# Multi-Objective Optimization in STAR-RIS-Aided SWIPT with RSMA via Meta-
Learning

Mojtaba Amiri, Elaheh Vaezpour, Sepideh Javadi, Mohammad Robat Mili, Halim
Yanikomeroglu, Fellow, IEEE, and Mehdi Bennis, Fellow, IEEE Mojtaba Amiri is
with the School of Electrical and Computer Engineering, University of Tehran,
Tehran, Iran (e-mail: mojtaba.amiri@ut.ac.ir). Elaheh Vaezpour is with the
Communication Department, ICT Research Institute, Tehran, Iran (e-mail:
e.vaezpour@itrc.ac.ir). Sepideh Javadi and Mohammad Robat Mili are with the
Pasargad Institute for Advanced Innovative Solutions (PIAIS), Tehran
1991633361, Iran<EMAIL_ADDRESS>mohammad.robatmili@gmail.com). Halim Yanikomeroglu is with the Department of
Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada,
(e-mail: halim@sce.carleton.ca). Mehdi Bennis is with the Centre for Wireless
Communications, University of Oulu, 90570 Oulu, Finland (e-mail:
mehdi.bennis@oulu.fi).

###### Abstract

Simultaneously transmitting and reflecting reconfigurable intelligent surface
(STAR-RIS) is a cutting-edge concept for the sixth-generation (6G) wireless
networks. In this letter, we propose a novel system that incorporates STAR-RIS
with simultaneous wireless information and power transfer (SWIPT) using rate
splitting multiple access (RSMA). The proposed system facilitates
communication from a multi-antenna base station (BS) to single-antenna users
in a downlink transmission. The BS concurrently sends energy and information
signals to multiple energy harvesting receivers (EHRs) and information data
receivers (IDRs) with the support of a deployed STAR-RIS. Furthermore, a
multi-objective optimization is introduced to strike a balance between users’
sum rate and the total harvested energy. To achieve this, an optimization
problem is formulated to optimize the energy/information beamforming vectors
at the BS, the phase shifts at the STAR-RIS, and the common message rate.
Subsequently, we employ a meta deep deterministic policy gradient (Meta-DDPG)
approach to solve the complex problem. Simulation results validate that the
proposed algorithm significantly enhances both data rate and harvested energy
in comparison to conventional DDPG.

###### Index Terms:

Meta-learning, RSMA, SWIPT, STAR-RIS.

##  I Introduction

The concept of reconfigurable intelligent surface (RIS) is a key sixth-
generation (6G) technology, attracting noteworthy attention. More
specifically, each RIS consists of plenty of reflection units capable of
adjusting amplitude and phase shift independently using a smart controller
[1]. Typically, RISs can only reflect the received incident signal and thus
working in one half-plane. Therefore, a new advanced type of RIS,
simultaneously transmitting and reflecting reconfigurable intelligent surface
(STAR-RIS), is recommended to overcome this limitation by enabling the full-
space coverage [2]. More specifically, authors in [3] considered the
deployment of STAR-RIS in a full-duplex system aiming to maximize the average
weighted sum rate. Moreover, a joint deployment and beamforming design
algorithm was proposed in [4] to address the sum rate maximization problem in
a downlink STAR-RIS-assisted communication system.

On the other hand, rate-splitting multiple access (RSMA) is a novel framework,
offering substantial gains in spectral and energy efficiency. Motivated by the
advantages of the integration of RIS/STAR-RIS with RSMA, a few existing works
have explored the performance of this integration [5, 6, 7]. More
specifically, the authors in [5] focus on maximizing the energy efficiency
(EE) in a downlink RIS-aided communication system with RSMA. In [6], a
proximal policy optimization method is introduced to address the sum-rate
maximization problem in a downlink STAR-RIS-assisted network with RSMA.
Additionally, the authors in [7] explore a two-user RIS-aided RSMA system in
the context of uplink transmission, with the goal of maximizing the achievable
rate.

Simultaneous wireless information and power transfer (SWIPT) is a cutting-edge
technology addressing energy limitations in battery-powered devices [8].
Furthermore, optimizing both wireless communication performance and energy
transfer is crucial, leading to multi-objective optimization approaches [9].
For instance, the sum rate maximization problem of multi-user SWIPT system
with the deployment of an intelligent reflecting surface (IRS) is studied in
[10], while the authors in [11] aim to maximize the weighted sum rate of the
information users in an active IRS-assisted SWIPT system. It should be noted
that this is the first work investigating a multi-user STAR-RIS-assisted SWIPT
system with RSMA. In this setup, users are divided into two main groups, i.e.,
information decoding receivers (IDRs) and energy harvesting receivers (EHRs),
to receive the signals reflected by a deployed STAR-RIS. The system
performance is evaluated by jointly maximizing the total sum rate of IDRs and
harvested energy by EHRs. Additionally, different from the conventional DDPG,
a Meta-learning-based DDPG algorithm is formulated to solve the non-convex NP-
hard problem.

Figure 1: System model of an STAR-RIS SWIPT system with RSMA.

##  II System Model

In this section, we consider a downlink STAR-RIS-assisted multi-user SWIPT
system with RSMA, where an STAR-RIS composed of $M$ elements is implemented to
assist the transmission between an $N$-antenna base station (BS) and $K$
single-antenna mobile users. The $K$ receivers are separated into two groups,
i.e., $U$ IDRs and $V$ EHRs, such that $U\cup V=K$ and $U\cap V=\emptyset$.
The sets of RIS elements, BS antennas, IDRs and EHRs are represented by
$\mathcal{M}=\left\\{{1,2,...,M}\right\\}$,
$\mathcal{N}=\left\\{{1,2,...,N}\right\\}$,
$\mathcal{U}=\left\\{{1,2,...,U}\right\\}$, and
$\mathcal{V}=\left\\{{1,2,...,V}\right\\}$, respectively. The signal is
divided into the refracted and reflected components by each element of a
configurable and programmable STAR-RIS. The transmitted and reflected signals
by the $m$-th element of STAR-RIS can be respectively described as

$s^{t}_{m}=\sqrt{\beta^{t}_{m}}e^{j\theta^{t}_{m}}s_{m},~{}~{}\textrm{and}~{}~{}s^{r}_{m}=\sqrt{\beta^{r}_{m}}e^{j\theta^{r}_{m}}s_{m},$
(1)

where $s_{m}$ denotes the signal incident on the $m$-th element of STAR-RIS
and $j$ represents the imaginary unit. Furthermore, the amplitude and phase
shift of the transmission and reflection coefficients of the $m$-th element of
the deployed STAR-RIS are respectively denoted by
$\\{\sqrt{\beta^{t}_{m}},\sqrt{\beta^{r}_{m}}\\}$ and
$\\{\theta^{t}_{m},\theta^{r}_{m}\\}$. It is worth pointing out that the
amplitude and phase shift of $m$-th element of STAR-RIS in both reflection and
refraction sides are within $[0,2\pi]$ and $[0,1]$, respectively. It is
assumed that the reflection and transmission phase shifts
$\\{\theta^{t}_{m},\theta^{r}_{m}\\}$ are independent, while the amplitude
coefficients for both reflection and transmission are coupled. Moreover, the
condition $\beta^{r}_{m}+\beta^{t}_{m}=1$ should be satisfied, as the incident
signal’s energy should be equal to aggregate energy of reflected and
transmitted signals, i.e.,
${\left|{s_{m}^{t}}\right|^{2}}+{\left|{s_{m}^{r}}\right|^{2}}={\left|{{s_{m}}}\right|^{2}}$.

With the one layer RSMA scheme at the BS, the total message of each IDR
$u\in\mathcal{U}$ is divided into two main parts, i.e., a common part
${\mathbb{w}_{c}}$ and a private part $\mathbf{w}_{k}$. Then, the total
transmit signal at the BS by considering both IDRs and EHRs is given by

$\mathbf{x}={{\mathbb{w}_{c}}}{s^{\text{ID}}_{c}}+\sum_{u=1}^{U}\mathbf{w}_{u}s^{\text{ID}}_{u}+\sum_{v=1}^{V}\mathbf{p}_{v}s^{\text{EH}}_{v},$
(2)

where ${s^{\text{ID}}_{c}}$, ${s^{\text{ID}}_{u}}$, and
${s^{\text{EH}}_{v}}\in\mathbb{C}$ are the common information signal, the
private information signal of IDR $u\in\mathcal{U}$, and the energy signal of
EHR $v\in\mathcal{V}$ in the proposed system model. It is assumed that both
common and private information signals are independent and identically
distributed (i.i.d), i.e., $\mathbb{E}\\{|s^{\text{ID}}_{c}|^{2}\\}=1$ and
$\mathbb{E}\\{|s^{\text{ID}}_{u}|^{2}\\}=1$, while the energy signals are
independently generated from an arbitrary distribution with
$\mathbb{E}\\{|s^{\text{EH}}_{v}|^{2}\\}=1$. Moreover,
$\mathbf{w}_{c}\in\mathbb{C}^{N\times 1}$ and
$\mathbf{w}_{u}\in\mathbb{C}^{N\times 1}$ denote the common and private
information beam for the $u$-th IDR, while
$\mathbf{p}_{v}\in\mathbb{C}^{N\times 1}$ represents the transmit energy beam
for the $v$-th EHR. We presume that the links of channel experience quasi-
static flat-fading, and the channel state information is accessible at both BS
and the deployed STAR-RIS. Subsequently, the received signal at IDR $u$ can be
formulated as

${y}^{\text{ID}}_{u}=\mathbf{h}^{H}_{u}\mathbf{x}+z^{\text{ID}}_{u},$ (3)

where
${z^{\text{ID}}_{u}}\sim\mathcal{C}\mathcal{N}\left({0,\sigma_{u}^{2}}\right)$
is the white Gaussian noise of the $u$-th IDR with zero mean and variance
$\sigma_{u}^{2}$. The equivalent channel from the BS to the STAR-RIS, the
STAR-RIS to the $u$-th IDR, as well as the direct path from the BS to IDR $u$
is given by

$\mathbf{h}^{H}_{u}\triangleq\mathbf{h}^{H}_{d,u}+\mathbf{h}^{H}_{\text{S-RIS},u}\boldsymbol{\Theta}^{l}\mathbf{G},$
(4)

where $\mathbf{G}\in\mathbb{C}^{M\times N}$,
$\mathbf{h}_{d,u}\in\mathbb{C}^{N\times 1}$, and
$\mathbf{h}_{\text{S-RIS},u}\in\mathbb{C}^{M\times 1}$ denote the channel
matrix from the BS to the STAR-RIS, the channel vector from the BS to IDR $u$,
and the channel vector from the STAR-RIS to the $u$-th IDR, respectively.
Furthermore, the transmission ($l=t$) or reflection ($l=r$) beamforming vector
is denoted by
${\boldsymbol{\phi}^{l}}=\left[{\sqrt{\beta_{1}^{l}}{e^{j\theta_{1}^{l}}},\sqrt{\beta_{2}^{l}}{e^{j\theta_{2}^{l}}},...,\sqrt{\beta_{M}^{l}}{e^{j\theta_{M}^{l}}}}\right]$,
in which ${\boldsymbol{\Theta}^{l}}=\text{diag}({\phi^{l}})$,
$l\in\left\\{{t,r}\right\\}$ represents the corresponding diagonal beamforming
matrix of the STAR-RIS in transmission/reflection region. It is assumed that
the number of $V$ out of $K$ receivers can harvest energy. Subsequently, the
received signal at the EHR $v\in\mathcal{V}$ can be written as

${y}^{\text{EH}}_{v}=\mathbf{f}^{H}_{v}\mathbf{x}+z^{\text{EH}}_{v},$ (5)

where
${z^{\text{EH}}_{v}}\sim\mathcal{C}\mathcal{N}\left({0,\sigma_{v}^{2}}\right)$
is the AWGN of the $v$-th EHR with zero mean and variance $\sigma_{v}^{2}$.
Additionally, the combined channel of the BS-(STAR-RIS)-user link for the
$v$-th EHR is given by

$\mathbf{f}^{H}_{v}=\mathbf{f}^{H}_{d,v}+\mathbf{f}^{H}_{\text{S-RIS},v}\boldsymbol{\Theta}^{l}\mathbf{G},$
(6)

where $\mathbf{f}_{d,v}\in\mathbb{C}^{N\times 1}$ is the channel vector
between the BS and $v$-th EHR, while
$\mathbf{f}_{\text{S-RIS},v}\in\mathbb{C}^{M\times 1}$ denotes the channel
vector between the STAR-RIS and EHR $v$.

Following this, each IDR decodes its private stream after eliminating the
common stream using successive interference cancellation (SIC). For the sake
of simplicity, it is assumed that all IDRs can mitigate interference caused by
the energy signals [10]. Hence, the received information rates of the common
stream and the private stream at the $u$-th IDR can be respectively stated as
follows:

$R_{c,u}={\log_{2}}\left(1+{\frac{|\mathbf{h}^{H}_{u}\mathbf{w}_{c}|^{2}}{\sum_{i=1}^{U}|\mathbf{h}^{H}_{u}\mathbf{w}_{i}|^{2}+\sigma_{u}^{2}}}\right),$
(7)

and

$R_{p,u}={\log_{2}}\left(1+{\frac{|\mathbf{h}^{H}_{u}\mathbf{w}_{u}|^{2}}{\sum_{i=1,i\neq
u}^{U}|\mathbf{h}^{H}_{u}\mathbf{w}_{i}|^{2}+\sigma_{u}^{2}}}\right).$ (8)

To guarantee the successful decoding of the common message at all IDRs, the
achievable rate of the common message should not exceed the minimum rate of
all IDRs, i.e., $R_{c}\leq\min\\{R_{c,1},R_{c,2},\dots,R_{c,U}\\}$. The common
rate $R_{c}=\sum_{u=1}^{U}C_{u}$ is shared by all $U$ IDRs, while $C_{u}$
denotes the $u$-th IDR’s portion of common rate. Then, we have

$\sum_{u=1}^{U}C_{u}\leq\min_{u}\\{R_{c,u}\\},\forall u\in\mathcal{U}.$ (9)

Furthermore, the total harvested energy at the $v$-th EHR is expressed as [12]

$P^{\text{Har}}_{v}(\mathbf{w}_{u},\mathbf{p}_{v},\mathbb{\Theta}^{l})=\lambda_{v}\mathbb{E}\\{\sum_{u=1}^{U}|\mathbf{f}^{H}_{v}\mathbf{w}_{u}|^{2}+\sum_{v=1}^{V}|\mathbf{f}^{H}_{v}\mathbf{p}_{v}|^{2}\\},$
(10)

where $\lambda_{v}\in[0,1]$ represents the energy conversion efficiency of the
EHR $v$.

##  III Optimization Problem Formulation

In multi-objective optimization, the goal is to simultaneously address
multiple conflicting objectives. The challenge in this optimization lies in
finding a set of solutions that represents a trade-off among the conflicting
objectives. In this letter, we specifically address a multi-objective
maximization problem for optimizing both the sum rate of IDRs and the total
harvested energy at the EHRs by jointly optimizing the active beamforming
vectors at the BS, i.e., $\mathbf{W}=\\{\mathbf{w}_{c},\mathbf{w}_{u},\forall
u\\}$ and $\mathbf{P}=\\{\mathbf{p}_{v},\forall v\\}$ , data rate for
receiving the common message $\mathbf{c}=\\{C_{u},\forall u\\}$, and
transmission/reflection coefficients of STAR-RIS. The problem is defined as
follows:

$\displaystyle\max_{\mathbf{W},\mathbf{P},\boldsymbol{\Theta}^{l},\mathbf{c}}\quad$
$\displaystyle\sum_{u=1}^{U}(R_{c,u}+R_{p,u}),$
$\displaystyle\max_{\mathbf{W},\mathbf{P},\boldsymbol{\Theta}^{l}}\quad$
$\displaystyle\sum_{v=1}^{V}P^{\text{Har}}_{v},$ s.t.
$\displaystyle\text{C}_{1}:\sum_{u=1}^{U}C_{u}\leq\min_{u}\\{R_{c,u}\\},\quad\forall
u\in\mathcal{U},$ $\displaystyle\text{C}_{2}:C_{u}+R_{u}\geq
R_{\text{min},u},\quad\forall u\in\mathcal{U},$
$\displaystyle\text{C}_{3}:{\parallel\mathbf{w}_{c}\parallel^{2}}+\sum_{u=1}^{U}{\parallel\mathbf{w}_{u}\parallel^{2}}+\sum_{v=1}^{V}\parallel\mathbf{p}_{v}\parallel^{2}\leq
P_{\text{max}},$ $\displaystyle\text{C}_{4}:P^{\text{Har}}_{v}\geq
E_{\text{min},v},\quad\forall v\in\mathcal{V},$
$\displaystyle\text{C}_{5}:{{{\beta_{m}^{t}},{\beta_{m}^{r}}\in\left[{0,1}\right]}},\quad\beta_{m}^{t}+\beta_{m}^{r}=1,~{}\forall
m\in\mathcal{M},$
$\displaystyle\text{C}_{6}:{\theta_{m}^{t},\theta_{m}^{r}\in\left[{0,2\pi}\right)},\quad\forall
m\in\mathcal{M},$ (11)

where constraints $\text{C}_{1}$ and $\text{C}_{2}$ ensure that the common
message can be successfully decoded at all the IDRs in both reflection and
transmission spaces. Moreover, constraint $\text{C}_{3}$ denotes the maximum
power budget at the BS, while $\text{C}_{4}$ represents the minimum harvested
energy for the $v$-th EHR. Finally, the constraints for amplitude and phase
shift coefficients of the deployed STAR-RIS are denoted by $\text{C}_{5}$ and
$\text{C}_{6}$, respectively. The above non-convex optimization problem is NP-
hard, making it challenging to solve using conventional solutions. Therefore,
a Meta-DDPG approach is applied to solve this problem.

##  IV Meta-DDPG based Learning Solution

The Meta-DDPG approach [13] is a reinforcement learning (RL) method employed
in this letter due to the extensive dimensionality of both the state and
action spaces. Conventional DDPG algorithm cannot quickly adapt to new and
dynamic environments. Instead, Meta-learning is used to address this issue.
This algorithm integrates conventional DDPG with Meta-learning to enable
learning a model which rapidly adapts to a new environment. This method is
based on the agent, state, action, and reward [14, 13] and utilizes actor and
critic networks to learn a suitable model.

In each time step $t\in[0,T]$, the agent observes a candidate state of the
environment represented as $s_{t}\in\mathcal{S}$, and subsequently selects an
action based on the current state $s_{t}$ which is denoted by
$a_{t}\in\mathcal{A}$. After taking action $a_{t}$ in state $s_{t}$ by the
agent, the environment transits to a new state $s_{t+1}\in\mathcal{S}$, in
which the agent receives a numerical reward $r_{t+1}\in\mathcal{R}$. As a
result, a sequence of interactions, i.e.,
$s_{0},a_{0},r_{1},s_{1},a_{1},r_{2},\ldots$ unfolds over time. The value of
$r$ plays a crucial role as it guides the agent in selecting better actions.
The policy, denoted as $\pi$, defines how the agent maps states to the
actions. In the Meta-DDPG algorithm, the main purpose is to find an optimal
policy denoted as $\pi^{*}$, with the ultimate aim of maximizing the reward
function. To characterize the Meta-DDPG algorithm, we examine its components
as follows
1) State: It is defined as a set of channel information, ensuring that the
states have the necessary information to make proper decisions.
2) Action: At each time step $t$, the agent selects the action
$a_{t}=[\mathbf{W},\mathbf{P},\boldsymbol{\Theta}^{l},\mathbf{c}]$ based on
the current state $s_{t}$.
3) Reward function: As the reward is an important part of RL, it should be
designed properly to ensure the learning model, aimed at solving the
optimization problem, satisfies the constraints. Therefore, the reward is
defined as follows:

$\displaystyle r_{t}=\left\\{\begin{array}[]{lc}\Xi,&\text{if constraints are
satisfied},\\\ 0,&\text{otherwise},\\\ \end{array}\right.$ (14)

where
$\Xi=\alpha\sum_{u=1}^{U}{(R_{c,u}+R_{p,u})}+(1-\alpha)\sum_{v=1}^{V}P^{\textrm{Har}}_{v}$
and a penalty with the value of $0$ is imposed to discourage the selection of
actions that do not satisfy the constraints.

1 Input: The maximum number of time steps (T), the number of episodes (E),
$\xi$ and $X$.

2 Initialize the actor and critic networks, $\pi(s|\theta^{\pi})$ and
$Q(s,a|\theta^{Q})$.

3 Initialize the target critic and actor networks with parameters of
${\bar{\theta}}^{Q}\leftarrow\theta^{Q}$ and
$\bar{\theta}^{\pi}\leftarrow\theta^{\pi}$.

4 Initialize a replay buffer $B$.

5 for  _$episode=1$ to $E$_ do

6 initialize observation state $s_{0}$

7 for  _$t=1$ to $T$_ do

8 Choose action $a_{t}$ based on the current policy and exploration noise.

9 Execute action $a_{t}$, observe reward $r_{t}$ and record the new state
$s_{t+1}$ by storing the transition $(s_{t},a_{t},r_{t},s_{t+1})$ in $B$.

10 for  _each step of gradient descent to solve problem(11)_ do

11 A mini-batch is randomly selected from the set $B$.

12 $\theta^{Q}=\theta^{Q}-\text{lr}_{\text{critic}}\nabla L(\theta^{Q})$

13 $\theta^{\pi}_{\text{old}}=\theta^{\pi}-\text{lr}_{\text{actor}}\nabla
J(\theta^{\pi})$

14
$\theta^{\pi}_{\text{new}}=\theta^{\pi}_{\text{old}}-\text{lr}_{\text{actor}}\nabla
F_{\text{new}}(\theta^{\pi},\mu)$

15 $\theta^{\pi}\leftarrow\theta^{\pi}_{\text{new}}$

16
$\mu=\mu-\text{lr}_{\text{meta}}\nabla\text{tanh}(J(\theta^{\pi}_{\text{new}})-J(\theta^{\pi}_{\text{old}}))$

17 if  _$\text{mod}(t,X)=0$_ then

18
${{\bar{\theta}}^{Q}}_{t+1}=(1-\xi){\bar{\theta}^{Q}}_{t}+{\xi{\theta_{t}^{Q}}}$

19
${{\bar{\theta}}^{\pi}}_{t+1}=(1-\xi){{\bar{\theta}}^{\pi}}_{t}+\xi{\theta_{t}^{\pi}}$

20 end if

21

22 end for

23

24 end for

25

26 end for

Algorithm 1 Meta-DDPG algorithm

Here, the Meta-DDPG solution is proposed, merging traditional DDPG with Meta-
learning for swift adaptation to new environments. Actually, Meta-DDPG extends
traditional DDPG by adding Meta-learning. It has inner loop for fine-tuning
and outer loop for faster adaptation to new environments. Note that the
parameters of four neural networks are represented as follows: critic network
$\theta^{Q}$, actor network $\theta^{\pi}$, target critic network
${\bar{\theta}}^{Q}$, and target actor network $\bar{\theta}^{\pi}$.
The actor network parameters are adjusted in a manner that minimizes the
subsequent loss function:

$J(\theta^{\pi})=-Q^{\pi}(s_{t},a_{t}=\pi(s|\theta^{\pi})|\theta^{Q}),$ (15)

where $\pi(s|\theta^{\pi})$ and $Q(s,a|\theta^{Q})$ are actor and critic
networks, respectively. In order to learn the parameters of critic network,
the subsequent loss function is minimized:

$L(\theta^{Q})=\mathbb{E}[(Q(s_{t},a_{t}|\theta^{Q})-Y)^{2}],$ (16)

where $Y=R+\tau Q(s_{t+1},\pi(s_{t+1}|\theta^{\pi})|\theta^{Q})$ and
$R=\sum_{t=0}^{T}\tau^{t}r_{t}$ with $\tau\in[0,1)$. Subsequently, to update
the parameters of the target critic and target actor networks,
${{\bar{\theta}}^{Q}}_{t+1}=(1-\xi){\bar{\theta}^{Q}}_{t}+{\xi{\theta_{t}^{Q}}}$
and
${{\bar{\theta}}^{\pi}}_{t+1}=(1-\xi){{\bar{\theta}}^{\pi}}_{t}+\xi{\theta_{t}^{\pi}}$
with $\xi\ll 1$ are defined.

A tuple $(s_{t},a_{t},r_{t},s_{t+1})$ is stored into a buffer pool and then is
updated within different episodes. The bi-level optimization problem is
defined as follows:

$\mu=\text{arg}~{}\underset{\mathbf{\mu}}{\min}\
F_{\text{meta}}(\theta^{*\pi}),$ (17a) subject to
$\displaystyle\theta^{*\pi}=\text{arg}~{}\min_{\theta^{\pi}}(J(\theta^{\pi})+F_{\text{new}}(\theta^{\pi},\mu)),$
(17b)

where
$F_{\text{meta}}(\theta^{*\pi})=\text{tanh}(J(\theta^{\pi}_{\text{new}})-J(\theta^{\pi}_{\text{old}}))$
and $F_{\text{new}}(\theta^{\pi},\mu)=\mathbb{E}[\mu
log(1+e^{\pi(s|\theta^{\pi})}]$. In the context of the DDPG algorithm, the
actor parameters are updated according to the equation
$\theta^{\pi}_{old}=\theta^{\pi}-\text{lr}_{\text{actor}}\nabla
J(\theta^{\pi})$. Furthermore, in the Meta-DDPG variant, the actor parameters
are derived from a distinct process denoted as
$\theta^{\pi}_{\text{new}}=\theta^{\pi}_{\text{old}}-\text{lr}_{\text{actor}}\nabla
F_{\text{new}}(\theta^{\pi},\mu)$. The incorporation of these additional steps
yields superior performance and enhanced convergence compared to the
conventional DDPG algorithm.

##  V Simulation Results

The effectiveness of the proposed method is empirically assessed with a
comparison to classical methods. In the simulations, we set $M=20$, while $K$
is equal to 4. Additionally, it is supposed that the BS is equipped with $N=4$
antennas, positioned at coordinates (0,0) m. Meanwhile, the RIS is located at
(100,0) m. Furthermore, the system operates within defined constraints,
including $P_{\text{max}}=40~{}\textrm{dBm}$, $E_{\text{min}}=0.0004$, and
$R_{\text{min}}=0.1$ bps/Hz.

In Fig. 2, the Meta-DDPG depicts faster and smoother convergence compared to
the DDPG approach. Moreover, combining the Meta-DDPG with STAR-RIS results in
even faster convergence than with conventional RIS, highlighting the
significant improvement in convergence speed. It is noteworthy that the Meta-
DDPG with STAR-RIS outperforms the DDPG with STAR-RIS, emphasizing the
advantages of Meta-learning with STAR-RIS.

Figure 2: Convergence of the Meta-DDPG-based algorithm with $\alpha=0.5$.

In Fig. 3, the objective function versus maximum transmit power in the BS for
$\alpha=0.5$ is shown. We observe that the objective improves for different
methods by increasing the maximum transmit power. The maximum transmit power
of the BS is varied across a range of values, spanning from 8 dBm to 20 dBm.
The objective function for the STAR-RIS scenario consistently outperformed
that of the conventional RIS scenario. Notably, the performance of this
approach is close to the optimal scenario, where the exhaustive search method
is considered as the optimal approach. This indicates that STAR-RIS
demonstrates superior performance compared to conventional RIS, and it is
remarkably competitive with the theoretically optimal scenario.

Figure 3: The objective versus maximum transmit power with $\alpha=0.5$.

In Fig. 4, data rates are calculated for vast values of $\alpha$. When
combined with STAR-RIS, the Meta-DDPG approach consistently achieves the
highest data rates compared to the Meta-DDPG approach coupled with
conventional RIS. This observation indicates that integrating STAR-RIS with
the Meta-DDPG algorithm significantly enhances data rate performance, with
differences of nearly 1 (bps/Hz) between the two approaches. Indeed, it is
significant to point out that data rate obtained when the DDPG approach is
combined with STAR-RIS is lower than the data rate achieved by the Meta-DDPG
approach in conjunction with STAR-RIS. This observation implies that the
utilization of the Meta-learning variant of DDPG, in combination with STAR-
RIS, yields superior data rate performance compared to the standard DDPG
approach.

Figure 4: The data rate versus $\alpha$.

In the following, a simulation is conducted to assess the average normalized
harvested energy (ANHE) across a range of different values for $\alpha$, as
depicted in Fig. 5. This figure reinforces the point that the integration of
the Meta-DDPG approach with STAR-RIS yields acceptable results and enhances
energy harvesting performance.

Figure 5: The ANHE versus $\alpha$.

In Fig. 6, the tradeoff between the data rate and ANHE for various scenarios
is investigated. As can be observed, the data rate decreases with increasing
ANHE. This result validates the conflict between the goal of maximizing ANHE
and maximizing the data rate. It is noteworthy that by integrating the Meta-
DDPG approach with STAR-RIS, a more favorable trade-off condition appears.

Figure 6: Data rate versus ANHE.

##  VI Conclusion

In this letter, a novel approach was proposed to enhance the performance of
wireless system via the deployment of an STAR-RIS. Specifically, an STAR-RIS
SWIPT-assisted wireless MISO system with RSMA is considered, in which the
Meta-DDPG approach is applied to solve the non-convex NP-hard multi-objective
problem. The users are separated into two groups, namely IDRs and EHRs which
respectively receive the information and energy from the multiple-antenna BS.
Furthermore, the IDRs’s sum rate maximization problem as well as the
maximization of total harvested energy at the EHRs is solved by applying the
Meta-DDPG algorithm.

### VI-A

## References

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Cotton Solitons on Almost Kenmotsu 3-$h$-Manifolds

Dibakar Dey and Pradip Majhi

Department of Pure Mathematics

University of Calcutta, 35, Ballygunge Circular Road

Kolkata-700019, West Bengal, India

E-mail<EMAIL_ADDRESS>and Pradip Majhi

Abstract: In this paper, we consider the notion of Cotton soliton within the
framework of almost Kenmotsu 3-$h$-manifolds. First we consider that the
potential vector field is pointwise collinear with the Reeb vector field and
prove a non-existence of such Cotton soliton. Next we assume that the
potential vector field is orthogonal to the Reeb vector field. It is proved
that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-$h$-manifold
such that the Reeb vector field is an eigen vector of the Ricci operator is
steady and the manifold is locally isometric to
$\mathbb{H}^{2}(-4)\times\mathbb{R}$.

Mathematics Subject Classification 2010: Primary 53D15; Secondary 35Q51.

Keywords: Almost Kenmotsu manifolds, Cotton solitons, Non-unimodular Lie
group.

##  1 Introduction

An almost contact metric manifold is an odd dimensional differentiable
manifold $M^{2n+1}$ together with a structure $(\varphi,\xi,\eta,g)$
satisfying ([1], [2])

$\varphi^{2}X=-X+\eta(X)\xi,\;\;\eta(\xi)=1,$ (1.1) $g(\varphi X,\varphi
Y)=g(X,Y)-\eta(X)\eta(Y)$ (1.2)

for all vector fields $X$, $Y$ on $M^{2n+1}$, where $g$ is the Riemannian
metric, $\varphi$ is a $(1,1)$-tensor field, $\xi$ is a unit vector field
called the Reeb vector field and $\eta$ is a 1-form defined by
$\eta(X)=g(X,\xi)$. Here also $\phi\xi=0$ and $\eta\circ\phi=0$; both can be
derived from (1.1) easily. The fundamental 2-form $\Phi$ on an almost contact
metric manifold is defined by $\Phi(X,Y)=g(X,\varphi Y)$ for all vector fields
$X$, $Y$ on $M^{2n+1}$. The condition for an almost contact metric manifold
being normal is equivalent to vanishing of the $(1,2)$-type torsion tensor
$N_{\varphi}$, defined by $N_{\varphi}=[\varphi,\varphi]+2d\eta\otimes\xi$,
where $[\varphi,\varphi]$ is the Nijenhuis tensor of $\varphi$ [1]. Almost
contact metric manifold such that $\eta$ is closed and $d\Phi=2\eta\wedge\Phi$
are called almost Kenmotsu manifolds (see [6],[7]). Obviously, a normal almost
Kenmotsu manifold is a Kenmotsu manifold. Also Kenmotsu manifolds can be
characterized by $(\nabla_{X}\varphi)Y=g(\varphi X,Y)\xi-\eta(Y)\varphi X$,
for any vector fields $X$, $Y$ on $M^{2n+1}$.

The $(0,3)$-Cotton tensor of a 3-dimensional Riemannian manifold $(M^{3},g)$
is defined as

$\displaystyle C(X,Y,Z)$ $\displaystyle=$
$\displaystyle(\nabla_{X}S)(Y,Z)-(\nabla_{Y}S)(X,Z)$ (1.3)
$\displaystyle-\frac{1}{4}[(X(r)g(Y,Z)-Y(r)g(X,Z)],$

where $S$ is the Ricci tensor and $r$ is the scalar curvature of $M^{3}$. The
Cotton tensor is skew-symmetric in first two indices and totally trace free.
It is well known that for $n\geq 4$, an $n$-dimensional Riemannian manifold is
conformally flat if the Weyl tensor vanishes. For $n=3$, the Weyl tensor
always vanishes but the Cotton tensor does not vanish in general.

In 2008, Kicisel, Sarioğlu and Tekin [8] introduced the notion of Cotton flow
as a analogy of the Ricci flow. The Cotton flow is based on the conformally
invariant Cotton tensor and defined exclusively foe 3-simension as

$\displaystyle\frac{\partial g}{\partial t}=C,$

where $C$ is the $(0,2)$-Cotton tensor of $g$. From the Cotton flow, they
defined the notion of Cotton soliton as follows:

######  Definition 1.1.

A Cotton soliton is a metric $g$ defined on 3-dimensional smooth manifold
$M^{3}$ such that the following equation

$\displaystyle(\mathcal{L}_{V}g)(X,Y)+C(X,Y)-\sigma g(X,Y)=0,$ (1.4)

holds for a constant $\sigma$ and a vector field $V$, called the potential
vector field, where $\mathcal{L}_{V}$ denotes the Lie derivative along $V$ and
$C$ is the $(0,2)$-Cotton tensor defined by

$\displaystyle C_{ij}=\frac{1}{2\sqrt{g}}C_{nmi}\epsilon^{nml}g_{lj}$ (1.5)

in a local frame of $M^{3}$, where $g=\operatorname{det}(g_{ij})$, $C_{ijk}$
is the $(0,3)$-Cotton tensor and $\epsilon$ is a tensor density.

In an orthonormal frame, $\epsilon^{123}=1$. Also exchange of any two indices
will give rise to minus sign and it will be zero if there has two same
indices. For example, $\epsilon^{231}=-\epsilon^{213}$ and
$\epsilon^{112}=\epsilon^{122}=\epsilon^{223}=0$. Cotton solitons are fixed
points of the Cotton flow upto diffeomorphisms and rescaling. The Cotton
soliton is said to be shrinking, steady or expanding according as $\sigma$ is
positive, zero or negative respectively. As far as we know, the Cotton soliton
was studied by Chen [3] on certain almost contact metric manifold, precisely
on almost coKähler 3-manifolds. Motivated by the study of Chen [3], we
consider the notion of Cotton soliton on an almost Kenmotsu 3-$h$-manifold and
prove some related results.

##  2 Almost Kenmotsu 3-$h$-Manifolds

Let $(M^{3},\varphi,\xi,\eta,g)$ be a 3-dimensional almost Kenmotsu manifold.
We denote by $l=R(\cdot,\xi)\xi$, $h=\frac{1}{2}\mathcal{L}_{\xi}\varphi$ and
$h^{\prime}=h\circ\varphi$ on $M^{3}$, where $R$ is the Riemannian curvature
tensor. The tensor fields $l$ and $h$ are symmetric operators and satisfy the
following relations ([6], [7]):

$\displaystyle h\xi=0,\;l\xi=0,\;tr(h)=0,\;tr(h\varphi)=0,\;h\varphi+\varphi
h=0,$ (2.1) $\displaystyle\nabla_{X}\xi=X-\eta(X)\xi-\varphi
hX(\Rightarrow\nabla_{\xi}\xi=0),$ (2.2)
$\displaystyle\nabla_{\xi}h=-\varphi-2h-\varphi h^{2}-\varphi l.$ (2.3)

######  Definition 2.1.

[10] A 3-dimensional almost Kenmotsu manifold is called an almost Kenmotsu
3-$h$-manifold if it satisfies $\nabla_{\xi}h=0$.

Let $\mathcal{U}_{1}$ be the maximal open subset of a 3-dimensional almost
Kenmotsu manifold $M^{3}$ such that $h\neq 0$ and $\mathcal{U}_{2}$ be the
maximal open subset on which $h=0$. Then $\mathcal{U}_{1}\cup\mathcal{U}_{2}$
is an open and dense subset of $M^{3}$. Then $\mathcal{U}_{1}$ is non-empty
and there is a local orthonormal basis $\\{e_{1}=\xi,e_{2}=e,e_{3}=\varphi
e\\}$ on $\mathcal{U}_{1}$ such that $he=\lambda e$ and $h\varphi
e=-\lambda\varphi e$ for some positive function $\lambda$. Since
$\nabla_{\xi}h=0$ for an almost Kenmotsu 3-$h$-manifold, then using Lemma 6 of
[5] and (2.3), a direct calculation gives $\xi(\lambda)=a=0$. Therefore, Lemma
6 of [5] can be rewritten for an almost Kenmotsu 3-$h$-manifold as

######  Lemma 2.2.

On $\mathcal{U}_{1}$, the coefficients of the Riemannian connection $\nabla$
of an almost Kenmotsu 3-$h$-manifold with respect to a local orthonormal basis
$\\{\xi,e,\varphi e\\}$ is given by

$\nabla_{\xi}\xi=0,\;\;\nabla_{\xi}e=0,\;\;\nabla_{\xi}\phi e=0,$
$\nabla_{e}\xi=e-\lambda\varphi e,\;\;\nabla_{e}e=-\xi-b\varphi
e,\;\;\nabla_{e}\varphi e=\lambda\xi+be,$ $\nabla_{\varphi e}\xi=-\lambda
e+\varphi e,\;\;\nabla_{\varphi e}e=\lambda\xi+c\varphi e,\;\;\nabla_{\varphi
e}\varphi e=-\xi-ce,$

where $b$ and $c$ are smooth functions.

From Lemma 2.2, the Lie brackets can be calculated as follows:

$\displaystyle[e,\xi]=e-\lambda\varphi e,\;\;[e,\varphi e]=be-c\varphi
e\;\;\mathrm{and}\;\;[\varphi e,\xi]=-\lambda e+\varphi e.$ (2.4)

From Lemma 3.2 of [11], the Ricci operator $Q$ of an almost Kenmotsu
3-$h$-manifold can be obtained as

######  Lemma 2.3.

On $\mathcal{U}_{1}$, the Ricci operator of an almost Kenmotsu 3-$h$-manifold
with respect to a local orthonormal basis $\\{\xi,e,\varphi e\\}$ is given by

$Q\xi=-2(\lambda^{2}+1)\xi-[\varphi e(\lambda)+2\lambda
b]e-[e(\lambda)+2\lambda c]\varphi e,$ $Qe=-[\varphi e(\lambda)+2\lambda
b]\xi-fe+2\lambda\varphi e,$ $Q\varphi e=-[e(\lambda)+2\lambda c]\xi+2\lambda
e-f\varphi e,$

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

The scalar curvature $r$ of an almost Kenmotsu 3-$h$-manifold is given by

$\displaystyle r=g(Qe_{i},e_{i})=-2(\lambda^{2}+1)-2f.$ (2.5)

Using Lemma 2.3, we obtain

$\displaystyle\begin{cases}S(\xi,\xi)=-2(\lambda^{2}+1),\;S(\xi,e)=-[\varphi
e(\lambda)+2\lambda b],\\\ S(\xi,\varphi e)=-[e(\lambda)+2\lambda
c],\;S(e,e)=-f,\\\ S(e,\varphi e)=2\lambda,\;S(\varphi e,\varphi
e)=-f.\end{cases}$ (2.6)

It is well known that an almost Kenmotsu 3-manifold is Kenmotsu if and only if
$h=0$. Thus a Kenmotsu metric always admits a almost Kenmotsu 3-$h$-metric
structure. We now give an example of a non-Kenmotsu almost Kenmotsu
3-$h$-manifold.

######  Example 2.4.

[12] Let $M^{3}$ be a 3-dimensional non-unimodular Lie group with a left
invariant local orthonormal frame $\\{e_{1},e_{2},e_{3}\\}$ satisfying

$[e_{1},e_{2}]=\alpha e_{2}+\beta
e_{3},\;[e_{2},e_{3}]=0\;\mathrm{and}\;[e_{1},e_{3}]=\beta
e_{2}+(2-\alpha)e_{3}$

for $\alpha,\beta\in\mathbb{R}$. If either $\alpha\neq 1$ or $\beta\neq 0$,
then $M^{3}$ admits a non-Kenmotsu almost Kenmotsu 3-$h$-metric struicture.

##  3 Cotton Soliton

In this section, we consider the notion of Cotton soliton within the framework
of almost Kenmotsu 3-$h$-manifolds. To study the notion of Cotton soliton, we
need to compute the components of the $(0,2)$-Cotton tensor. In this regard,
we prove the following Lemma:

######  Lemma 3.1.

The components of the $(0,2)$-Cotton tensor $C$ with respect to an orthonormal
frame $\\{\xi,e,\varphi e\\}$ of a non-Kenmotsu almost Kenmotsu 3-$h$-manifold
$M^{3}$ can be expressed as follows:

$\displaystyle C_{11}=C(\xi,\xi)$ $\displaystyle=$ $\displaystyle b[\varphi
e(\lambda)+2\lambda b]-c[e(\lambda)+2\lambda c]$ (3.1)
$\displaystyle-e(e(\lambda)+2\lambda c)+\varphi e(\varphi e(\lambda)+2\lambda
b),$ $\displaystyle C_{12}=C(\xi,e)$ $\displaystyle=$ $\displaystyle
2[e(\lambda)-3\lambda\varphi e(\lambda)+2\lambda c-2\lambda^{2}b]$ (3.2)
$\displaystyle+\xi(e(\lambda)+2\lambda c)-\frac{1}{4}\varphi e(r),$
$\displaystyle C_{13}=C(\xi,\varphi e)$ $\displaystyle=$
$\displaystyle-2[\varphi e(\lambda)-3\lambda e(\lambda)+2\lambda
b-2\lambda^{2}c]$ (3.3) $\displaystyle-\xi(\varphi e(\lambda)+2\lambda
b)+\frac{1}{4}e(r),$ $\displaystyle
C_{22}=C(e,e)=2\lambda^{3}-f\lambda+c[e(\lambda)+2\lambda c]-\varphi e(\varphi
e(\lambda)+2\lambda b),$ (3.4) $C_{23}=C(e,\varphi e)=-\xi(f)-f+2+e(\varphi
e(\lambda)+2\lambda b)+b[e(\lambda)+2\lambda c]-\frac{1}{4}\xi(r),$ (3.5)
$C_{33}=C(\varphi e,\varphi e)=-2\lambda^{3}+f\lambda-b[\varphi
e(\lambda)+2\lambda b]+e(e(\lambda)+2\lambda c).$ (3.6)

###### Proof.

The components of the metric tensor $g$ with respect to an orthonormal frame
$\\{\xi,e,\varphi e\\}$ of a non-Kenmotsu almost Kenmotsu 3-$h$-manifold
$M^{3}$ is given by

$\displaystyle(g_{ij})=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right)$ (3.10)

and hence $\operatorname{det}(g_{ij})=1$. Therefore, equation (1.5) reduces to

$\displaystyle C_{ij}=\frac{1}{2}C_{nmi}\epsilon^{nmj},\;\;i,j=1,2,3,$

where $C_{ijk}=C(e_{i},e_{j},e_{k})$. Also, $C_{ijk}=-C_{jik}$ and $C_{iik}=0$
for all $i,j,k=1,2,3$. It can be easily obtained that (see [3])

$\displaystyle
C_{11}=C_{231},\;C_{12}=C_{311},\;C_{13}=C_{121},\;C_{22}=C_{312},\;C_{23}=C_{122},\;C_{33}=C_{123}.$

Making use of (1.3), we get the followings:

$\displaystyle C_{11}=C_{231}=C(e,\varphi e,\xi)=(\nabla_{e}S)(\varphi
e,\xi)-(\nabla_{\varphi e}S)(e,\xi),$ (3.11) $\displaystyle C_{12}=C(\varphi
e,\xi,\xi)=(\nabla_{\varphi e}S)(\xi,\xi)-(\nabla_{\xi}S)(\varphi
e,\xi)-\frac{1}{4}\varphi e(r),$ (3.12) $\displaystyle
C_{13}=C(\xi,e,\xi)=(\nabla_{\xi}S)(e,\xi)-(\nabla_{e}S)(\xi,\xi)+\frac{1}{4}e(r),$
(3.13) $\displaystyle C_{22}=C(\varphi e,\xi,e)=(\nabla_{\varphi
e}S)(\xi,e)-(\nabla_{\xi}S)(\varphi e,e),$ (3.14) $\displaystyle
C_{23}=C(\xi,e,e)=(\nabla_{\xi}S)(e,e)-(\nabla_{e}S)(\xi,e)-\frac{1}{4}\xi(r),$
(3.15) $\displaystyle C_{33}=C(\xi,e,\varphi e)=(\nabla_{\xi}S)(e,\varphi
e)-(\nabla_{e}S)(\xi,\varphi e).$ (3.16)

Using (2.6), Lemma 2.2 and $\xi(\lambda)=0$, we now obtain the followings:

$\displaystyle\begin{cases}(\nabla_{e}S)(\varphi
e,\xi)=2\lambda^{3}-f\lambda+b[\varphi e(\lambda)+2\lambda
b]-e(e(\lambda)+2\lambda c),\\\ (\nabla_{\varphi
e}S)(e,\xi)=2\lambda^{3}-f\lambda+c[e(\lambda)+2\lambda c]-\varphi e(\varphi
e(\lambda)+2\lambda b).\end{cases}$ (3.17)
$\displaystyle\begin{cases}(\nabla_{\varphi
e}S)(\xi,\xi)=2[e(\lambda)-3\lambda\varphi e(\lambda)+2\lambda
c-2\lambda^{2}b],\\\ (\nabla_{\xi}S)(\varphi e,\xi)=-\xi(e(\lambda)+2\lambda
c).\end{cases}$ (3.18)
$\displaystyle\begin{cases}(\nabla_{\xi}S)(e,\xi)=-\xi(\varphi
e(\lambda)+2\lambda b),\\\ (\nabla_{e}S)(\xi,\xi)=2[\varphi
e(\lambda)-3\lambda e(\lambda)+2\lambda b-2\lambda^{2}c].\end{cases}$ (3.19)
$\displaystyle\begin{cases}(\nabla_{\varphi
e}S)(\xi,e)=2\lambda^{3}-f\lambda+c[e(\lambda)+2\lambda c]-\varphi e(\varphi
e(\lambda)+2\lambda b),\\\ (\nabla_{\xi}S)(\varphi e,e)=0.\end{cases}$ (3.20)
$\displaystyle\begin{cases}(\nabla_{\xi}S)(e,e)=-\xi(f),\\\
(\nabla_{e}S)(\xi,e)=f-2-e(\varphi e(\lambda)+2\lambda
b)-b[e(\lambda)+2\lambda c].\end{cases}$ (3.21)
$\displaystyle\begin{cases}(\nabla_{\xi}S)(e,\varphi e)=0,\\\
(\nabla_{e}S)(\xi,\varphi e)=2\lambda^{3}-f\lambda+b[\varphi
e(\lambda)+2\lambda b]-e(e(\lambda)+2\lambda c).\end{cases}$ (3.22)

We now complete the proof by substituting the equations (3.17)-(3.22) in the
equations (3.11)-(3.16) respectively. ∎

######  Proposition 3.2.

If the Reeb vector field of a non-Kenmotsu almost Kenmotsu 3-$h$-manifold
$M^{3}$ is an eigen vector of the Ricci operator, then $M^{3}$ is locally
isometric to a non-unimodular Lie group equipped with a left invariant non-
Kenmotsu almost Kenmotsu structure.

###### Proof.

Since $\xi$ is an eigen vector of $Q$, then Lemma 2.3 implies

$\displaystyle\begin{cases}\varphi e(\lambda)+2\lambda b=0,\\\
e(\lambda)+2\lambda c=0.\end{cases}$ (3.23)

It is well known that

$\frac{1}{2}X(r)=(\operatorname{div}Q)X=\sum_{i=1}^{3}g((\nabla_{e_{i}}Q)X,e_{i}).$

From the preceding equation, we can write

$\displaystyle\frac{1}{2}X(r)=(\nabla_{\xi}S)(X,\xi)+(\nabla_{e}S)(X,e)+(\nabla_{\varphi
e}S)(X,\varphi e).$ (3.24)

Making use of (2.6), (3.23) and $\xi(\lambda)=0$, we obtain the followings:

$(\nabla_{\xi}S)(\xi,\xi)=0,\;(\nabla_{e}S)(\xi,e)=f-2,\;(\nabla_{\varphi
e}S)(\xi,\varphi e)=f-2,$ (3.25)
$(\nabla_{\xi}S)(e,\xi)=0,\;(\nabla_{e}S)(e,e)=4\lambda
b-e(f),\;(\nabla_{\varphi e}S)(e,\varphi e)=-4\lambda b,$ (3.26)
$(\nabla_{\xi}S)(\varphi e,\xi)=0,\;(\nabla_{e}S)(\varphi e,e)=-4\lambda
c,\;(\nabla_{\varphi e}S)(\varphi e,\varphi e)=4\lambda c-\varphi e(f).$
(3.27)

Now, substituting $X=\xi,\;e\;\mathrm{and}\;\varphi e$ in (3.24) and then
using (3.25), (3.26) and (3.27) respectively, we obtain

$\displaystyle\xi(r)=4(f-2),\;\;e(r)=-2e(f),\;\;\varphi e(r)=-2\varphi e(f).$
(3.28)

Using (3.23) and $\xi(\lambda)=0$, we get from (2.5)

$\displaystyle\xi(r)=-2\xi(f),\;e(r)=-2e(f)+8\lambda^{2}c,\;\varphi
e(r)=-2\varphi e(f)+8\lambda^{2}b.$ (3.29)

Since $\lambda$ is a positive function, then the second and third equations of
(3.28) and (3.29) implies $b=c=0$. From Lemma 2.3, we get $f=2$. Also from
(3.23), we get $e(\lambda)=\varphi e(\lambda)=0$ and therefore $\lambda$ is a
constant. Now, the Lie brackets given in (2.4) reduces to

$[e,\xi]=e-\lambda\varphi e,\;\;[e,\varphi e]=0\;\;\mathrm{and}\;\;[\varphi
e,\xi]=-\lambda e+\varphi e.$

Therefore, according to Milnor [9], $M^{3}$ is locally isometric to a non-
unimodular Lie group equipped with a left invariant non-Kenmotsu almost
Kenmotsu structure. ∎

Combining Lemma 3.1 and Proposition 3.2, the components of the Cotton tensor
can be written as given in the follwing Corollary.

######  Corollary 3.3.

If the Reeb vector field of a non-Kenmotsu almost Kenmotsu 3-$h$-manifold
$M^{3}$ is an eigen vector of the Ricci operator, then the components of the
$(0,2)$-Cotton tensor $C$ with respect to an orthonormal frame
$\\{\xi,e,\varphi e\\}$ on $M^{3}$ can be expressed as follows:

$C_{11}=C(\xi,\xi)=0,\;C_{12}=C(\xi,e)=0,\;C_{13}=C(\xi,\varphi e)=0,$
$C_{22}=C(e,e)=2\lambda^{3}-2\lambda,\;C_{23}=C(e,\varphi
e)=0,\;C_{33}=C(\varphi e,\varphi e)=-2\lambda^{3}+2\lambda.$

We first consider the Cotton soliton with potential vector field $V$ pointwise
collinear with the Reeb vector field. In this regard, we prove the following
non-existing result.

######  Theorem 3.4.

On a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the Reeb vector
field is an eigen vector of the Ricci operator, there exist no Cotton soliton
with potential vector field pointwise collinear with the Reeb vector field.

###### Proof.

Suppose that the potential vector field $V$ is pointwise collinear with the
Reeb vector field $\xi$. Then there exist a non-zero smooth function $\alpha$
on $M^{3}$ such that $V=\alpha\xi$. Now, substituting $X=e$ and $Y=\varphi e$
in (1.4) and using Lemma 2.2 and Corollary 3.3, we get $2\lambda\alpha=0$.
This gives either $\lambda=0$ or $\alpha=0$. In either cases, we get a
contradiction. This completes the proof. ∎

From Theorem 3.4 and Proposition 3.2, we can say the following:

######  Corollary 3.5.

On a 3-dimensional non-unimodular Lie group equipped with a left invariant
non-Kenmotsu almost Kenmotsu structure, there exist no Cotton soliton with
potential vector field pointwise collinear with the Reeb vector field.

It is now quite tempting to consider the potential vector field $V$ as
orthogonal to the Reeb vector field. In this setting, we prove the following:

######  Theorem 3.6.

Let $(M^{3},g)$ be a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the
Reeb vector field is an eigen vector of the Ricci operator. If $g$ is a Cotton
soliton with potential vector field orthogonal to the Reeb vector field, then
$M^{3}$ is locally isometric to $\mathbb{H}^{2}(-4)\times\mathbb{R}$ and the
Cotton soliton is steady.

###### Proof.

For a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the Reeb vector
field is an eigen vector of the Ricci operator, Proposition 3.2 gives
$b=c=0,\;f=2,\;\lambda=\mathrm{constant}\;\mathrm{and}\;r=\mathrm{constant}$.
Since $V$ is orthogonal to $\xi$, then there exist two smooth functions
$\alpha_{1}$ and $\alpha_{2}$ on $M^{3}$ such that
$V=\alpha_{1}e+\alpha_{2}\varphi e$. With the help of Lemma 2.2, we now obtain
the components of $\mathcal{L}_{V}g$ as follows:

$\displaystyle\begin{cases}(\mathcal{L}_{V}g)(\xi,\xi)=0,\;(\mathcal{L}_{V}g)(\xi,e)=\xi(\alpha_{1})-\alpha_{1}+\lambda\alpha_{2},\\\
(\mathcal{L}_{V}g)(\xi,\varphi
e)=\xi(\alpha_{2})-\alpha_{2}+\lambda\alpha_{1},\;(\mathcal{L}_{V}g)(e,e)=2e(\alpha_{1}),\\\
(\mathcal{L}_{V}g)(e,\varphi e)=e(\alpha_{2})+\varphi
e(\alpha_{1}),\;(\mathcal{L}_{V}g)(\varphi e,\varphi e)=2\varphi
e(\alpha_{2}).\end{cases}$ (3.30)

We now use Corollary 3.3 and the set of equations (3.30). Substituting
$X=Y=\xi$ in (1.4), we get $\sigma=0$. This shows that the Cotton soliton is
steady. Now, substitution of $X=\xi$, $Y=e$ in (1.4) yields

$\displaystyle\xi(\alpha_{1})-\alpha_{1}+\lambda\alpha_{2}=0.$ (3.31)

Replacing $X$ by $\xi$ and $Y$ by $\varphi e$ in (1.4), we get

$\displaystyle\xi(\alpha_{2})-\alpha_{2}+\lambda\alpha_{1}=0.$ (3.32)

Putting $X=Y=e$ in (1.4), we obtain

$\displaystyle 2e(\alpha_{1})+2\lambda^{3}-2\lambda=0.$ (3.33)

Substitution of $X=e$ and $Y=\varphi e$ in (1.4) yields

$\displaystyle e(\alpha_{2})+\varphi e(\alpha_{1})=0.$ (3.34)

Putting $X=Y=\varphi e$ in (1.4), we infer

$\displaystyle 2\varphi e(\alpha_{2})-2\lambda^{3}+2\lambda=0.$ (3.35)

Since $b=c=0$, the Lie brackets given in (2.4) reduces to

$\displaystyle[e,\xi]=e-\lambda\varphi e,\;\;[e,\varphi
e]=0\;\;\mathrm{and}\;\;[\varphi e,\xi]=-\lambda e+\varphi e.$ (3.36)

Since $\lambda$ is a positive constant, then from (3.31) and (3.33), we obtain

$e(\xi(\alpha_{1}))=e(\alpha_{1})-\lambda
e(\alpha_{2})\;\;\mathrm{and}\;\;\xi(e(\alpha_{1}))=0.$

Applying the first Lie bracket of (3.36) in the preceding eqaution, we get
$\varphi e(\alpha_{1})=e(\alpha_{2})$. Hence, equation (3.34) implies $\varphi
e(\alpha_{1})=e(\alpha_{2})=0$. Now, from (3.32), we get
$e(\xi(\alpha_{2}))=-\lambda e(\alpha_{1})$. Also, we have
$\xi(e(\alpha_{2}))=0$. Again, using these two in the first Lie bracket of
(3.36) yields $\varphi e(\alpha_{2})=e(\alpha_{1})$. Applying (3.33) and
(3.35) in the preceding relation and using the fact that $\lambda$ is a
positive function, we obtain $\lambda=1$. Now, it is easy to check that
$\nabla Q=0$. Notice that, a Riemannian 3-manifold is Ricci parallel if and
only if it is locally symmetric. The rest of the proof follows from Theorem 5
of [4] which says that “A non-Kenmotsu almost Kenmotsu 3-manifold $M^{3}$ is
locally symmetric if and only if $M^{3}$ is locally isometric to the product
space $\mathbb{H}^{2}(-4)\times\mathbb{R}$”. ∎

As a combination of Proposition 3.2 and Theorem 3.6, we have the following:

######  Corollary 3.7.

If $g$ is a Cotton soliton with potential vector field orthogonal to the Reeb
vector field on a 3-dimensional non-unimodular Lie group $M^{3}$ equipped with
a left invariant non-Kenmotsu almost Kenmotsu structure, then $M^{3}$ is
locally isometric to $\mathbb{H}^{2}(-4)\times\mathbb{R}$ and the Cotton
soliton is steady.

### Acknowledgement

The author Dibakar Dey is thankful to the Council of Scientific and Industrial
Research, India (File no: 09/028(1010)/2017-EMR-1) for their assistance.

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