crystal
could be useful for designing a perfect lens, a long-standing goal in optics,
since the information of an object carried by evanescent modes can be
effectively amplified, resulting in a highly-resolved image of the object.
While our example metasurface operates in the infrared spectrum, it can be
extended to the visible spectrum with other materials. In addition, the shape
of the meta-atoms is not restricted to spherical, and they can be deployed
over a substrate. We anticipate and encourage experimental endeavors to
facilitate the proposed approach.

##  IV Materials and methods

###  IV.1 Calculation of eigenmodes of a metasurface with time-varying spheres

Eigenmodes of a scattering structure are self-standing modes that exist
without incident excitation. We use the T-matrix method to evaluate the
eigenmodes of the time-varying metasurface. For details on the method, see
Sec. 5 in Supplementary Material sup . Combining Eqs.(S29) and (S30), we write

$\begin{split}\mathbf{A}^{\mathrm{sca}}=\left(\mathbf{\hat{U}}-\mathbf{\hat{T}^{\mathrm{(s)}}(\omega)}\cdot\sum_{\mathbf{R^{\prime}}\neq\mathbf{0}}\mathbf{\hat{C}}^{(3)}(-\mathbf{R^{\prime}})\hskip
1.0pt\mathrm{e}^{-j\mathbf{k_{{||}}}\cdot\mathbf{R^{\prime}}}\right)^{-1}\\\
\cdot\mathbf{\hat{T}^{\mathrm{(s)}}}(\omega)\cdot\mathbf{A}^{\mathrm{inc}}\,.\end{split}$
(3)

Next, we use $\mathbf{A}^{\mathrm{inc}}=0$ in Eq. (3). Therefore, we can
rewrite Eq. (3) as

$\displaystyle\left(\mathbf{\hat{U}}-\mathbf{\hat{T}^{\mathrm{(s)}}(\omega)}\cdot\sum_{\mathbf{R^{\prime}}\neq\mathbf{0}}\mathbf{\hat{C}}^{(3)}(-\mathbf{R^{\prime}})\hskip
1.0pt\mathrm{e}^{-j\mathbf{k_{{||}}}\cdot\mathbf{R^{\prime}}}\right)\cdot\mathbf{A}^{\mathrm{sca}}=0\,.$
(4)

Finally, for Eq. (4) to have a non-trivial solution i.e.,
$\mathbf{A}^{\mathrm{sca}}\neq 0$, we arrive at the condition

$\left|\mathbf{\hat{U}}-\mathbf{\hat{T}^{\mathrm{(s)}}(\omega)}\cdot\sum_{\mathbf{R^{\prime}}\neq\mathbf{0}}\mathbf{\hat{C}}^{(3)}(-\mathbf{R^{\prime}})\hskip
1.0pt\mathrm{e}^{-j\mathbf{k_{{||}}}\cdot\mathbf{R^{\prime}}}\right|=0\,.$ (5)

Here, the values of $\omega$ and $\mathbf{k_{||}}$ for which Eq. (5) is
satisfied correspond to the location of the eigenmodes of the time-varying
metasurface. Note that instead of calculating the determinant $D$, we minimize
the lowest singular value $S_{\mathrm{min}}$ of the matrix in Eq. (5) to
identify the eigenmodes of the system. The metric of the $S_{\mathrm{min}}$ is
advantageous over that of the determinant (see Sec. 6 in Supplementary
Material sup  for more details).

## Acknowledgments

###### Acknowledgements.

X.W. and C.R. acknowledge support by the Helmholtz Association via the
Helmholtz program “Materials Systems Engineering” (MSE). P.G., A.G.L., and
C.R. are part of the Max Planck School of Photonics, supported by the
Bundesministerium für Bildung und Forschung, the Max Planck Society, and the
Fraunhofer Society. P.G. acknowledges support from the Karlsruhe School of
Optics and Photonics (KSOP). P.G. and C.R. acknowledge support by the German
Research Foundation within the SFB 1173 (Project-ID No. 258734477). The
authors would like to thank Prof. Sergei Tretyakov for the fruitful
discussions on the application of the designed photonic time crystals for
constructing a perfect lens.

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# Almost Ricci-Yamabe solitons on Almost Kenmotsu Manifolds

Mohan Khatri  Mohan Khatri
Department of Mathematics and Computer Science
Mizoram University
Aizawl-796004, India<EMAIL_ADDRESS>and  Jay Prakash Singh
Jay Prakash Singh
Department of Mathematics and Computer Science
Mizoram University
Aizawl-796004, India<EMAIL_ADDRESS>
###### Abstract.

This manuscript examines almost Kenmotsu manifolds (briefly, AKMs) endowed
with the almost Ricci-Yamabe solitons (ARYSs) and gradient ARYSs. The
condition for an AKM with ARYS to be $\eta$-Einstein is established. We also
show that an ARYS on Kenmotsu manifold becomes Ricci-Yamabe soliton under
certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional
$(\kappa,\mu)^{\prime}$-AKM equipped with a gradient ARYS is either locally
isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ or the Reeb vector
field and the soliton vector field are codirectional. The properties of
$3$-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.

###### Key words and phrases:

Almost Ricci-Yamabe soliton, Kenmotsu manifold, Almost Kenmotsu manifold,
Ricci soliton, Yamabe soliton

###### 2020 Mathematics Subject Classification:

Primary 53C20, 53C25, 53C15

††volume-info: Volume , Number 0, ††copyright: ©: Korean Mathematical Society

##  1\. Introduction

The Ricci flow is a well-known geometric flow, introduced by Hamilton [16] in
80’s, and it is used to prove the three-dimensional sphere theorem [17]. The
Ricci flow plays a crucial role in the proof of Poincaré and Thurston’s
conjectures. The Ricci soliton is a special self-similar solution of the
Hamilton’s Ricci flow: $\frac{\partial}{\partial t}g(t)=-2S(t),$ with initial
condition $g(0)=g_{0}$. A Ricci soliton $(g,V,\lambda)$ on an $m$-dimensional
Riemannian manifold $(M,g)$ is defined by

$\displaystyle(\mathcal{L}_{V}g)(\Omega_{1},\Omega_{2})+2S(\Omega_{1},\Omega_{2})=2\lambda
g(\Omega_{1},\Omega_{2}),$

for any vector fields $\Omega_{1},\Omega_{2}$ on $M$. Here, $\mathcal{L}_{V}g$
denotes the Lie-derivative of $g$ along a potential vector field $V$, $S$ is
the Ricci tensor of $M$ and $\lambda$, real constant (soliton constant). It is
said to be shrinking, steady and expanding accordingly as $\lambda$ is
positive, zero and negative, respectively. When the vector field $V$ is zero,
it is said to be trivial and when $V$ is the gradient of a smooth function $f$
on $M$, that is, $V=\nabla f$, where $\nabla$ is the covarient operator of
$g$, then we say that the Ricci soliton reduces to the gradient Ricci soliton.
For details see [6, 5, 26].

Similar to Ricci flow, Hamilton introduced Yamabe flow to tackle the Yamabe’s
problem on manifolds of positive conformal Yamabe invariant. The Yamabe
soliton is a self-similar solution to the Yamabe flow. On a Riemannian
manifold $(M,g)$, a Yamabe soliton is given by

$\displaystyle(\mathcal{L}_{V}g)(\Omega_{1},\Omega_{2})=2(r-\lambda)g(\Omega_{1},\Omega_{2})$

for any vector fields $\Omega_{1},\Omega_{2}$ on $M$. Here, $r$ is the scalar
curvature of the manifold and $\lambda$, real constant. The Yamabe soliton
preserves the conformal class of the metric but the Ricci soliton does not in
general. Moreover, in dimension $n=2$, both the solitons are similar. If
$\lambda$ is a smooth function on $M$, then the Ricci solitons and the Yamabe
solitons are called almost Ricci solitons [27] and almost Yamabe solitons [2],
respectively.

In [15], Güler and Crasmareanu gave the notion of Ricci-Yamabe flow. A Ricci-
Yamabe soliton (RYS) of type $(\alpha,\beta)$ on $M$ emerges as the limit of
the solution of Ricci-Yamabe flow, and it is goverened by the following
equation:

(1) $\displaystyle\mathcal{L}_{V}g+2\alpha S=(2\lambda-\beta r)g,$

where $\lambda,\alpha,\beta\in\mathbb{R}$. If the soliton vector $V$ is a
gradient of some smooth function $f$ on $M$, then (1) becomes gradient RYS
equation, that is,

(2) $\displaystyle\nabla^{2}f+\alpha S=(\lambda-\frac{1}{2}\beta r)g,$

where $\nabla^{2}f$ is the Hessian of $f$. An RYS is said to be expanding,
shrinking or steady if $\lambda$ is negative, positive or zero, respectively.
Particularly, $(1,0)$, $(0,1)$, $(1,-1)$ and $(1,-2\rho)$-type RYSs are Ricci
solitons, Yamabe solitons, Einstein solitons and $\rho$-Einstein solitons,
respectively. This infers that the notion of RYS is a natural generalization
of a large class of soliton-like equations. If we replace $\lambda$ in
equation (1) by a smooth function, then we say that $(M,g)$ is an ARYS.

Recently, in [8], the author studied RYSs on almost Kenmotsu manifolds. He
showed that a $(k,\mu)^{\prime}$-AKM admitting a RYS or gradient RYS is
locally isometric to a product manifold. Siddiqi and Akyol [29] have
introduced the notion of $\eta$-RYS and established the geometrical bearing on
Riemannian submersions in terms of $\eta$-RYS with the potential field. Yoldaş
[35] studied the geometrical properties of Kenmotsu manifolds with
$\eta$-RYSs. In [18, 19], authors established several results of RYSs within
the framework of indefinite Kenmotsu manifolds and Riemannian manifolds,
respectively. In [30], authors explored the applications of RYS in perfect
fluid spacetimes. Singh and Khatri analyzed ARYS in almost contact metric
manifolds and obtained several results. The last authors also analysed ARYS
with certain vector fields [23]. In this sequel, we study ARYSs and gradient
ARYSs within the framework of AKMs.

The structure of this manuscript is as follows: In Section 2 is preliminaries.
Normal AKMs endowed with ARYSs and gradient ARYSs are discussed in Section 3.
It is shown that a Kenmotsu manifold with ARYS is $\eta$-Einstein. Also, we
prove that an ARYS in Kenmotsu manifold reduces to an RYS, provided
$Hess_{\lambda}(\zeta,\zeta)$ is invariant along $\zeta$. In Section 4, we
prove that if a $(\kappa,\mu)^{\prime}$-AKM with $h^{\prime}\neq 0$ admits a
gradient ARYS, then either $M$ is locally isometric to
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ or the potential vector field is
pointwise collinear with the Reeb vector field. In Section 5, we investigate
gradient ARYS in 3-dimensional non-Kenmotsu AKM.

##  2\. Preliminaries

A smooth manifold $M^{2n+1}$ (dim$M=2n+1$) satisfying equations

(3)
$\displaystyle\varphi^{2}\Omega_{1}=-\Omega_{1}+\eta(\Omega_{1})\zeta,~{}~{}~{}~{}~{}~{}\eta(\Omega_{1})=g(\Omega_{1},\zeta),\forall\,\,\Omega_{1}\in\mathfrak{X}(M),$

for a $(1,1)$-tensor field $\varphi$, a unit vector field $\zeta$ (called the
Reeb vector field) and a 1-form $\eta$, is termed as an almost contact
manifold [4]. Here $\mathfrak{X}(M)$ represents the collection of smooth
vector fields of $M^{2n+1}$. If the Riemannian metric $g$ of $M^{2n+1}$
satisfies

(4)
$g(\varphi\Omega_{1},\varphi\Omega_{2})=g(\Omega_{1},\Omega_{2})-\eta(\Omega_{1})\eta(\Omega_{2}),\,\,g(\Omega_{1},\zeta)=\eta(\Omega_{1}),\forall\,\,\Omega_{1},\Omega_{2}\in\mathfrak{X}(M),$

then $M^{2n+1}$ is said to be an almost contact metric manifold (ACMM) and the
structure $(\phi,\zeta,\eta,g)$ on $M^{2n+1}$ is called as an almost contact
metric structure [4].

An ACMM with $d\eta=0$ and $\Phi=2\eta\wedge\Phi$ is defined as an AKM, where
the fundamental 2-form $\Phi$ of ACMM is defined by
$\Phi(\Omega_{1},\Omega_{2})=g(\Omega_{1},\varphi\Omega_{2})$ and $d$ stands
for exterior derivative [20]. On the product $M^{2n+1}\times\mathbb{R}$ of an
ACMM $M^{2n+1}$ and $\mathbb{R}$, there exists an almost complex structure $J$
defined by
$J\big{(}\Omega_{1},\mathcal{F}\frac{d}{dt}\big{)}=\big{(}\varphi\Omega_{1}-\mathcal{F}\zeta,\eta(\Omega_{1})\frac{d}{dt}\big{)},$
where $\Omega_{1}$ denotes a vector field tangent to $M^{2n+1}$, $t$ is the
coordinate of $\mathbb{R}$ and $\mathcal{F}$ is a $C^{\infty}$-function on
$M^{2n+1}\times\mathbb{R}$. If $J$ is integrable, then $(\phi,\zeta,\eta,g)$
on $M^{2n+1}$ is said to be normal. A normal ACMM is called a Kenmotsu
manifold [22]. An ACMM is a Kenmotsu manifold if and only if

$(\nabla_{\Omega_{1}}\varphi)\Omega_{2}=g(\varphi\Omega_{1},\Omega_{2})\zeta-\eta(\Omega_{2})\varphi\Omega_{1},$

which infers that

(5) $\displaystyle\nabla_{\Omega_{1}}\zeta=\Omega_{1}-\eta(\Omega_{1})\zeta,$
(6) $\displaystyle
R(\Omega_{1},\Omega_{2})\zeta=\eta(\Omega_{1})\Omega_{2}-\eta(\Omega_{2})\Omega_{1},$
(7) $\displaystyle Q\zeta=-2n\zeta,$

Here $R$ is the curvature tensor of $g$ and $Q$ the Ricci operator associated
with the Ricci tensor $S$ as
$S(\Omega_{1},\Omega_{2})=g(Q\Omega_{1},\Omega_{2})$. It is shown that a
Kenmotsu manifold is locally a warped product $I\times_{f}N^{2n}$, where $I$
is an open interval with coordinate $t$, $f=ce^{t}$ is the warping function
for some positive constant $c$ and $N^{2n}$ is a K$\ddot{a}$hlerian manifold
[22].

On an AKM the following formula is valid [9, 10]:

(8) $\displaystyle\nabla_{\Omega_{1}}\zeta=-\varphi^{2}\Omega_{1}-\varphi
h\Omega_{1},,\forall\,\,\Omega_{1}\in\mathfrak{X}(M).$

Let define operators $h$ and $\ell$ as:

$h=\frac{1}{2}\mathcal{L}_{\zeta}\varphi,\,\,\ell=R(\cdot,\zeta)\zeta.$

Then we have $h\zeta=h^{\prime}\zeta=0$,
$Tr(h)=Tr(h^{\prime})=0,~{}~{}~{}h\varphi=-\varphi h$, where
$h^{\prime}=h\cdot\varphi$ and $Tr$ denotes trace.

##  3\. Normal almost Kenmotsu manifold

In this section, we deal with normal AKM, that is, Kenmotsu manifold admitting
ARYS and gradient ARYS. Firstly, we give some examples of gradient ARYS.

######  Example 3.1.

Let $(N,J,g_{0})$ be a K$\ddot{a}$hler manifold of dimension $2n$. Consider
the warped product
$(M,g)=(\mathbb{R}\times_{\sigma}N,dt^{2}+\sigma^{2}g_{0})$, where $t$ is the
coordinate on $\mathbb{R}$. We set $\eta=dt$, $\zeta=\frac{\partial}{\partial
t}$ and (1,1) tensor field $\varphi$ by $\varphi\Omega_{1}=J\Omega_{1}$ for
vector field $\Omega_{1}$ on $N$ and $\varphi\Omega_{1}=0$ if $\Omega_{1}$ is
tangent to $\mathbb{R}$. The above warped product with the structure
$(\varphi,\zeta,\eta,g)$ is a Kenmotsu manifold [22]. In particular, if we
take $N=\mathbb{CH}^{2n}$, then $N$ being Einstein, the Ricci tensor of $M$
becomes $S^{M}=-2ng$. Then it is easy to verify that $(M,f,g,\lambda)$ is an
ARYS for $f(x,t)=ke^{t},k>0$ and $\lambda(x,t)=-2n\alpha-n\beta(2n+1)+ke^{t}$.

Therefore, a large number of examples can be constructed by considering
different potential functions $f$ on warped product spaces. Next, we
constructed an example by using Kanai’s result [21].

######  Example 3.2.

Let $N^{2n}$ be a complete Einstein K$\ddot{a}$hler manifold with
$S^{N}=-(2n-1)g_{0}$. Now consider the warped product
$M^{2n+1}=\mathbb{R}\times_{cosht}N^{2n}$ with the metric
$g=dt^{2}+(cosht)^{2}g_{0}$. Then by using result by Kanai [21], there exists
a function $f$ on $M$ without critical points satisfying $\nabla^{2}f=-fg$.
Then it is easy to see that $(M,g,\nabla f,\lambda)$ is an ARYS for
$\lambda=-2n\alpha-f-n\beta(2n+1)$.

Ghosh [11] initiated the study of Ricci soliton in Kenmotsu 3-manifold. He
later studied gradient ARYS in Kenmotsu manifold and obtained Theorem 3 (see
[12]). Here, we generalized these results for ARYS and prove.

######  Theorem 3.3.

If the metric of a Kenmotsu manifold $M^{2n+1}(\varphi,\zeta,\eta,g)$ admits a
gradient ARYS with $\alpha\neq 0$, then it is $\eta$-Einstein. Moreover, if
$\zeta$ leaves the scalar curvature invariant then $\lambda$ can be expressed
locally as $\lambda=Acosht+Bsinht-2n\alpha-n\beta(2n+1)$, where $A,B$ are
constants on $M$.

###### Proof.

Suppose the metric $g$ of Kenmtosu manifold admits gradient RYS, then from (2)
we have

(9) $\displaystyle\nabla_{\Omega_{1}}Df=\sigma\Omega_{1}-\alpha Q\Omega_{1},$

for any vector field $\Omega_{1}$ on $M$ and $\sigma=\lambda-\frac{\beta
r}{2}$ is a smooth function on $M$.
Taking an inner product of (9) along arbitrary vector field $\Omega_{2}$, we
obtain:

(10)
$\displaystyle\nabla_{\Omega_{2}}\nabla_{\Omega_{1}}Df=(\Omega_{2}\sigma)\Omega_{1}+\sigma(\nabla_{\Omega_{2}}\Omega_{1})-\alpha(\nabla_{\Omega_{2}}Q)\Omega_{1}-\alpha
Q(\nabla_{\Omega_{2}}\Omega_{1}).$

Making use of (10) in the well-known formula
$R(\Omega_{1},\Omega_{2})Df=\nabla_{\Omega_{1}}\nabla_{\Omega_{2}}Df-\nabla_{\Omega_{2}}\nabla_{\Omega_{1}}Df-\nabla_{[\Omega_{1},\Omega_{2}]}Df$
yields

(11) $\displaystyle
R(\Omega_{1},\Omega_{2})Df=(\Omega_{1}\sigma)\Omega_{2}-(\Omega_{2}\sigma)\Omega_{1}-\alpha[(\nabla_{\Omega_{1}}Q)\Omega_{2}-(\nabla_{\Omega_{2}}Q)\Omega_{1}].$

Taking a covariant derivative of (7) and using (5), we get
$(\nabla_{\Omega_{1}}Q)\zeta=-2n(\Omega_{1}-\eta(\Omega_{1})\zeta)$. Because
of this in the inner product of (11) with $\zeta$ gives

(12) $\displaystyle
g(R(\Omega_{1},\Omega_{2})Df,\zeta)=(\Omega_{1}\sigma)\eta(\Omega_{2})-(\Omega_{2}\sigma)\eta(\Omega_{1}).$

Now, taking an inner product of (6) with $Df$ yields

(13) $\displaystyle
g(R(\Omega_{1},\Omega_{2})\zeta,Df)=(\Omega_{2}f)\eta(\Omega_{1})-(\Omega_{1}f)\eta(\Omega_{2}).$

Combining (12) and (13) and replacing $\Omega_{2}$ by $\zeta$ in the obtain
relations we obtain

(14) $\displaystyle d(\sigma-f)=\zeta(\sigma-f)\eta,$

where $d$ is the exterior derivative. This means that $\sigma-f$ is invariant
along the distribution $\mathcal{D}$ (i.e., $\mathcal{D}=ker\eta)$ hence
$\sigma-f$ is constant for all $\Omega_{1}\in\mathcal{D}$.
Contracting (11) infer

(15) $\displaystyle
S(\Omega_{2},Df)=-2n(\Omega_{2}\sigma)+\frac{\alpha}{2}(\Omega_{2}r),$

for any vector field $\Omega_{2}$ on $M$. Replacing $\Omega_{2}$ by $\zeta$ in
(11) and taking an inner product with $\Omega_{2}$ gives

(16) $\displaystyle
g(R(\Omega_{1},\zeta)Df,\Omega_{2})=(\Omega_{1}\sigma)\eta(\Omega_{2})-(\zeta\sigma)g(\Omega_{1},\Omega_{2})-\alpha
S(\Omega_{1},\Omega_{2})+2n\alpha g(\Omega_{1},\Omega_{2}).$

In consequence of (5) and (6) in (16) we get

(17)
$\displaystyle[(\Omega_{1}f)-(\Omega_{1}\sigma)]\eta(\Omega_{2})+\zeta(\sigma-f)g(\Omega_{1},\Omega_{2})+\alpha
S(\Omega_{1},\Omega_{2})+2n\alpha g(\Omega_{1},\Omega_{2})=0.$

Contracting (17) over $\Omega_{1}$ gives

(18) $\displaystyle 2n\zeta(\sigma-f)+\alpha[r+2n(2n+1)]=0.$

Replacing $\Omega_{2}$ by $\zeta$ in (15) and making use of (18) and (7), we
see that $\zeta r=-2(r+2n(2n+1))$, for $\alpha\neq 0$. In consequence, (18) in
(14) gives

(19) $\displaystyle d(\sigma-f)=-\alpha(\frac{r}{2n}+2n+1)\eta.$

Applying Poincare lemma and using the fact that $d\eta=0$ on (19), we obtain
$-\alpha dr\wedge\eta=0$, and making use of the value of $\zeta r$ we have

(20) $\displaystyle Dr=-2(r+2n(2n+1))\zeta.$

Taking an inner product of (19) with vector field $\Omega_{1}$, then inserting
it along with (18) in (17) we get

(21) $\displaystyle
Q\Omega_{1}=(\frac{r}{2n}+1)\Omega_{1}-(\frac{r}{2n}+2n+1)\eta(\Omega_{1})\zeta,$

for any vector field $\Omega_{1}$ on $M$. Therefore, $M$ is $\eta$-Einstein.
Suppose that $\zeta r=0$, i.e., $\zeta$ leaves the scalar curvature invariant.
In consequence of this we get $r=-2n(2n+1)$, a constant. Inserting this in
(21) implies $Q\Omega_{1}=-2n\Omega_{1}$ i.e., $M$ is Einstein. As $r$ is
constant, (19) gives $Df=D\lambda$. In consequence of this, (9) becomes

(22) $\displaystyle\nabla_{\Omega_{1}}D\lambda=(\lambda+k)\Omega_{1},$

where $k=n(2\alpha+\beta(2n+1))$. Replacing $\Omega_{1}$ by $\zeta$ and taking
inner product with $\zeta$, (22) gives $\zeta(\zeta\lambda)=\lambda+k$. But as
we know that a Kenmotsu manifold is locally isometric to the warped product
$(-\epsilon,\epsilon)\times_{ce^{t}}N$, where $N$ is a K$\ddot{a}$hler
manifold of dimension $2n$ and $(-\epsilon,\epsilon)$ is an open interval.
Using the local parametrization: $\zeta=\frac{\partial}{\partial t}$ (where
$t$ is the coordinate on $(-\epsilon,\epsilon)$) we get from (22)

$\frac{\partial^{2}\lambda}{\partial t^{2}}=\lambda+2n\alpha+n\beta(2n+1)$

Its solution can be exhibited as
$\lambda=Acosht+Bsinht-2n\alpha-n\beta(2n+1)$, where $A,B$ are constants on
$M$. This completes the proof. ∎

######  Lemma 3.4.

If the metric of a Kenmotsu manifold $M^{2n+1}(\varphi,\zeta,\eta,g)(n>1)$
admits ARYS then
1\. $\zeta(\zeta\lambda)+\zeta\lambda=2(2n\alpha+\lambda+n\beta(2n+1)).$
2\. $D\lambda=(\zeta\lambda)\zeta+\beta\\{(r+2n(2n+1))\zeta+\frac{Dr}{2}\\}.$

###### Proof.

Taking the covariant derivative of (1) along arbitrary vector field
$\Omega_{1}$, we get

(23)
$\displaystyle(\nabla_{\Omega_{1}}\mathcal{L}_{V}g)(\Omega_{2},\Omega_{3})=2(\Omega_{1}\sigma)g(\Omega_{2},\Omega_{3})-2\alpha(\nabla_{\Omega_{1}}S)(\Omega_{2},\Omega_{3}),$

where $\sigma=\lambda-\frac{\beta r}{2}$. We know the following commutative
formula (see [34]):

$\displaystyle(\mathcal{L}_{V}\nabla_{\Omega_{1}}g-\nabla_{\Omega_{1}}\mathcal{L}_{V}g-\nabla_{[V,\Omega_{1}]}g)(\Omega_{2},\Omega_{3})$
(24)
$\displaystyle=-g((\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{2}),\Omega_{3})-g((\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{3}),\Omega_{2}),$

for all vector fields $\Omega_{1},\Omega_{2},\Omega_{3}$ on $M$. Since $g$ is
parallel with respect to Levi-Civita connection $\nabla$, the above relation
becomes:

(25)
$\displaystyle(\nabla_{\Omega_{1}}\mathcal{L}_{V}g)(\Omega_{2},\Omega_{3})=g((\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{2}),\Omega_{3})+g((\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{3}),\Omega_{2}).$

We know that $\mathcal{L}_{V}\nabla$ is a symmetric tensor of type (1,2) and
so it follows from (25) that

(26) $\displaystyle
2g((\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{2}),\Omega_{3})=(\nabla_{\Omega_{1}}\mathcal{L}_{V}g)(\Omega_{2},\Omega_{3})+(\nabla_{\Omega_{2}}\mathcal{L}_{V}g)(\Omega_{3},\Omega_{1})-(\nabla_{\Omega_{3}}\mathcal{L}_{V}g)(\Omega_{1},\Omega_{2}).$

Inserting (23) in (26), then replacing $\Omega_{2}$ by $\zeta$ we obtain

(27) $\displaystyle(\mathcal{L}_{V}\nabla)(\Omega_{1},\zeta)=2\alpha
Q\Omega_{1}+(4n\alpha+\zeta\sigma)\Omega_{1}+g(\Omega_{1},D\sigma)\zeta-\eta(\Omega_{1})D\sigma.$

Taking the covariant derivative of (27) along arbitrary vector field
$\Omega_{2}$ gives

$\displaystyle(\nabla_{\Omega_{2}}\mathcal{L}_{V}\nabla)(\Omega_{1},\zeta)+(\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{2})-\eta(\Omega_{2})(\mathcal{L}_{V}\nabla)(\Omega_{1},\zeta)$
$\displaystyle=2\alpha(\nabla_{\Omega_{2}}Q)\Omega_{1}+\Omega_{2}(\zeta\sigma)\Omega_{1}+g(\Omega_{1},\nabla_{\Omega_{2}}D\sigma)\zeta$
(28)
$\displaystyle-g(\Omega_{1},D\sigma)\varphi^{2}{\Omega_{2}}+g(\Omega_{1},\varphi^{2}\Omega_{2})D\sigma-\eta(\Omega_{1})(\nabla_{\Omega_{2}}D\sigma).$

Making use of this in the formula (see [34])

$(\mathcal{L}_{V}R)(\Omega_{1},\Omega_{2})\Omega_{3}=(\nabla_{\Omega_{1}}\mathcal{L}_{V}\nabla)(\Omega_{2},\Omega_{3})-(\nabla_{\Omega_{2}}\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{3})$

we obtain

$\displaystyle(\mathcal{L}_{V}R)(\Omega_{1},\Omega_{2})\zeta=2\alpha\\{(\nabla_{\Omega_{1}}Q)\Omega_{2}-(\nabla_{\Omega_{2}}Q)\Omega_{1}\\}+\Omega_{1}(\zeta\sigma)\Omega_{2}-\Omega_{2}(\zeta\sigma)\Omega_{1}$
$\displaystyle+g(\Omega_{2},D\sigma)\Omega_{1}-g(\Omega_{1},D\sigma)\Omega_{2}+\eta(\Omega_{1})\nabla_{\Omega_{2}}D\sigma-\eta(\Omega_{2})\nabla_{\Omega_{1}}D\sigma$
(29)
$\displaystyle+2\alpha\\{\eta(\Omega_{1})Q\Omega_{2}-\eta(\Omega_{2})Q\Omega_{1}\\}+(4\alpha
n+\zeta\sigma)\\{\eta(\Omega_{1})\Omega_{2}-\eta(\Omega_{2})\Omega_{1}\\}.$

Now differentiating $\zeta\sigma=g(\zeta,D\sigma)$ along vector field
$\Omega_{1}$ and using (5) we get

(30)
$\displaystyle\Omega_{1}(\zeta\sigma)=g(\Omega_{1},D\sigma)-(\zeta\sigma)\eta(\Omega_{1})+g(\nabla_{\Omega_{1}}D\sigma,\zeta).$

Replacing $\Omega_{2}$ by $\zeta$ in (1), then inserting it in the Lie-
derivative of (6) yields

$\displaystyle(\mathcal{L}_{V}R)(\Omega_{1},\Omega_{2})\zeta+R(\Omega_{1},\Omega_{2})\mathcal{L}_{V}\zeta=g(\Omega_{1},\mathcal{L}_{V}\zeta)\Omega_{2}$
(31)
$\displaystyle-g(\Omega_{2},\mathcal{L}_{V}\zeta)\Omega_{1}+2(\sigma+2\alpha
n)\\{\eta(\Omega_{1})\Omega_{2}-\eta(\Omega_{2})\Omega_{1}\\}.$

Combining (3), (30) and (3), we obtain

$\displaystyle
g(\Omega_{1},\mathcal{L}_{V}\zeta)\Omega_{2}-g(\Omega_{2},\mathcal{L}_{V}\zeta)\Omega_{1}-R(\Omega_{1},\Omega_{2})\mathcal{L}_{V}\zeta$
$\displaystyle=2\alpha\\{(\nabla_{\Omega_{1}}Q)\Omega_{2}-(\nabla_{\Omega_{2}}Q)\Omega_{1}+\eta(\Omega_{1})Q\Omega_{2}$
$\displaystyle-\eta(\Omega_{2})Q\Omega_{1}\\}+g(\nabla_{\Omega_{1}}D\sigma,\zeta)\Omega_{2}-g(\nabla_{\Omega_{2}}D\sigma,\zeta)\Omega_{1}$
(32)
$\displaystyle+\eta(\Omega_{1})\nabla_{\Omega_{2}}D\sigma-\eta(\Omega_{2})\nabla_{\Omega_{1}}D\sigma-2\sigma\\{\eta(\Omega_{1})\Omega_{2}-\eta(\Omega_{2})\Omega_{1}\\}.$

Replacing $\Omega_{1}$ and $\Omega_{2}$ by $\varphi\Omega_{1}$ and
$\varphi\Omega_{2}$ in (3) then contracting the obtain equation and using
Lemma 4.2 (see [13]) results in

$\displaystyle
S(\Omega_{2},\mathcal{L}_{V}\zeta)+2ng(\Omega_{2},\mathcal{L}_{V}\zeta)=\alpha(\Omega_{2}r)+2\alpha(r+4n^{2}+2n)\eta(\Omega_{2})-g(\nabla_{\zeta}D\lambda,\varphi^{2}\Omega_{2}).$

Contracting (3) and combining it with the forgoing equation yields

(33) $\displaystyle
2(n-1)g(\nabla_{\zeta}D\sigma,\Omega_{2})+\zeta(\zeta\sigma)\eta(\Omega_{2})+\eta(\Omega_{2})divD\sigma=4n(2n\alpha+\sigma)\eta(\Omega_{2}).$

Replacing $\Omega_{2}$ by $\zeta$ in (33), we get
$(2n-1)\zeta(\zeta\sigma)+divD\sigma=4n(2n\alpha+\sigma)$. In view of this in
(33) infer
$g(\nabla_{\zeta}D\sigma,\Omega_{1})=\zeta(\zeta\sigma)\eta(\Omega_{1})$ for
$n>1$. Now taking $\zeta$ instead of $\Omega_{2}$ in (3) and making use of
above relations we obtain

(34)
$\displaystyle\nabla_{\Omega_{1}}D\sigma=-2(2n\alpha+\sigma)\varphi^{2}\Omega_{1}+\zeta(\zeta\sigma)\varphi^{2}\Omega_{1}+\zeta(\zeta\sigma)\eta(\Omega_{1})\zeta.$

In consequence of (34), the expression of the curvature tensor is as follows:

$\displaystyle
R(\Omega_{1},\Omega_{2})D\sigma=2(\Omega_{2}\sigma)\varphi^{2}\Omega_{1}-2(\Omega_{1}\sigma)\varphi^{2}\Omega_{2}+\Omega_{2}(\zeta(\zeta\sigma))\Omega_{1}$
(35)
$\displaystyle-\Omega_{1}(\zeta(\zeta\sigma))\Omega_{2}+2(\zeta(\zeta\sigma)-\sigma-2n\alpha)\\{\eta(\Omega_{2})\Omega_{1}-\eta(\Omega_{1})\Omega_{2}\\}.$

Replacing $\Omega_{2}$ by $\zeta$ in (3), then inserting the obtain equation
back in (3) gives

$\displaystyle
R(\Omega_{1},\Omega_{2})D\sigma=(\Omega_{1}\sigma)\Omega_{2}-(\Omega_{2}\sigma)\Omega_{1}-2\\{(\Omega_{1}\sigma)\eta(\Omega_{2})\zeta$
$\displaystyle-(\Omega_{2}\sigma)\eta(\Omega_{1})\zeta\\}+\\{\zeta(\zeta(\zeta\sigma))-\zeta\sigma\\}\\{\eta(\Omega_{2})\Omega_{1}-\eta(\Omega_{1})\Omega_{2}\\}$
(36)
$\displaystyle+2\\{\zeta(\zeta\sigma)-\sigma-2n\alpha\\}\\{\eta(\Omega_{2})\Omega_{1}-\eta(\Omega_{1})\Omega_{2}\\}.$

Replacing $\Omega_{1}$ and $\Omega_{2}$ by $\varphi\Omega_{1}$ and
$\varphi\Omega_{2}$ in (3) then contracting the obtained result yields

$\displaystyle S(\Omega_{2},D\sigma)=-2ng(\Omega_{2},D\sigma).$

In consequence of this in the contraction of (3) and further replacing
$\Omega_{2}$ by $\varphi\Omega_{2}$ in the obtained expression yields $\varphi
D\sigma=0$. Differentiating this along vector field $\Omega_{1}$ and inserting
it in (34) along with the fact that $\sigma=\lambda-\frac{\beta r}{2}$ and (5)
gives

(37)
$\displaystyle\zeta(\zeta\lambda)+\zeta\lambda=2(2n\alpha+\lambda+n\beta(2n+1)).$

This completes the proof. ∎

######  Theorem 3.5.

Let $M^{2n+1}(\varphi,\zeta,\eta,g)(n>1)$ be a Kenmotsu manifold whose metric
represents an ARYS. If $Hess_{\lambda}(\zeta,\zeta)$ is constant along Reeb
vector field then it reduces to RYS with $\lambda=-2n\alpha-n\beta(2n+1)$.

###### Proof.

By hypothesis, $Hess_{\lambda}(\zeta,\zeta)$ is constant along the Reeb vector
field $\zeta$ i.e., $\zeta(\zeta(\zeta\lambda))=0$ implies
$\zeta(\zeta\lambda)$ is constant along $\zeta$. In view of this in the
covariant derivative of first relation in Lemma (3.4) along $\zeta$, we get
$\zeta(\zeta\lambda)=2(\zeta\lambda)$. Again differentiating this along
$\zeta$ yields $\zeta\lambda=0$, that is, $\lambda$ is constant along $\zeta$.
Making use of this in first relation of Lemma (3.4) gives
$\lambda=-2n\alpha-n\beta(2n+1)$. Therefore ARYS reduces to RYS. This
completes the proof. ∎

######  Remark 3.6.

The above Theorem 3.5 is a generalization of Theorem 4.1 in Ghosh [13], where
he obtained the condition under which a Ricci almost soliton reduces to an
expanding Ricci soliton with $\lambda=-2n$. It is easy to see that for
$\alpha=1$ and $\beta=0$ Theorem 4.1 [13] can be obtained from Theorem 3.5.
Moreover, the first condition of Theorem 4.1 [13] is also true for this choice
of scalars.

A Kenmotsu manifold is said to be $\eta$-Einstein if there exists smooth
functions $a$ and $b$ such that

(38) $\displaystyle
S(\Omega_{1},\Omega_{2})=ag(\Omega_{1},\Omega_{2})+b\eta(\Omega_{1})\eta(\Omega_{2}),$

for all vector field $\Omega_{1},\Omega_{2}$ on $M$. If $b=0$, then $M$
becomes an Einstein manifold.

######  Theorem 3.7.

If the metric of an $\eta$-Einstein Kenmotsu manifold
$M^{2n+1}(\varphi,\zeta,\eta,g)(n>1)$ admits RYS with $\alpha\neq 0$ then it
is Einstein with constant scalar curvature $r=-2n(2n+1)$, provided
$2\alpha+n\beta\neq 0$.

###### Proof.

Replacing $\Omega_{2}$ by $\zeta$ in (38) and using (7), we get $a+b=-2n$.
Then contracting (38) gives $r=(2n+1)a+b$. In view of this (38) becomes

(39) $\displaystyle
S(\Omega_{1},\Omega_{2})=(\frac{r}{2n}+1)g(\Omega_{1},\Omega_{2})-(\frac{r}{2n}+2n+1)\eta(\Omega_{1})\eta(\Omega_{2}),$

for any vector field $\Omega_{1},\Omega_{2}$ on $M$. Making use of (39) in (1)
yields

(40) $\displaystyle(\mathcal{L}_{V}g)(\Omega_{2},\Omega_{3})=\\{2\lambda-\beta
r-2\alpha(\frac{r}{2n}+1)\\}g(\Omega_{2},\Omega_{3})+2\alpha\\{(2n+1)+\frac{r}{2n}\\}\eta(\Omega_{2})\eta(\Omega_{3}).$

Taking the covariant derivative of (40) along arbitrary vector field
$\Omega_{1}$ we obtain

$\displaystyle(\nabla_{\Omega_{1}}\mathcal{L}_{V}g)(\Omega_{2},\Omega_{3})=-(\frac{\alpha}{n}+\beta)(\Omega_{1}r)g(\Omega_{2},\Omega_{3})+\frac{\alpha}{n}(\Omega_{1}r)\eta(\Omega_{2})\eta(\Omega_{3})$
(41)
$\displaystyle+2\alpha(\frac{r}{2n}+2n+1)\\{g(\Omega_{1},\Omega_{2})\eta(\Omega_{3})+g(\Omega_{1},\Omega_{3})\eta(\Omega_{2})-2\eta(\Omega_{1})\eta(\Omega_{2})\eta(\Omega_{3})\\}.$

Making use of (3) in (26) yields

$\displaystyle
2(\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{2})=-(\frac{\alpha}{n}+\beta)\\{(\Omega_{1}r)\Omega_{2}+(\Omega_{2}r)\Omega_{1}-g(\Omega_{1},\Omega_{2})Dr\\}$
$\displaystyle+\frac{\alpha}{n}\\{(\Omega_{1}r)\eta(\Omega_{2})\zeta+(\Omega_{2}r)\eta(\Omega_{1})\zeta-\eta(\Omega_{1})\eta(\Omega_{2})Dr\\}$
(42)
$\displaystyle+4\alpha(2n+1+\frac{r}{2n})\\{g(\Omega_{1},\Omega_{2})\zeta-\eta(\Omega_{1})\eta(\Omega_{2})\zeta\\}.$

Setting $\Omega_{1}=\Omega_{2}=e_{i}$ where $e_{i}:i=1,2,..,2n+1$ is an
orthonormal frame in (3) and summing over $i$, we get

$\displaystyle
2\sum_{i=1}^{2n+1}\varepsilon_{i}(\mathcal{L}_{V}\nabla)(e_{i},e_{i})=-\\{\beta(1-2n)+\frac{\alpha}{n}(1-n)\\}Dr$
(43) $\displaystyle+\frac{2\alpha}{n}(\zeta
r)\zeta+8n\alpha(\frac{r}{2n}+2n+1)\zeta.$

Now taking the covariant derivative of (40) give

$\displaystyle(\nabla_{\Omega_{1}}\mathcal{L}_{V}g)(\Omega_{2},\Omega_{3})=-\beta(\Omega_{1}r)g(\Omega_{2},\Omega_{3})-2\alpha(\nabla_{\Omega_{1}}S)(\Omega_{2},\Omega_{3}),$

which on contracting yields
$2\sum_{i=1}^{2n+1}\varepsilon_{i}(\mathcal{L}_{V}\nabla)(e_{i},e_{i})=\beta(2n-1)Dr$.
In consequence of this in (3), we obtain

(44) $\displaystyle 2\alpha(n-1)Dr+2\alpha(\zeta
r)\zeta+8n\alpha(\frac{r}{2n}+2n+1)\zeta=0.$

Taking an inner product of (44) with $\zeta$, we get $\zeta r=-2(r+2n(2n+1))$
for $\alpha\neq 0$. In view of this, (44) yields $Dr=(\zeta r)\zeta$ for
$n>1$. Then, replacing $\Omega_{2}$ by $\zeta$ in (3) results in

(45) $\displaystyle
2(\mathcal{L}_{V}\nabla)(\Omega_{1},\zeta)=-\beta(\Omega_{1}r)\zeta+(\frac{\alpha}{n}+\beta)(\zeta
r)\varphi^{2}\Omega_{1}.$

Taking the covariant derivate of (45), then inserting it in Yano’s result
[34]:

$(\mathcal{L}_{V}R)(\Omega_{1},\Omega_{2})\Omega_{3}=(\nabla_{\Omega_{1}}\mathcal{L}_{V}\nabla)(\Omega_{2},\Omega_{3})-(\nabla_{\Omega_{2}}\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{3})$

, for $\Omega_{3}=\zeta$ yields

$\displaystyle
2(\mathcal{L}_{V}R)(\Omega_{1},\Omega_{2})\zeta=-\beta\\{\Omega_{1}(\Omega_{2}r)\zeta-\Omega_{2}(\Omega_{1}r)\zeta\\}-\beta\\{(\Omega_{1}r)\varphi^{2}\Omega_{2}-(\Omega_{2}r)\varphi^{2}\Omega_{1}\\}$
(46) $\displaystyle+(\frac{\alpha}{n}+\beta)\\{\Omega_{1}(\zeta
r)\varphi^{2}\Omega_{2}-\Omega_{2}(\zeta r)\varphi^{2}\Omega_{1}+2(\zeta
r)\\{\eta(\Omega_{2})\Omega_{1}-\eta(\Omega_{1})\Omega_{2}\\}\\}.$

Contracting (3) over $\Omega_{1}$ gives
$(\mathcal{L}_{V}S)(\Omega_{2},\zeta)=-n\beta(\zeta r)\eta(\Omega_{2})$. In
consequence of this in the Lie-derivative of (7), we obtain

$\displaystyle(\frac{r}{2n}+1)g(\Omega_{2},\mathcal{L}_{V}\zeta)-(\frac{r}{2n}+2n+1)\eta(\Omega_{2})\eta(\mathcal{L}_{V}\zeta)=$
(47) $\displaystyle n\beta(\zeta r)\eta(\Omega_{2})-2n(2\lambda-\beta
r+4\alpha n)\eta(\Omega_{2})-2ng(\Omega_{2},\mathcal{L}_{V}\zeta).$

Replacing $\Omega_{2}$ by $\zeta$ in (3) then inserting back in (3) gives
$\lambda=-2\alpha n-n\beta(2n+1)$. In view of this in (40) we get
$\eta(\mathcal{L}_{V}\zeta)=-\frac{\beta}{2}(r+2n(2n+1))$. In consequence of
this and $Dr=(\zeta r)\zeta$ in the Lie-derivative of
$S(\Omega_{1},\zeta)=-2n\eta(\Omega_{1})$ we get

(48) $\displaystyle(r+2n(2n+1))\\{2\mathcal{L}_{V}\zeta-\frac{\beta}{2}(\zeta
r)\zeta\\}=0.$

Thus we get either $r=-2n(2n+1)$ in this case $M$ is Einstein or
$\mathcal{L}_{V}\zeta=\frac{\beta}{4}(\zeta r)\zeta$. Suppose $r\neq-2n(2n+1)$
in some open set $\mathcal{O}$ on $M$. Then using (5) and (48) implies

(49) $\displaystyle\nabla_{\zeta}V=V-\eta(V)\zeta-\frac{\beta}{4}(\zeta
r)\zeta.$

Taking $\Omega_{2}=\zeta$ in the commutative formula
$(\mathcal{L}_{V}\nabla)(\Omega_{1},\Omega_{2})=\nabla_{\Omega_{1}}\nabla_{\Omega_{2}}V-\nabla_{\nabla_{\Omega_{1}}\Omega_{2}}V+R(V,\Omega_{1})\Omega_{2}$
and using (49), (48) and (45), we obtain

$\displaystyle(2\alpha+n\beta)(\zeta r)\varphi^{2}\Omega_{1}=0,$

for any vector field $\Omega_{1}$ on $\mathcal{O}$. This shows that $\zeta
r=0$, that is, $r=-2n(2n+1)$, a contradiction. This completes the proof. ∎

In particular, if we take scalar $\alpha=1$ and $\beta=0$ then in regard of
Theorem 3.7, we can state the following:

######  Corollary 3.8.

[14] If the metric of an $\eta$-Einstein Kenmotsu manifold
$M^{2n+1}(\varphi,\zeta,\eta,g),n>1$ is a Ricci soliton then it is Einstein
and the soliton is expanding.

For the case $\alpha=1$ and $\beta=-2\rho$, we can state the following:

######  Corollary 3.9.

If the metric of an $\eta$-Einstein Kenmotsu manifold
$M^{2n+1}(\varphi,\zeta,\eta,g)(n>1)$ admits $\rho$-Einstein soliton then it
is Einstein with constant scalar curvature $r=-2n(2n+1)$, provided
$\rho\neq\frac{1}{n}$.

It is known that the warped product $\mathbb{R}\times_{ce^{t}}V(k)$ where
$V(k)$ is a K$\ddot{a}$hler manifold of constant holomorphic sectional
curvature of dimension $2n$ admits Kenmotsu structure (see [22]). Moreover,
its sectional tensor is given by [3]

$\displaystyle
R(\Omega_{1},\Omega_{2})\Omega_{3}=H\\{g(\Omega_{2},\Omega_{3})\Omega_{1}-g(\Omega_{1},\Omega_{3})\Omega_{2}\\}+(H+1)\\{g(\Omega_{1},\Omega_{3})\eta(\Omega_{2})\zeta$
$\displaystyle-g(\Omega_{2},\Omega_{3})\eta(\Omega_{1})\zeta+\eta(\Omega_{1})\eta(\Omega_{3})\Omega_{2}-\eta(\Omega_{2})\eta(\Omega_{3})\Omega_{1}+g(\Omega_{1},\varphi\Omega_{3})\varphi\Omega_{2}$
$\displaystyle-g(\Omega_{2},\varphi\Omega_{3})\varphi\Omega_{1}+2g(\Omega_{1},\varphi\Omega_{2})\varphi\Omega_{3}\\}.$

Contracting the above equation we see that

$S(\Omega_{1},\Omega_{2})=2\\{(n-1)H-1\\}g(\Omega_{1},\Omega_{2})-2(n-1)(H+1)\eta(\Omega_{1})\eta(\Omega_{2}),$

that is, it is $\eta$-Einstein. Now making use of Theorem (3.7) we get $H=-1$.
Hence, we can state the following:

######  Corollary 3.10.

If the metric of the warped product $\mathbb{R}\times_{ce^{t}}V(k)$, $(n>1)$
is a RYS with $\alpha\neq 0$ then it is of constant curvature -1, provided
$2\alpha+n\beta\neq 0$.

##  4\. Non-Normal almost Kenmotsu manifolds

In this section, we give some examples of AKMs admitting ARYS and also
investigated it in $(\kappa,\mu)^{\prime}$-AKM. An AKM
$M^{2n+1}(\varphi,\zeta,\eta,g)$ is said to be a $(\kappa,\mu)^{\prime}$-AKM
if $\zeta$ belongs to the $(\kappa,\mu)$-nullity distribution, i.e.,

(50) $\displaystyle
R(\Omega_{1},\Omega_{2})\zeta=\kappa[\eta(\Omega_{2})\Omega_{1}-\eta(\Omega_{1})\Omega_{2}]+\mu[\eta(\Omega_{2})h^{\prime}\Omega_{1}-\eta(\Omega_{1})h^{\prime}\Omega_{2}],$

for all vector fields $\Omega_{1},\Omega_{2}$ on $M$, where $\kappa,\mu$ are
constants. Moreover if both $\kappa$ and $\mu$ are smooth functions in (50),
then $M$ is called a generalized $(\kappa,\mu)^{\prime}$-AKM (see [10, 25,
31]). On generalized $(\kappa,\mu)$ or $(\kappa,\mu)^{\prime}$-AKM with $h\neq
0$ (equivalently, $h^{\prime}\neq 0$), the following relations hold [10]:

(51) $\displaystyle h^{\prime
2}=(\kappa+1)\varphi^{2},~{}~{}~{}~{}~{}h^{2}=(\kappa+1)\varphi^{2},$ (52)
$\displaystyle Q\zeta=2n\kappa\zeta.$

It follows from (51) that $\kappa\leq-1$ and $\nu=\pm\sqrt{-\kappa-1}$, where
$\nu$ is an eigenvalue corresponding to eigenvector
$\Omega_{1}\in\mathcal{D}~{}~{}(\mathcal{D}=Ker(\eta))$ of $h^{\prime}$. The
equality holds if and only if $h=0$ (equivalently, $h^{\prime}=0$). Thus
$h^{\prime}\neq 0$ if and only if $\kappa<-1$. First, we give some examples of
AKM admitting ARYS.

######  Example 4.1.

Let $(N,J,\bar{g})$ be a strictly almost K$\ddot{a}$hler Einstein manifold. We
take $\eta=dt$, $\zeta=\frac{\partial}{\partial t}$ and (1,1)-tensor $\varphi$
by $\varphi\Omega_{1}=J\Omega_{1}$ for vector field $\Omega_{1}$ on $N$ and
$\varphi\Omega_{1}=0$ if $\Omega_{1}$ is tangent to $\mathbb{R}$. Then it is
known that $(M,g)=(\mathbb{R}\times_{ce^{t}}N,g_{0}+ce^{2t}\bar{g})$ together
with the structure $(\varphi,\zeta,\eta,g)$ is an AKM (see [22]). Also since
$N$ is Einstein, we see $S^{M}=-2ng$. We define a smooth function
$f(x,t)=t^{2}$ then it is easy to see that $(M,g,f,\lambda)$ is an ARYS for
$\lambda(x,t)=-2n\alpha-n\beta(2n+1)+2$.

We can also construct an example of ARYS in AKM constructed by Barbosa-Ribeiro
[2].

######  Example 4.2.

On the warped product $M=\mathbb{R}\times_{\sigma(t)}\mathbb{H}^{2n}$ consider
the metric $g=dt^{2}+\sigma^{2}(t)g_{0}$, where $g_{0}$ is the standard metric
on the hyperbolic space $\mathbb{H}^{2n}$. Then by Algere et al. [1] result,
it is easy to see that it is AKM. Let $\sigma(t)=cosht$ and $f(x,t)=sinht$
then $(M,g,f,\lambda)$ is an ARYS with $\lambda=sinht-n\beta(2n+1)-2n\alpha$.

Next, we restate some of the results obtained by Wang and Liu [31] which will
be used later in the prove of main theorems.

######  Lemma 4.3.

[31] Let $M^{2n+1}(\varphi,\zeta,\eta,g)$ be a generalized
$(\kappa,\mu)^{\prime}$-AKM with $h^{\prime}\neq 0$. For $n>1$, the Ricci
operator $Q$ of $M$ can be expressed as

$\displaystyle
Q\Omega_{1}=-2n\Omega_{1}+2n(\kappa+1)\eta(\Omega_{1})\zeta-[\mu-2(n-1)h^{\prime}]\Omega_{1},$

for any vector field $\Omega_{1}$ on $M$. Further, if $\kappa$ and $\mu$ are
constants and $n\geq 1$, then $\mu=-2$ and hence

(53) $\displaystyle
Q\Omega_{1}=-2n\Omega_{1}+2n(\kappa+1)\eta(\Omega_{1})\zeta-2nh^{\prime}\Omega_{1},$

for any vector field $\Omega_{1}$ on $M$. In both cases, the scalar curvature
of $M$ is $2n(\kappa-2n)$.

Here, we consider gradient ARYS in the context of $(\kappa,\mu)^{\prime}$-AKM
and generalized Theorem 3.5 [8] and Theorem 3.1 [32]. We state and prove the
following:

######  Theorem 4.4.

If $M^{2n+1}(\varphi,\zeta,\eta,g)$ be a $(\kappa,\mu)^{\prime}$-AKM with
$h^{\prime}\neq 0$ admitting gradient ARYS then either $M$ is locally
isometric to $\mathbb{H}^{2n+1}(-4)\times\mathbb{R}^{n}$ or potential vector
field is pointwise collinear with the Reeb vector field.

###### Proof.

Suppose that $(\kappa,\mu)^{\prime}$-AKM admits gradient ARYS then Eq.
(9)-(11) is valid. Taking an inner product of (11) with $\zeta$ and inserting
Lemma 4.3, we obtain

(54) $\displaystyle
g(R(\Omega_{1},\Omega_{2})Df,\zeta)=(\Omega_{1}\lambda)\eta(\Omega_{2})-(\Omega_{2}\lambda)\eta(\Omega_{1})-\alpha\\{g(Qh^{\prime}\Omega_{2},\Omega_{1})-g(Qh^{\prime}\Omega_{1},\Omega_{2})\\}.$

Taking an inner product of (50) with $Df$, then inserting it in (54) and
replacing $\Omega_{1}$ by $\zeta$ gives

(55) $\displaystyle-(\zeta\lambda)\zeta+D\lambda=\kappa\\{(\zeta f)\zeta-
Df\\}-\mu h^{\prime}Df.$

Contracting (11) over $\Omega_{1}$, we get $QDf=-2nD\lambda$. In consequence
of this in Lemma 4.3 gives

(56) $\displaystyle D\lambda-Df+(\kappa+1)(\zeta f)\zeta=h^{\prime}Df.$

Combining (55) and (56), we obtain

(57) $\displaystyle\kappa\\{(\zeta f)\zeta-
Df\\}+D\lambda+(\zeta\lambda)\zeta-2Df+2(\kappa+1)(\zeta f)\zeta=0.$

Operating the forgoing equation by $\varphi$ yields

$\displaystyle\varphi D\lambda-(\kappa+2)\varphi Df=0,$

implies,

$D\lambda-(\kappa+2)Df\in\mathbb{R}\zeta.$

Therefore, we can write $D\lambda=(\kappa+2)Df+s\zeta$, where $s$ is a smooth
function. In view of this in (55) infer

(58) $\displaystyle 2(\kappa+1)Df+(s-\zeta\lambda-\kappa\zeta
f)\zeta=2h^{\prime}Df.$

Operating (58) by $h^{\prime}$ then inserting the obtain expression in (58)
gives

$\displaystyle(\kappa+2)\varphi Df=0.$

Thus we have either $\kappa=-2$ or $Df=(\zeta f)\zeta$.
Suppose $\kappa=-2$. Then without loss of generality, we may choose $\nu=1$.
Then we have from Theorem 5.1 of [25] we get

$R({\Omega_{1}}_{\nu},{\Omega_{2}}_{\nu}){\Omega_{3}}_{\nu}=-4[g({\Omega_{2}}_{\nu},{\Omega_{3}}_{\nu}){\Omega_{1}}_{\nu}-g({\Omega_{1}}_{\nu},{\Omega_{3}}_{\nu}){\Omega_{2}}_{\nu}],$
$R({\Omega_{1}}_{-\nu},{\Omega_{2}}_{-\nu}){\Omega_{3}}_{-\nu}=0,$

for any
${\Omega_{1}}_{\nu},{\Omega_{2}}_{\nu},{\Omega_{3}}_{\nu}\in[\nu]^{\prime}$
and
${\Omega_{1}}_{-\nu},{\Omega_{2}}_{-\nu},{\Omega_{3}}_{-\nu}\in[-\nu]^{\prime}$.
In consequence of this the Proposition 4.1 and Proposition 4.3 of [25] along
with $\nu=1$ shows that it is locally isometric to
$\mathbb{H}^{2n+1}(-4)\times\mathbb{R}^{n}$. This completes the proof. ∎

Suppose $\kappa\neq-2$. Then in regard of Theorem 4.4 we have $V=Df=(\zeta
f)\zeta$. Take $F=\zeta f$ and taking the covariant derivative of $V=F\zeta$
along arbitrary vector field $\Omega_{1}$ we get

$\displaystyle\nabla_{\Omega_{1}}V=(\Omega_{1}F)\zeta+F(-\varphi^{2}\Omega_{1}+h^{\prime}\Omega_{1}).$

Making use of this in (1) yields

$\displaystyle(\Omega_{1}F)\eta(\Omega_{2})+(\Omega_{2}F)\eta(\Omega_{1})+2Fg(\Omega_{1},\Omega_{2})-2F\eta(\Omega_{1})\eta(\Omega_{2})$
(59) $\displaystyle+2Fg(h^{\prime}\Omega_{1},\Omega_{2})=(2\lambda-\beta
r)g(\Omega_{1},\Omega_{2})-2\alpha S(\Omega_{1},\Omega_{2}).$

Replacing $\Omega_{2}$ by $\zeta$ in (4) gives

(60) $\displaystyle\Omega_{1}F=(2\lambda-\beta r-4n\alpha\kappa-\zeta
F)\eta(\Omega_{1}),$

for any vector field $\Omega_{1}$ on $M$. Contracting (4) then inserting it in
(60) and replacing $\Omega_{1}$ by $\zeta$ in the obtained expression we
obtain

(61) $\displaystyle F=\lambda-\frac{\beta r}{2}+2n\alpha.$

Inserting (60) in (4) and comparing it with Lemma 4.3 yields
$(F-2n\alpha)(\kappa+1)\varphi^{2}\Omega_{1}=0$ for any $\Omega_{1}$ on $M$.
As $\kappa<-1$, we see that $F=2n\alpha$, in view of this in (61) implies
$\lambda=\frac{\beta r}{2}$ i.e., a constant. Therefore, $M$ reduces to
gradient RYS. Hence using Corollary 3.7 of [8] we can state the following:

######  Corollary 4.5.

Let $M^{2n+1}(\varphi,\zeta,\eta,g)$ be a non-Kenmotsu
$(\kappa,\mu)^{\prime}$-AKM with $\kappa\neq-2$ admitting a gradient ARYS then
1\. The potential vector field $V$ is a constant multiple of $\zeta$.
2\. $V$ is a strict infinitisimal contact transformation.
3\. $V$ leaves $h^{\prime}$ invariant.

Consider a generalized $(\kappa,\mu)^{\prime}$-AKM of dimension three with
$\kappa<-1$. If we assume that $\kappa$ is invariant along $\zeta$, then from
(Proposition 3.2, [25]) we have $\zeta(\kappa)=-2(\kappa+1)(\mu+2)$ implies
$\mu=-2$. Moreover, from (Lemma 3.3, [28]), we have
$h^{\prime}(grad\mu)=grad\kappa-\zeta(\kappa)\zeta$ which implies $\kappa$ is
constant under our assumption. Therefore $M^{3}$ becomes a
$(\kappa,-2)^{\prime}$-AKM. By applying Theorem 4.4, we can conclude the
following:

######  Corollary 4.6.

Let $M^{3}(\varphi,\eta,\zeta,g)$ be a generalized non-Kenmotsu
$(\kappa,\mu)^{\prime}$-AKM with $\kappa<-1$ invariant along the Reeb vector
field admitting gradient ARYS then it is either locally isometric to the
product space $\mathbb{H}^{2n+1}(-4)\times\mathbb{R}^{n}$ or potential vector
field is pointwise collinear with the Reeb vector field.

##  5\. 3-dimensional Almost Kenmotsu manifolds

On a 3-dimensional AKM $M^{3}$, we define the following open subsets:

$\displaystyle\mathcal{U}_{1}=\\{p\in M^{3}:h\neq 0~{}~{}\mbox{in a
neighbourhood of}~{}~{}p\\},$ $\displaystyle\mathcal{U}_{1}=\\{p\in
M^{3}:h=0~{}~{}\mbox{in a neighbourhood of}~{}~{}p\\}.$

Then, $\mathcal{U}_{1}\cup\mathcal{U}_{2}$ is an open and dense subset of
$M^{3}$ and there exists a local orthonormal basis $\\{e,\varphi e,\zeta\\}$
of three smooth unit eigenvectors of $h$ for any point
$p\in\mathcal{U}_{1}\cup\mathcal{U}_{2}$. On $\mathcal{U}_{1}$ we may set
$he_{1}=\vartheta e_{1}$ and $he_{2}=-\vartheta e_{2}$, where $\vartheta$ is a
positive function.

######  Lemma 5.1.

[7] On $\mathcal{U}_{1}$ we have

$\displaystyle\nabla_{\zeta}\zeta=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{\zeta}e=a\varphi
e,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{\zeta}\varphi
e=-ae,$ $\displaystyle\nabla_{e}\zeta=e-\vartheta\varphi
e,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{e}e=-\zeta-b\varphi
e,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{e}\varphi e=\vartheta\zeta+be,$
$\displaystyle\nabla_{\varphi e}\zeta=-\vartheta e+\varphi
e,~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{\varphi e}e=\vartheta\zeta+c\varphi
e,~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{\varphi e}\varphi e=-\zeta-ce,$

where $a,b,c$ are smooth functions.

From Lemma 5.1, the poisson brackets for $\\{e,\varphi e,\zeta\\}$ are as
follows:

(62) $\displaystyle[\zeta,e]=(a+\vartheta)\varphi e-e,[e,\varphi
e]=be-c\varphi e,[\varphi e,\zeta]=(a-\vartheta)e+\varphi e.$

In view of lemma 5.1, the expressions for Ricci operator are as follows:

######  Lemma 5.2.

On $\mathcal{U}_{1}$, we have

(63) $\displaystyle\begin{cases}Q\zeta=-2(\vartheta^{2}+1)\zeta-(\varphi
e(\vartheta)+2\vartheta b)e-(e(\vartheta)+2\vartheta c)\varphi e,\\\
Qe=-(\varphi e(\vartheta)+2\vartheta b)\zeta-(A+2\vartheta
a)e+(\zeta(\vartheta)+2\vartheta)\varphi e,\\\ Q\varphi
e=-(e(\vartheta)+2\vartheta
c)\zeta+(\zeta(\vartheta)+2\vartheta)e-(A-2\vartheta a)\varphi e,\end{cases}$

where $A=e(c)+\varphi e(b)+b^{2}+c^{2}+2$.

Suppose that the non-trivial potential vector field of gradient ARY soliton
$(\alpha\neq 0)$ is orthogonal to the Reeb vector field $\zeta$, then we can
write $V=f_{1}e+f_{2}\varphi e$, where $f_{1},f_{2}$ are smooth functions.
Replacing $\Omega_{1}$ by $\zeta$ in (9) and making use of Lemmas 5.1 and 5.2,
we get

(64) $\displaystyle\begin{cases}-2\alpha(\vartheta^{2}+1)=\lambda-\frac{\beta
r}{2},\\\ \varphi e(\vartheta)=-2b\vartheta,\\\
e(\vartheta)=-2c\vartheta.\end{cases}$

Similarly, taking $\Omega_{1}=e$ in (9) gives

(65) $\displaystyle\begin{cases}\vartheta f_{1}-f_{2}=0,\\\
e(f_{1})+bf_{2}-\alpha(A+2a\vartheta)=\lambda-\frac{\beta r}{2},\\\
e(f_{2})-bf_{1}+\alpha(\zeta(\vartheta)+2\vartheta)=0.\end{cases}$

Also for $\Omega_{1}=\varphi e$, we obtain

(66) $\displaystyle\begin{cases}\vartheta f_{1}-f_{2}=0,\\\ \varphi
e(f_{1})-cf_{2}+\alpha(\zeta(\vartheta)+2\vartheta)=0,\\\ cf_{1}+\varphi
e(f_{2})-\alpha(A-2a\vartheta)=\lambda-\frac{\beta r}{2}.\end{cases}$

Comparing the first arguments of (65) and (66), we see that
$(\vartheta^{2}-1)f_{1}=0$. If $f_{1}=0$, then from first argument of (66) we
get $f_{2}=0$, which further implies $V=0$, a contradiction. Therefore, we
must have $\vartheta=1$. In consequence of this in second and third statement
of (64) yields $b=c=0$ and first argument of (65) gives $f_{1}=f_{2}$.
Combining the first equation of (64) with second eqn. (65) and third eqn.
(66), then making use of the fact that $\vartheta=1$ yields

(67) $\displaystyle e(f_{1})-\varphi e(f_{1})=cf_{1}-bf_{1}+4\alpha a=0,$

where we use $f_{1}=f_{2}$. Similarly from (65) and (66), one can get

(68) $\displaystyle\varphi e(f_{1})-cf_{1}=e(f_{1})-bf_{1}.$

Making use of (67) and (68), together with $b=c=0$ gives $a=0$. In
consequence, Eq. (62) becomes

$[\zeta,e]=\varphi e-e,~{}~{}~{}[e,\varphi e]=0,~{}~{}~{}[\varphi
e,\zeta]=-e+\varphi e.$

Using Milnor’s result [24] we can conclude that $M^{3}$ is locally isometric
to a non-unimodular Lie group with a left-invariant almost Kenmotsu structure.
Moreover, it is obvious that $\nabla_{\zeta}h=0$ and it is conformally flat
with constant scalar curvature $r=-8$. Now making use of Wang’s result [33]
which state that, “An almost Kenmotsu 3-manifold satisfying
$\nabla_{\zeta}h=0$ is conformally flat with constant scalar curvature if and
only if it is locally isometric to either the hyperbolic space
$\mathbb{H}^{3}(-1)$ or the Riemannian product
$\mathbb{H}^{2}(-4)\times\mathbb{R}$” we can state the following:

######  Theorem 5.3.

If a 3-dimensional non-Kenmotsu AKM admits a gradient ARYS $(\alpha\neq 0)$
whose non-trivial potential vector field is orthogonal to the Reeb vector
field, then it is locally isometric to the Riemannian product
$\mathbb{H}^{2}(-4)\times\mathbb{R}$.

## Acknowledgments

The first author is thankful to the Department of Science and Technology, New
Delhi, India for financial support in the form of INSPIRE Fellowship
(DST/INSPIRE Fellowship/2018/IF180830).

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