# Benchmarking Optimizers for Qumode State Preparation with Variational
Quantum Algorithms

Shuwen Kan1*, Miguel Palma1*, Zefan Du 1
Samuel A Stein2, Chenxu Liu2, Juntao Chen1, Ang Li2, and Ying Mao1
1 Computer and Information Science Department, Fordham University,
{sk107, mip2, zdu19, jchen504<EMAIL_ADDRESS>
2Pacific Northwest National Laboratory (PNNL), {samuel.stein, chenxu.liu,
<EMAIL_ADDRESS>
*Authors contributed equally

###### Abstract

Quantum state preparation involves preparing a target state from an initial
system, a process integral to applications such as quantum machine learning
and solving systems of linear equations. Recently, there has been a growing
interest in qumodes due to advancements in the field and their potential
applications. However there is a notable gap in the literature specifically
addressing this area. This paper aims to bridge this gap by providing
performance benchmarks of various optimizers used in state preparation with
Variational Quantum Algorithms. We conducted extensive testing across multiple
scenarios, including different target states, both ideal and sampling
simulations, and varying numbers of basis gate layers. Our evaluations offer
insights into the complexity of learning each type of target state and
demonstrate that some optimizers perform better than others in this context.
Notably, the Powell optimizer was found to be exceptionally robust against
sampling errors, making it a preferred choice in scenarios prone to such
inaccuracies. Additionally, the Simultaneous Perturbation Stochastic
Approximation optimizer was distinguished for its efficiency and ability to
handle increased parameter dimensionality effectively.

##  I introduction

Quantum computing has the potential to solve computational problems that are
simply intractable on classical computers[1]. Superposition, an exponentially
large computational Hilbert space, and entanglement, give rise to a new model
of computation. Seminal algorithms such as Shor’s Algorithm [2] or Digital
Quantum Simulation of physical systems[3], motivate the field to develop
quantum computers powerful enough to tackle these problems.

Despite this significant potential, hardware and software problems limit the
physical realization of quantum computers[4]. Arguably, the largest challenge
in quantum computing is the inability to escape noise[5]. Imperfect control,
decoherence, and undesirable coupling, are problems that plague quantum
computing platforms today but quantum error correction provides a path towards
exponential error suppression [6].

Quantum computing diverges from today’s classical computers, which rely solely
on transistor-based technology, as researchers have not yet standardized a
single approach for constructing the perfect qubit. Currently, the field
features multiple architectures, each with its own set of strengths and
weaknesses. Superconducting circuits, for instance, utilize supercooled
circuits to create an anharmonic oscillator [7], offering rapid operation
times. However, these systems face challenges such as short decoherence times
and limited qubit connectivity. On the other hand, trapped ion quantum
computers employ lasers to manipulate and control charged atoms held in place
by magnetic fields [8]. While these systems boast longer decoherence times and
are generally more manageable, they operate more slowly and present
difficulties in optical and electrical control.

Qumodes represent a promising approach in quantum computing, characterized by
their use of continuous variable systems over infinite dimensional matrices
[9]. Typically associated with bosonic modes, these systems theoretically can
reach infinitely high Fock levels. However, such levels are not physically
realistic, prompting the truncation of these matrices to a manageable, finite
dimension that reflects the highest achievable Fock level. Despite being less
explored than other quantum systems, qumodes are particularly noteworthy due
to their natural alignment with real-world applications. Examples include
optimizing graph cliques [10], analyzing point processes [11], and simulating
molecular vibrational spectra [12], showcasing their potential in practical
scenarios. There have been notable developments in qumode control and readout
[13, 14, 15, 16]. Beamsplitter operations with gate fidelities exceeding
99.98%[13] have been reported and show promise that high-reliability physical
implementations are possible in the near future. Additionally, a compiler
optimzation framework for bosonic quantum acheived a 39.6% reduction of
Beamsplitter gate counts while maintaining fidelity at 98.00% [14], improving
the qumode software and compiler state of the art.

Variational Quantum Algorithms (VQA) have garnered interest due to their
ability to mitigate the effects of noise in contemporary quantum hardware [17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. VQAs operate by utilizing quantum
hardware to execute parameterized quantum circuits, alongside classical
systems that run optimization algorithms to adjust these parameters. This
dual-system approach enables the application of VQAs in a variety of fields,
including solving optimization and eigenvalue problems that are prevalent in
quantum physics and chemistry simulations.

State preparation is a crucial subroutine in various quantum algorithms,
notably in Variational Quantum Algorithms (VQA)-based quantum learning,
Hamiltonian simulation, and solving systems of linear equations [28]. This
process, which involves loading classical information into quantum devices
before computation, is notably resource-intensive. Specifically, preparing
N-dimensional vectors necessitates a circuit with depth proportional to O(N)
[29]. For practical applications to realize Quantum Advantage, the development
of an efficient state preparation method is paramount. While traditional
quantum state preparation has been extensively studied [30, 31, 32, 33],
qumode state preparation has received less attention.

In this paper, we focus on VQA-based applications and investigate the quantum
learning of a gate sequence tailored for arbitrary qumode target states, aimed
at enhancing qumode state preparation efficiency. The key contributions of
this research are as follows:

  * •

We conduct a comprehensive benchmark study on the performance of various
optimizers in qumode state preparation with different settings.

  * •

Our evaluations compare various optimizers in the task of Local and Non-local
Gaussian states as well as Non Gaussian using a 2-gateset VQA.

  * •

The experiments demonstrate that learning Non-Gaussian states posed a greater
challenge, requiring up to 10 layers to minimize state preparation error.

##  II Related Work

When discussing how quantum computers outperform classical devices, a crucial
component is the loading of classical data onto these quantum devices. The No
Cloning Theorem states that that arbitrary quantum states cannot be duplicated
the same way data can be duplicated on classical devices [34], making it a
non-trivial problem. The key problem can be stated as determining the correct
quantum gate sequence that transforms an initial quantum state to the desired
target state with minimal error and in the least amount of time possible.

Quantum state preparation schemes are classified into two broad categories:
arithmetic decomposition (AD) and variational quantum state preparation (VQSP)
[35]. AD approaches employ a more analytical approach to generating circuits
to produce the target state, however they are affected by quantum noise on
real devices that throw off the results. VQSP, on the other hand, uses the
variational method to approximate the target state by iteratively updating a
set of parameters. This approach is more suited to NISQ devices because of its
limited circuit depth and gate count.

Variational quantum algorithms (VQA) utilize classical optimizers to determine
which direction to adjust parameters. Optimizers are also organized under two
categories: gradient-based and gradient-free optimizers. The gradient is the
derivative of the objective function that specifies the rate of change of
input to the output. Gradient-free methods sample the function at various
points to derive an estimated value of the gradient. The choice of optimizer
to use affects the quality of the output and the number of iterations needed
to approach a reasonable approximate solution.

Bosonic quantum is a separate branch of quantum computing based on bosonic
qumodes. The fundamental difference is that these qumodes are continuous
variable systems, as opposed to the 2-state qubit system. However, this
capacity for an infinite number of states is truncated when simulating on
classical devices. Bosonic quantum has immense potential to advance quantum
algorithms because it presents a simpler and more versatile way of encoding
and representing quantum information [36].

Despite the potential of bosonic qumodes and the necessity of state
preparation in quantum algorithms, there is barely any research on preparation
of target qumode states. A study shows that an energy-dependent barren plateau
phenomenon is exhibited in the cost function of training bosonic VQCs for
state preparation [37]. Our contribution is to provide an empirical benchmark
and comparison of different optimizers for this task.

##  III background

###  III-A Qumodes

Qumodes are a type of quantum system characterized by having an infinite
number of levels, which makes them capable of being in a superposition of
multiple Fock basis states. These states are represented by a linear
combination: $\rho=\alpha_{0}|0\rangle+...+\alpha_{n}|N\rangle$, where
$\alpha_{0}$, … , $\alpha_{n}$ are complex numbers that sum to 1 when squared.
When dealing with systems comprising multiple qumodes, their overall state can
be described by the tensor product of the individual qumodes’ states. However,
in practical applications, qumodes are often truncated to a manageable number
of Fock levels, ranging from 0 (the vacuum state) to $N$, where $N$ serves as
the cutoff value. This truncation simplifies calculations and makes
experimental realizations feasible.

###  III-B Variational Qumode Circuits

Variational Quantum Algorithms (VQAs) are hybrid algorithms that employ both
quantum and classical computing techniques to find approximate solutions to
problems. These algorithms are particularly effective in mitigating the
limitations of current quantum devices, which include limited qubit counts and
high susceptibility to noise. In VQAs, a parametrized quantum circuit encodes
the problem at hand, while a classical optimization algorithm iteratively
adjusts the circuit’s input parameters to refine the solution. However, it is
important to note that VQAs do not guarantee the discovery of an exact
solution.

The effectiveness of VQAs in practical applications is often hindered by
several limitations. One significant challenge is the Barren Plateau (BP)
phenomenon, where the landscape of the cost function is extremely flat across
vast regions, with the solution residing in a narrow, difficult-to-locate
gorge [18]. In such scenarios, the optimizer struggles to find adequate
gradient information to effectively navigate out of the BP, often resulting in
prolonged convergence times or the acceptance of suboptimal solutions. The
expressiveness of the parametrized circuit and the choice of initial
parameters are crucial factors that influence the likelihood of encountering
BPs.

###  III-C Optimizers

Optimizers play a crucial role in Variational Quantum Algorithms (VQAs) as
they navigate the cost landscape to minimize the cost function. Integral to
logistics planning, operations research, and machine learning, these
algorithms are categorized into two primary groups: gradient-based methods and
gradient-free methods. This classification highlights the varied approaches
used to tackle optimization problems in quantum computing environments.

The performance of these optimizers is assessed on several fronts. Accuracy is
crucial as it measures how closely the optimizer’s solution approaches the
global minimum and its robustness against noise, essential for reliable
quantum computing. Speed, determined by the number of iterations to
convergence and the number of samples per iteration, reflects the efficiency
of the optimizer. Scalability indicates the optimizer’s performance as the
number of parameters increases, important for complex systems. Lastly,
consistency evaluates the optimizer’s ability to deliver similar results
across different runs with randomized initial parameters, ensuring reliability
and repeatability in outcomes. Together, these factors determine the
effectiveness of optimizers in practical quantum computing applications.

##  IV evaluation

This section evaluates the performance of various optimizers for VQAs on
qumode systems, specifically in learning parameter values to approximate
arbitrary target states. The parameters are structured into layers, with each
layer comprising five real numbers:
$[v_{r},v_{i},\theta_{x},\theta_{y},\theta_{z}]$. Here, $v_{r}+v_{i}*i$ forms
the complex number for the VP gate, and $\theta_{x}$, $\theta_{y}$,
$\theta_{z}$ serve as input parameters for the RX, RY, and RZ gates,
respectively.

For the experiments, we implement testing cases with Qiskit Bosonic [38] along
with customized gates and QuTiP functions. This study employs the Python
libraries Scipy [39] and SPSA [40], renowned for their extensive use in
scientific computing and engineering. We have tested following optimizers
including L-BFGS-B [41], Conjugate Gradient (CG) [42], Sequential Least
Squares Programming (SLSQP) [43], Constrained Optimization by Linear
Approximation (COBYLA) [44], Nelder-Mead [45], Powell [46], Implicit Filtering
(ImFil) [47], Bound Optimization BY Quadratic Approximation (PyBobyQA) [48],
and Simultaneous Perturbation Stochastic Approximation (SPSA) [49].

Additionally, to evaluate the similarity between the learned and target
states, we utilize the SWAP test. Given the prohibitive cost of state
tomography, the SWAP test provides a more efficient means of approximating
state similarities due to its lower sampling requirements. In our study, which
focuses on learning latent states and reproducing them parametrically, the
SWAP test is particularly suitable. It measures the squared inner product
between two qumode state vectors, translating this measure into the
probability of observing a ’0’ on an ancilla coupled transmon, as determined
by the algorithm.

###  IV-A System Setup

####  IV-A1 Basis gates

Given a qubit-coupled bosonic quantum system, we opt to analyse applying
layers of following computationally complete sets of instructions:

$V_{p}(\alpha)=e^{i\sigma_{z}(\alpha a^{\dagger}-\alpha^{*}a)}$
$R(\hat{b},\theta)=e^{i\theta\hat{b}\cdot\hat{\sigma}}$

  * •

$\alpha$ is a complex number for $V_{p}$ gate

  * •

$\theta$ is a real number in the range $[0,2\pi)$ for rotation,

  * •

$\hat{b}$ is a unit 3D vector,

  * •

$\hat{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ represents the Pauli
matrices.

Layers are comprised of a complete U3 rotation on the coupled transmon
parameterised by two parameters, providing complete control over the transmon,
followed by an application of the parametric displation gate (Vp). This is
considered a layer in further discussion.

####  IV-A2 Objective Function

Following the gate layer applications, the swap test is performed between the
two qumodes. The swap test results in the transmon having a P($|0\rangle$) of
$0.5+0.5\times|\langle\Phi\Psi\rangle|^{2}$. Our goal is to maximize the
fidelity of our learnt state and a target state, hence optimizing directly
over the swap test results can lead to gradient scaling issues due to the
squared fidelity term. To address this, we applied a square root
transformation to the swap test results, resulting in
$1-\sqrt{2\times(P(|0\rangle)-0.5)}$. Consequently, our final objective
function becomes infidelity. In scenarios where sampling simulation produces
swap test results lower than 0.5, leading to negative values inside the square
root, a lower bound was imposed to ensure that the swap test result remains
greater than 0.5.

####  IV-A3 Target State of Interest

Given the impracticality of simulating infinite dimensions, we truncate the
bosonic systems to a 10-level system for general experiments, which we deemed
sufficient to represent the distributions of interest. Three types of target
states were explored in our experiments: Local Gaussian , Gaussian, and Non-
Gaussian states. We selected an identical target state for different
optimizers under the same settings. The target states we investigated were as
follows: Local Gaussian (mean = 0, std = 0.75), Gaussian distribution (mean =
5, std = 1), and Non-Gaussian distribution (0, 0.209, 0.417, 0.209, 0, 0.417,
0.626, 0.417, 0, 0).

####  IV-A4 Workload

In this study, two primary metrics, the final infidelity value and the number
of function evaluations (nfev) are evaluated for benchmarking purpose.
Infidelity directly relates to optimizer’s ability to approach global optimal.
The number of function evaluations directly relates to the runtime efficiency
for each optimizer, correlating to how many times one would need to optimize
over real life experiments. In the result table, a standard deviation of 0 may
be observed for COBYLA and SPSA. This occurs because Scipy enforces a
1000-iteration limit for the COBYLA optimizer, causing certain results to
display an nfev standard deviation of 0. Furthermore, a custom iteration count
of 1000 is utilized for the SPSA optimizer. We conducted experiments with
varying numbers of layers and truncation levels. Each experimental setup was
replicated 30 times.

Each target state is tested by two simulation methods, ideal simulation and
sampling simulation. In ideal simulation, the swap test results reflects the
use of the statevector directly, whereas sampling simulation incorporates a
sampling process, representing the real world experimental process.

####  IV-A5 Optimizers

The optimizers we tested fall into two categories: gradient-based optimizers
and derivative optimizers. Moreover, to effectively tackle the challenges
arising from the high-dimensional nature of the problem when multiple layers
are needed to approximate the target state closely, we utilized the
Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer.

In the context of gradient-based optimizers, the selection of step size for
finite difference gradient estimation is crucial. Smaller step sizes tend to
be heavily influenced by sampling error, while larger sample sizes may be able
to provide an unbiased estimator of the gradient. Throughout our experiments,
we discovered that an adaptive step size approach was prone to encountering
local optima. Consequently, we opted for a fixed step size of 0.03 for ideal
simulation. However, for sampling simulation, both the step size and sampling
size must be carefully addressed to ensure accurate results. Table I
illustrates the percentage of non-converging trials for L-BFGS-B, a gradient-
based optimizer, across various sampling sizes and step sizes. We conducted
200 trials with sample sizes ranging from 1024 to 8192, with increments of
1024 and step size selection of [0.03,0.05,0.08].

The findings indicate that a sampling size of 1024 proves insufficient for
providing a meaningful gradient estimation direction, resulting in non-
converging trials ranging from 15% to 38.5%. As sampling size reaches between
6144 and 8192, percentage of non-converging dropped below 5%. Given that
increasing the sampling size corresponds to an increase in the number of shots
of measurement in practical implementation, which can be costly, we opted to
use a sampling size of 6144 for our subsequent sampling simulations.

It is notable that in ideal simulation, as indicated in Table II, the L-BFGS-B
optimizer is expected to attain an infidelity of 0.003. For trials that
converge, the final infidelity is anticipated to be close to this value,
albeit subject to some sampling error. Hence, the standard deviation is
closely associated with the number of non-converging trials, potentially
yielding values within the range of [0.3, 0.9].

TABLE I: Evaluation of Various Step Sizes and Sample Sizes for gradient-based
optimizer for Sampling Simulation

trial | sample_size | step size | infidelity_mean | infidelity_std | None_converge(%)
---|---|---|---|---|---
200 | 1024 | 0.03 | 0.299 | 0.408 | 38.5
200 | 1024 | 0.05 | 0.198 | 0.354 | 25
200 | 1024 | 0.08 | 0.131 | 0.296 | 15
200 | 2048 | 0.03 | 0.210 | 0.362 | 25.5
200 | 2048 | 0.05 | 0.126 | 0.292 | 14.5
200 | 2048 | 0.08 | 0.113 | 0.281 | 11.5
200 | 3072 | 0.03 | 0.199 | 0.360 | 23
200 | 3072 | 0.05 | 0.0950 | 0.252 | 12
200 | 3072 | 0.08 | 0.0629 | 0.194 | 7
200 | 4096 | 0.03 | 0.198 | 0.361 | 23
200 | 4096 | 0.05 | 0.0604 | 0.184 | 9.5
200 | 4096 | 0.08 | 0.0418 | 0.115 | 5.5
200 | 5120 | 0.03 | 0.0844 | 0.233 | 10
200 | 5120 | 0.05 | 0.0510 | 0.159 | 5.5
200 | 5120 | 0.08 | 0.0427 | 0.128 | 7
200 | 6144 | 0.03 | 0.143 | 0.317 | 17
200 | 6144 | 0.05 | 0.0593 | 0.192 | 6
1000 | 6144 | 0.08 | 0.0332 | 0.192 | 4.8
1000 | 7168 | 0.08 | 0.0290 | 0.0948 | 3.5
1000 | 8192 | 0.08 | 0.0297 | 0.106 | 3.5

###  IV-B Local Gaussian State

####  IV-B1 Ideal Simulation

The results can be found in Table II and our we observe that the Powell
optimizer was able to minimize infidelity the most at the cost of also having
the highest average number of function evaluations. The mean and std dev of
infidelity remains more or less consistent even when the cutoff and layer
count values are changed while the mean and std dev of the number of
evaluations showed a lot more variation in the results. There is no observed
relationship between the number of layers and cutoff on the average
infidelity.

TABLE II: Local Gaussian State of Ideal Simulation

layers | method | infidelity_mean | infidelity_std | nfev_mean | nfev_mean
---|---|---|---|---|---
1 | CG | 0.00358 | 0.00192 | 34.58 | 22.08
2 | CG | 0.00309 | 0.000722 | 140.4 | 65.32
1 | L-BFGS-B | 0.00353 | 0.00308 | 21.34 | 15.17
2 | L-BFGS-B | 0.00385 | 0.00504 | 53.66 | 25.93
1 | SLSQP | 0.00405 | 0.00208 | 18.24 | 14.31
2 | SLSQP | 0.00355 | 0.000948 | 23.48 | 10.86
1 | SPSA | 0.00740 | 0.0111 | 1000 | 0
2 | SPSA | XXXX | XXXX
1 | Nelder-Mead | 0.00313 | 0.000442 | 254.9 | 60.77
2 | Nelder-Mead | 0.00300 | 0.000609 | 991.28 | 366.13
1 | Powell | 0.00330 | 0.000713 | 487.82 | 282.64
2 | Powell | 0.00339 | 0.000567 | 2415.06 | 1617.42
1 | COBYLA | 0.00369 | 0.00156 | 427.3 | 452.93
2 | COBYLA | 0.00425 | 0.000920 | 952.58 | 189.93

####  IV-B2 Sampling Simulation

The results are displayed in Table III. The Powell optimizer showcases the
highest robustness against sampling errors. Generally, derivative-free
optimizers outshine gradient-based ones, as sampling error markedly impacts
the gradient estimation process. Among the three gradient-based optimizers, CG
is the most susceptible to sampling error, while SPSA exhibits the most
robustness. Notably, increasing the layer from 1 to 2 for the CG optimizer
helps achieve results comparable to other gradient-based optimizers.

TABLE III: Local Gaussian State of Sampling Simulation

layers | method | infidelity_mean | infidelity_std | nfev_mean | nfev_mean
---|---|---|---|---|---
1 | CG | 0.156 | 0.162 | 32.85 | 8.526
2 | CG | 0.0188 | 0.0168 | 35.8 | 10.272
1 | L-BFGS-B | 0.0337 | 0.0336 | 17.2 | 5.205
2 | L-BFGS-B | 0.0142 | 0.0122 | 16.1 | 4.433
1 | SLSQP | 0.111 | 0.115 | 21.05 | 8.565
2 | SLSQP | 0.0152 | 0.0116 | 27 | 14.259
1 | SPSA | 0.0112 | 0.0161 | 1000 | 0
2 | SPSA | 0.00957 | 0.0119
1 | Nelder-Mead | 0.0943 | 0.0977 | 2773.15 | 67.729
2 | Nelder-Mead | 0.0797 | 0.209 | 2854.6 | 95.060
1 | Powell | 0.00516 | 0.00450 | 183.5 | 64.209
2 | Powell | 0.00778 | 0.00608 | 491.9 | 147.625
1 | COBYLA | 0.0154 | 0.0143 | 61.9 | 4.959
2 | COBYLA | 0.0250 | 0.0203 | 108.3 | 9.934

###  IV-C Gaussian State

####  IV-C1 Ideal Simulation

The results are displayed in Table IV. With an increased number of layers, the
dimensionality of parameters expands, creating a more intricate optimization
space. Consequently, the number of function evaluations grows nonlinearly with
each additional layer. An unusual trend observed in this table is that for
layer=1, CG and L-BFGS-B optimizers require a significantly higher number of
function evaluations. This anomaly arises because, with only one layer, the
final learned state fails to capture the full complexity of the Gaussian
distribution. Consequently, these two optimizers struggle to converge. Prior
to termination, they expend additional iterations attempting to achieve
convergence.

It’s important to note that Powell consistently outperforms other optimizers
despite this increase in complexity. While derivative-free optimizers require
a greater number of function evaluations, they maintain efficiency in terms of
time. It’s crucial to acknowledge that gradient-based optimizers (CG,
L-BFGS-B, SLSQP) entail additional objective function evaluations for finite
difference gradient estimation. This process involves evaluating the objective
function multiple times based on the dimensionality of the parameter space.
For example, with two layers, each gradient estimation requires 20 function
evaluations. Therefore, the overall number of objective function evaluations
for CG, L-BFGS-B, and SLSQP is 2499, 882, and 672, respectively. With three
layers, these numbers increase to 7920, 2759, and 1984, respectively, with CG
being the most inefficient. Since the number of objective function evaluations
directly affects the optimizer’s runtime, gradient-based optimizers become
less efficient as the number of layers increases, making derivative-free
optimizers more favorable in terms of time efficiency.

TABLE IV: Gaussian State of Ideal Simulation

layers | method | infidelity_mean | infidelity_std | nfev_mean | nfev_mean
---|---|---|---|---|---
1 | CG | 0.372 | 0.198 | 642.8 | 288.16
2 | CG | 0.128 | 0.0516 | 119.56 | 28.47
3 | CG | 0.0549 | 0.0489 | 264.7 | 94.92
1 | L-BFGS-B | 0.361 | 0.194 | 375.38 | 187.62
2 | L-BFGS-B | 0.140 | 0.0599 | 42.7 | 16.68
3 | L-BFGS-B | 0.0564 | 0.0501 | 88.9 | 37.05
1 | SLSQP | 0.363 | 0.196 | 115.8 | 33.50
2 | SLSQP | 0.131 | 0.0516 | 32.1 | 9.7
3 | SLSQP | 0.0502 | 0.0492 | 63.75 | 31.31
1 | SPSA | 0.384 | 0.227 | 1000 | 0
2 | SPSA | 0.180 | 0.0517
3 | SPSA | 0.119 | 0.0681
1 | Nelder-Mead | 0.382 | 0.200 | 227.05 | 41.73
2 | Nelder-Mead | 0.121 | 0.0438 | 1124.36 | 229.32
3 | Nelder-Mead | 0.0922 | 0.0406 | 1324.7 | 17.61
1 | Powell | 0.250 | 0.106 | 413.4 | 150.75
2 | Powell | 0.136 | 0.0552 | 2193.3 | 1101.89
3 | Powell | 0.0468 | 0.0274 | 8540.75 | 3671.44
1 | COBYLA | 0.313 | 0.173 | 80.7 | 10.56
2 | COBYLA | 0.142 | 0.0575 | 994.84 | 36.12
3 | COBYLA | 0.0977 | 0.0351 | 1000 | 0

(a) Learnt State layer = 1

(b) Learnt State layer = 5

(c) Learnt State layer = 10

(d) target state

Figure 1: Wigner Function of Learnt Stat at different Layer for Non-Gaussian
State

####  IV-C2 Sampling Simulation

The results are displayed in Table V. It’s noteworthy that Nelder-Mead
displays the lowest resilience to sampling errors, consistent with findings
for both local Gaussian and non-Gaussian states. Powell continues to
demonstrate superior performance among all optimizers, with SPSA also showing
favorable results. Despite this, all optimizers yield final infidelity values
of only 0.11 and 0.13 for 3 layers. Hence, achieving convergence in sampling
simulation for learning Gaussian states requires more than 3 layers.

TABLE V: Gaussian State of Sampling Simulation

layers | method | infidelity_mean | infidelity_std | nfev_mean | nfev_mean
---|---|---|---|---|---
1 | CG | 0.518 | 0.274 | 32.15 | 9.65
2 | CG | 0.198 | 0.0638 | 40.95 | 14.03
3 | CG | 0.179 | 0.0693 | 41 | 13.05
1 | L-BFGS-B | 0.395 | 0.179 | 18.2 | 9.16
2 | L-BFGS-B | 0.234 | 0.0742 | 17.55 | 4.87
3 | L-BFGS-B | 0.170 | 0.0877 | 26.13 | 9.14
1 | SLSQP | 0.414 | 0.198 | 42.05 | 19.15
2 | SLSQP | 0.208 | 0.0677 | 67.25 | 32.64
3 | SLSQP | 0.176 | 0.0549 | 37.47 | 13.52
1 | SPSA | 0.341 | 0.183 | 1000 | 0
2 | SPSA | 0.175 | 0.0533
3 | SPSA | 0.118 | 0.0511
1 | Nelder-Mead | 0.659 | 0.299 | 2849.95 | 408.62
2 | Nelder-Mead | 0.429 | 0.223 | 2816.55 | 71.52
3 | Nelder-Mead | 0.447 | 0.259 | 2955.13 | 157.06
1 | Powell | 0.242 | 0.0179 | 189.8 | 68.24
2 | Powell | 0.183 | 0.0531 | 538.25 | 245.94
3 | Powell | 0.134 | 0.0503 | 1153.27 | 505.55
1 | COBYLA | 0.372 | 0.0978 | 64.2 | 9.71
2 | COBYLA | 0.225 | 0.0896 | 117.7 | 10.921
3 | COBYLA | 0.215 | 0.0894 | 163.8 | 11.37

###  IV-D Non-Gaussian State

####  IV-D1 Ideal Simulation

Learning a non-Gaussian state presents inherent challenges compared to
learning Gaussian states due to its more complex structure. Gaussian states
typically have simpler parameterizations based on mean and standard deviation,
while non-Gaussian states may necessitate more intricate parameterizations.

Another challenge associated with increasing layers is the resultant increase
in the number of parameters. Testing beyond 3 layers, particularly for
gradient-based optimizers, becomes computationally expensive. However, SPSA
stands out as an exception, reducing the number of function evaluations
required for each gradient estimation to just two, regardless of
dimensionality. Consequently, to investigate the required number of layers for
learning non-Gaussian states, we tested layers 1 to 10 using the SPSA
optimizer. The results shown in Table VI indicate a nearly linear relationship
for the SPSA optimizer from 1 layer to 10 layers.

####  IV-D2 Sampling Simulation

Figure 2 presents a comparison of the infidelity change over iterations for
SPSA in both ideal and sampling simulations for layer = 10. It highlights the
characteristics of sampling simulation, where the objective function exhibits
slight fluctuations due to sampling errors.

Similar to the ideal simulation, the error decreases almost linearly and
stabilizes as the number of layers approaches 10. In summary, our findings
suggest that employing 10 layers is adequate to capture the features of the
non-Gaussian state.

Furthermore, Figure 1 presents the Wigner function visualization to illustrate
the similarity between the learned and target states across various layers. It
distinctly depicts that with 1 layer, the learned state retains the shape of a
local Gaussian distribution. At 5 layers, corresponding to an infidelity value
of 0.125, the basic shape of a non-Gaussian state begins to emerge. Finally,
with 10 layers and an infidelity value of 0.04, the learned state closely
resembles the target state. These plots also aid in understanding how the
infidelity value correlates with the Wigner representation of Qumodes.

TABLE VI: Non-Gaussian State of Ideal Simulation

layers | method | infidelity_mean | infidelity_std | nfev_mean | nfev_mean
---|---|---|---|---|---
1 | CG | 0.340 | 0.187 | 656.45 | 274.83
2 | CG | 0.223 | 0.0964 | 2803.6 | 758.29
3 | CG | 0.112 | 0.0373 | 8209 | 2055.25
1 | L-BFGS-B | 0.340 | 0.187 | 335.5 | 187.34
2 | L-BFGS-B | 0.221 | 0.108 | 1173.9 | 489.98
3 | L-BFGS-B | 0.142 | 0.0353 | 3633.2 | 1456.66
1 | SLSQP | 0.341 | 0.189 | 113.45 | 28.14
2 | SLSQP | 0.205 | 0.0756 | 539.4 | 201.61
3 | SLSQP | 0.158 | 0.0305 | 1658.6 | 835.49
1 | Nelder-Mead | 0.361 | 0.196 | 240.45 | 43.37
2 | Nelder-Mead | 0.218 | 0.129 | 1025.85 | 306.71
3 | Nelder-Mead | 0.158 | 0.0252 | 1328.5 | 19.60
1 | Powell | 0.240 | 0.0892 | 319.85 | 124.35
2 | Powell | 0.170 | 0.0200 | 1769 | 774.26
3 | Powell | 0.148 | 0.0323 | 5774.4 | 2825.30
1 | COBYLA | 0.302 | 0.167 | 82.35 | 12.24
2 | COBYLA | 0.205 | 0.113 | 941.8 | 106.44
3 | COBYLA | 0.194 | 0.0542 | 1000 | 0
1 | SPSA | 0.480 | 0.0396 | 1000 | 0
2 | SPSA | 0.348 | 0.00937
3 | SPSA | 0.290 | 0.0111
4 | SPSA | 0.238 | 0.00330
5 | SPSA | 0.125 | 0.00330
6 | SPSA | 0.104 | 0.000678
7 | SPSA | 0.0780 | 0.00102
8 | SPSA | 0.0494 | 0.000446
9 | SPSA | 0.0408 | 0.000294
10 | SPSA | 0.0413 | 0.000179

TABLE VII: Non-Gaussian State of Sampling Simulation

layers | method | infidelity_mean | infidelity_std | nfev_mean | nfev_mean
---|---|---|---|---|---
1 | CG | 0.672 | 0.202 | 38.8 | 14.30
2 | CG | 0.555 | 0.205 | 33.1 | 14.21
3 | CG | 0.310 | 0.150 | 41.8 | 20.26
1 | L-BFGS-B | 0.618 | 0.225 | 24.05 | 8.83
2 | L-BFGS-B | 0.552 | 0.209 | 23 | 11.23
3 | L-BFGS-B | 0.356 | 0.185 | 19.4 | 3.95
1 | SLSQP | 0.661 | 0.226 | 32.9 | 12.85
2 | SLSQP | 0.503 | 0.200 | 62.05 | 29.61
3 | SLSQP | 0.337 | 0.207 | 32.4 | 13.43
1 | Nelder-Mead | 0.574 | 0.267 | 2757.65 | 59.85
2 | Nelder-Mead | 0.468 | 0.232 | 2871.3 | 91.10
3 | Nelder-Mead | 0.412 | 0.196 | 2931.5 | 120.39
1 | Powell | 0.277 | 0.146 | 258.1 | 111.74
2 | Powell | 0.198 | 0.0263 | 563.05 | 260.35
3 | Powell | 0.208 | 0.0449 | 820 | 424.66
1 | COBYLA | 0.402 | 0.200 | 64 | 12.43
2 | COBYLA | 0.294 | 0.136 | 107.5 | 11.11
3 | COBYLA | 0.251 | 0.0761 | 157.2 | 20.00
1 | SPSA | 0.474 | 0.0147 | 1000 | 0
2 | SPSA | 0.417 | 0.0167
3 | SPSA | 0.270 | 0.00319
4 | SPSA | 0.218 | 0.00137
5 | SPSA | 0.158 | 0.00603
6 | SPSA | 0.111 | 0.000829
7 | SPSA | 0.0839 | 0.000943
8 | SPSA | 0.0567 | 0.000399
9 | SPSA | 0.0546 | 0.000165
10 | SPSA | 0.0455 | 0.000497

Figure 2: Stochastic and non-Stochastic simulation

##  V conclusion

This paper evaluates various optimizers for preparing target qumode states,
focusing on three types: Local Gaussian, Non-local Gaussian, and Non-Gaussian
states. Each state type is evaluated with specific parameters, including step
size and sampling size. In ideal simulations, both Local and Non-local
Gaussian states were effectively approximated within three layers. However,
simulations involving sampling required additional layers to achieve similar
accuracy. The preparation of Non-Gaussian states proved more challenging,
necessitating up to 10 layers to minimize the state preparation error
effectively. The study also delves into the implications of increasing the
number of layers on runtime efficiency and optimization success. It reveals
that additional layers lead to more iterations needed to find optimal points
within an expanded parameter space. Furthermore, the analysis shows that
gradient-based optimizers tend to struggle more with increases in
dimensionality compared to derivative-free optimizers. An exception is the
SPSA optimizer, which maintains efficient gradient estimation capabilities
irrespective of the system’s dimensionality. This insight is crucial for
optimizing quantum state preparation processes, particularly in more complex
quantum systems.

##  VI Acknowledgement

This research was supported in part by the National Science Foundation (NSF)
under grant agreements 2329020, 2301884 and 2335788. This work was partially
supported by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Co-design Center for Quantum
Advantage (C2QA) under contract number DE-SC0012704, (Basic Energy Sciences,
PNNL FWP 76274). This research used resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility located at Lawrence Berkeley National Laboratory,
operated under Contract No. DE-AC02-05CH11231.

## References

  * [1] P. W. Shor, “Quantum computing,” _Documenta Mathematica_ , vol. 1, no. 1000, pp. 467–486, 1998.
  * [2] ——, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” _SIAM review_ , vol. 41, no. 2, pp. 303–332, 1999.
  * [3] D. S. Abrams and S. Lloyd, “Simulation of many-body fermi systems on a universal quantum computer,” _Physical Review Letters_ , vol. 79, no. 13, p. 2586, 1997.
  * [4] J. Preskill, “Quantum computing in the nisq era and beyond,” _Quantum_ , vol. 2, p. 79, 2018.
  * [5] S. Resch and U. R. Karpuzcu, “Benchmarking quantum computers and the impact of quantum noise,” _ACM Computing Surveys (CSUR)_ , vol. 54, no. 7, pp. 1–35, 2021.
  * [6] J. Roffe, “Quantum error correction: an introductory guide,” _Contemporary Physics_ , vol. 60, no. 3, pp. 226–245, 2019.
  * [7] S. P. Upadhyay, M. A. Alam, and S. Ghosh, “Architectures for quantum information processing,” 2022. [Online]. Available: https://api.semanticscholar.org/CorpusID:253511080
  * [8] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, “Trapped-ion quantum computing: Progress and challenges,” _Applied Physics Reviews_ , vol. 6, no. 2, 2019.
  * [9] S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” _Reviews of Modern Physics_ , vol. 77, no. 2, p. 513–577, Jun. 2005. [Online]. Available: http://dx.doi.org/10.1103/RevModPhys.77.513
  * [10] L. Banchi, M. Fingerhuth, T. Babej, C. Ing, and J. M. Arrazola, “Molecular docking with gaussian boson sampling,” _Science advances_ , vol. 6, no. 23, p. eaax1950, 2020.
  * [11] S. Jahangiri, J. M. Arrazola, N. Quesada, and N. Killoran, “Point processes with gaussian boson sampling,” _Phys. Rev. E_ , vol. 101, p. 022134, Feb 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevE.101.022134
  * [12] J. Huh, G. G. Guerreschi, B. Peropadre, J. R. McClean, and A. Aspuru-Guzik, “Boson sampling for molecular vibronic spectra,” _Nature Photonics_ , vol. 9, no. 9, pp. 615–620, 2015.
  * [13] Y. Lu, A. Maiti, J. W. O. Garmon, S. Ganjam, Y. Zhang, J. Claes, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “High-fidelity parametric beamsplitting with a parity-protected converter,” _Nature Communications_ , vol. 14, no. 1, Sep. 2023. [Online]. Available: http://dx.doi.org/10.1038/s41467-023-41104-0
  * [14] J. Zhou, Y. Liu, Y. Shi, A. Javadi-Abhari, and G. Li, “Bosehedral: Compiler optimization for bosonic quantum computing,” 2024.
  * [15] C. Fabre and N. Treps, “Modes and states in quantum optics,” _Reviews of Modern Physics_ , vol. 92, no. 3, p. 035005, 2020.
  * [16] G. Oberdieck and A. Pixton, “Quantum cohomology of the hilbert scheme of points on an elliptic surface,” _arXiv preprint arXiv:2312.13188_ , 2023.
  * [17] S. A. Stein, B. Baheri, D. Chen, Y. Mao, Q. Guan, A. Li, S. Xu, and C. Ding, “Quclassi: A hybrid deep neural network architecture based on quantum state fidelity,” _Proceedings of Machine Learning and Systems_ , vol. 4, pp. 251–264, 2022.
  * [18] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, “Variational quantum algorithms,” _Nature Reviews Physics_ , vol. 3, no. 9, p. 625–644, Aug. 2021. [Online]. Available: http://dx.doi.org/10.1038/s42254-021-00348-9
  * [19] S. A. Stein, B. Baheri, D. Chen, Y. Mao, Q. Guan, A. Li, B. Fang, and S. Xu, “Qugan: A quantum state fidelity based generative adversarial network,” in _2021 IEEE International Conference on Quantum Computing and Engineering (QCE)_. IEEE, 2021, pp. 71–81.
  * [20] S. Stein, N. Wiebe, Y. Ding, P. Bo, K. Kowalski, N. Baker, J. Ang, and A. Li, “Eqc: ensembled quantum computing for variational quantum algorithms,” in _Proceedings of the 49th annual international symposium on computer architecture_ , 2022, pp. 59–71.
  * [21] S. Stein, Y. Mao, J. Ang, and A. Li, “Qucnn: A quantum convolutional neural network with entanglement based backpropagation,” in _2022 IEEE/ACM 7th Symposium on Edge Computing (SEC)_. IEEE, 2022, pp. 368–374.
  * [22] A. Zhao, A. Tranter, W. M. Kirby, S. F. Ung, A. Miyake, and P. J. Love, “Measurement reduction in variational quantum algorithms,” _Physical Review A_ , vol. 101, no. 6, p. 062322, 2020.
  * [23] W. Mu, Y. Mao, L. Cheng, Q. Wang, W. Jiang, and P.-Y. Chen, “Iterative qubits management for quantum index searching in a hybrid system,” in _2022 IEEE International Performance, Computing, and Communications Conference (IPCCC)_. IEEE, 2022, pp. 283–289.
  * [24] B. Baheri, D. Chen, B. Fang, S. A. Stein, V. Chaudhary, Y. Mao, S. Xu, A. Li, and Q. Guan, “Tqea: temporal quantum error analysis,” in _2021 51st Annual IEEE/IFIP International Conference on Dependable Systems and Networks-Supplemental Volume (DSN-S)_. IEEE, 2021, pp. 65–67.
  * [25] C. N. Self, K. E. Khosla, A. W. Smith, F. Sauvage, P. D. Haynes, J. Knolle, F. Mintert, and M. Kim, “Variational quantum algorithm with information sharing,” _npj Quantum Information_ , vol. 7, no. 1, p. 116, 2021.
  * [26] A. D’Onofrio, A. Hossain, L. Santana, N. Machlovi, S. Stein, J. Liu, A. Li, and Y. Mao, “Distributed quantum learning with co-management in a multi-tenant quantum system,” in _2023 IEEE International Conference on Big Data (BigData)_. IEEE, 2023, pp. 221–228.
  * [27] R. L’Abbate, A. D’Onofrio, S. Stein, S. Y.-C. Chen, A. Li, P.-Y. Chen, J. Chen, and Y. Mao, “A quantum-classical collaborative training architecture based on quantum state fidelity,” _IEEE Transactions on Quantum Engineering_ , 2024.
  * [28] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,” _Nature_ , vol. 549, no. 7671, pp. 195–202, 2017.
  * [29] S. Aaronson, “Read the fine print,” _Nature Physics_ , vol. 11, no. 4, pp. 291–293, 2015.
  * [30] A. W. Smith, J. Gray, and M. Kim, “Efficient quantum state sample tomography with basis-dependent neural networks,” _PRX Quantum_ , vol. 2, no. 2, p. 020348, 2021.
  * [31] X.-M. Zhang, T. Li, and X. Yuan, “Quantum state preparation with optimal circuit depth: Implementations and applications,” _Physical Review Letters_ , vol. 129, no. 23, p. 230504, 2022.
  * [32] L. Lin and Y. Tong, “Near-optimal ground state preparation,” _Quantum_ , vol. 4, p. 372, 2020.
  * [33] X. Sun, G. Tian, S. Yang, P. Yuan, and S. Zhang, “Asymptotically optimal circuit depth for quantum state preparation and general unitary synthesis,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , 2023.
  * [34] W. K. Wootters and W. H. Zurek, “The no-cloning theorem,” _Physics Today_ , vol. 62, no. 2, pp. 76–77, 2009.
  * [35] H. Wang, Y. Liu, P. Liu, J. Gu, Z. Li, Z. Liang, J. Cheng, Y. Ding, X. Qian, Y. Shi, D. Z. Pan, F. T. Chong, and S. Han, “Robuststate: Boosting fidelity of quantum state preparation via noise-aware variational training,” 2023.
  * [36] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” _Rev. Mod. Phys._ , vol. 84, pp. 621–669, May 2012. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.84.621
  * [37] B. Zhang and Q. Zhuang, “Energy-dependent barren plateau in bosonic variational quantum circuits,” 2023.
  * [38] T. J. Stavenger, E. Crane, K. C. Smith, C. T. Kang, S. M. Girvin, and N. Wiebe, “C2qa-bosonic qiskit,” in _2022 IEEE High Performance Extreme Computing Conference (HPEC)_. IEEE, 2022, pp. 1–8.
  * [39] “SciPy - — scipy.org,” https://scipy.org/, [Accessed 30-04-2024].
  * [40] “Spsa,” https://pennylane.ai/qml/demos/tutorial_spsa/, [Accessed 30-04-2024].
  * [41] “L-bfgs,” https://en.wikipedia.org/wiki/Limited-memory_BFGS, [Accessed 30-04-2024].
  * [42] “Conjugate gradient,” https://en.wikipedia.org/wiki/Conjugate_gradient_method, [Accessed 30-04-2024].
  * [43] “Slsqp,” https://mdolab-pyoptsparse.readthedocs-hosted.com/en/latest/optimizers/SLSQP.html, [Accessed 30-04-2024].
  * [44] “cobyla,” https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_cobyla.html, [Accessed 30-04-2024].
  * [45] “Nelder-mead,” https://docs.quantum.ibm.com/api/qiskit/0.19/qiskit.aqua.components.optimizers.NELDER_MEAD, [Accessed 30-04-2024].
  * [46] “Powell,” https://docs.scipy.org/doc/scipy/reference/optimize.minimize-powell.html, [Accessed 30-04-2024].
  * [47] “Imfil,” https://docs.quantum.ibm.com/api/qiskit/0.37/qiskit.algorithms.optimizers.IMFIL, [Accessed 30-04-2024].
  * [48] “Bobyqa,” https://docs.quantum.ibm.com/api/qiskit/0.37/qiskit.algorithms.optimizers.BOBYQA, [Accessed 30-04-2024].
  * [49] “Spsa,” https://en.wikipedia.org/wiki/Simultaneous_perturbation_stochastic_approximation, [Accessed 30-04-2024].


# Characterization of Sasakian manifolds

Vladimir Rovenski111Department of Mathematics, University of Haifa, 3498838
Haifa, Israel.
e-mail<EMAIL_ADDRESS>
###### Abstract

Weak contact metric manifolds, i.e., the linear complex structure on the
contact distribution is replaced by a nonsingular skew-symmetric tensor,
defined by the author and R. Wolak, allowed us to take a new look at the
theory of contact manifolds. In this paper, we continue our study [16, 17] of
a structure of this type, called a weak nearly Sasakian structure, and prove
two theorems characterizing Sasakian manifolds. Our main result generalizes
the theorem by A. Nicola–G. Dileo–I. Yudin (2018) and provides a new criterion
for a weak almost contact metric manifold to be Sasakian.

Keywords: Almost contact structure, nearly Sasakian manifold, Riemannian
curvature.

Mathematics Subject Classifications (2010) 53C15, 53C25, 53D15

##  1 Introduction

Contact Riemannian geometry plays an important role in both pure mathematics
and theoretical physics. It considers a $(2n+1)$-dimensional smooth manifold
$M$ equipped with an almost contact metric structure $(\varphi,\xi,\eta,g)$,
i.e., $g$ is a Riemannian metric with the Levi-Civita connection $\nabla$,
$\varphi$ is a $(1,1)$-tensor, $\xi$ is a unit vector field and $\eta$ is its
dual 1-form ($\eta(\xi)=1$) satisfying $g(\varphi X,\varphi
Y)=g(X,Y)-\eta(X)\,\eta(Y)$ for any $X,Y\in\mathfrak{X}_{M}$, and
$\mathfrak{X}_{M}$ is the space of smooth vector fields on $M$. An important
class of almost contact metric manifolds $M^{\,2n+1}(\varphi,\xi,\eta,g)$
consists of Sasakian manifolds characterized by the equality, see [1, Theorem
6.3],

$(\nabla_{X}\,\varphi)Y=g(X,Y)\,\xi-\eta(Y)X,\quad X,Y\in\mathfrak{X}_{M}.$
(1)

D. Blair, D. Showers and Y. Komatu [2] defined nearly Sasakian structure
$(\varphi,\xi,\eta,g)$ using a condition similar to (1) for the symmetric part
of $\nabla\varphi$:

$(\nabla_{Y}\,\varphi)Y=g(Y,Y)\,\xi-\eta(Y)Y,\quad Y\in\mathfrak{X}_{M},$ (2)

and showed that a normal nearly Sasakian structure is Sasakian and hence is
contact. Nearly Sasakian structure is a counterpart of the nearly Kähler
structure $(J,g)$ defined by A. Gray [6] by the condition that the symmetric
part of $\nabla J$ vanishes, in contrast to the Kähler case where $\nabla
J=0$. An example of a nearly Sasakian (non-Sasakian) manifold is the sphere
$S^{\,5}$ endowed with an almost contact metric structure induced by the
almost Hermitian structure of $S^{\,6}$.

The curvature and topology of nearly Sasakian manifolds have been studied by
several authors, e.g., Z. Olszak, see [11, 12], B. Cappelletti-Montano, G.
Dileo, A.D. Nicola and I. Yudin, see [3, 10]. Any nearly Sasakian manifold of
dimension greater than 5 is Sasakian, see [10], and any 5-dimensional nearly
Sasakian manifold has Einstein metric of positive scalar curvature, see [3];
and a 3-dimensional nearly Sasakian manifold is Sasakian, see [12].

In [13, 14, 15] and [19, Section 5.3.8], we introduced metric structures that
generalize the almost contact, Sasakian, etc. metric structures. In [16, 17]
we investigated new structures of this type, called weak nearly Sasakian
structure and weak nearly Kähler structure, and asked the question: under what
conditions are weak nearly Sasakian manifolds Sasakian?

These so-called “weak” structures (the linear complex structure on the contact
distribution is replaced by a nonsingular skew-symmetric tensor) made it
possible to take a new look at the classical theory and find new applications,
e.g., in Sasakian twistor spinors theory, see E.C. Kim [8]. For an oriented
Riemannian manifold, its twistor space can be seen as the space of linear
complex structures on the tangent space that are compatible with the metric
$g$ and orientation. Some authors, for example, A.C. Herrera in [7], studied
the problem of finding parallel skew-symmetric 2-tensors different from
almost-complex structures on a Riemannian manifold and classified such
tensors. We believe that theories like twistor spinors can be naturally
extended by considering non-singular skew-symmetric tensors instead of linear
complex structures.

In this article, we continue our study of the geometry of weak nearly Sasakian
manifolds and find new criterions for a weak almost contact metric manifold to
be Sasakian. Section 2, following the introductory Section 1, reviews the
basics of weak almost contact manifolds and auxiliary results on the geometry
of weak almost Sasakian structure with conditions (5)–(6). Section 3 presents
some properties of the curvature tensor and covariant derivatives of tensors
$\varphi$ and $h$. In Theorem 2, we consider weak almost contact metric
manifolds with the condition (5) and characterize Sasakian manifolds in this
class using the property (1). In Theorem 3 (generalizing Theorem 3.3 from
[10], see also Theorem 4.9 from [4]) we characterize Sasakian manifolds of
dimension greater than five in the class of weak almost contact metric
manifolds with conditions (5)–(6). In the proof of Theorem 3 we also use our
result from [17] (see Theorem 1) that under certain conditions the weak nearly
Sasakian structure foliates into two types of totally geodesic foliations. In
Section 4, we prove two lemmas (formulated in Section 3) that allow us to
obtain the main results. Our proofs use the properties of new tensors, as well
as classical constructions.

##  2 Preliminaries

A weak almost contact metric structure on a smooth manifold $M^{2n+1}\ (n>0)$
is a set $(\varphi,Q,\xi,\eta,g)$, where $\varphi$ is a $(1,1)$-tensor, $\xi$
is a vector field, $\eta$ is a dual 1-form ($\eta(\xi)=1$), $g$ is a
Riemannian metric and $Q$ is a nonsingular $(1,1)$-tensor on $M$, satisfying,
see [13, 14],

$\displaystyle g(\varphi X,\varphi Y)=g(X,Q\,Y)-\eta(X)\,\eta(Y),\quad
X,Y\in\mathfrak{X}_{M}.$ (3)

Assume that a $2n$-dimensional distribution $\ker\eta$ is $\varphi$-invariant,
i.e., $\varphi(\ker\eta)\subset\ker\eta$, as in the classical theory [1],
where $Q={\rm id}_{\,TM}$. By this and (3), $\ker\eta$ is $Q$-invariant and
the following hold:

$\displaystyle\varphi\,\xi=0,\quad\eta\circ\varphi=0,\quad\eta\circ
Q=\eta,\quad[Q,\,\varphi]:={Q}\circ\varphi-\varphi\circ{Q}=0.$

For a weak almost contact metric structure, $\varphi$ is skew-symmetric and
$Q$ is self-adjoint and positive definite. Setting $Y=\xi$ in (3), we obtain
$\eta(X)=g(\xi,X)$. We get a weak contact metric structure is the following is
true:

$d\eta(X,Y)=g(X,\varphi Y),\quad X,Y\in\mathfrak{X}_{M},$

where the exterior derivative $d\eta$ of $\eta$ is given by
$d\eta(X,Y)=\frac{1}{2}\,\\{X(\eta(Y))-Y(\eta(X))-\eta([X,Y])\\}$, for
example, [1]. A 1-form $\eta$ on a manifold $M^{\,2n+1}$ is said to be contact
if $\eta\wedge(d\eta)^{n}\neq 0$, e.g., [1]. For a weak contact metric
structure $(\varphi,Q,\xi,\eta,g)$, the 1-form $\eta$ is contact, see [18,
Lemma 2.1].

######  Definition 1 (see [16]).

A weak almost contact metric manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ is
called weak nearly Sasakian if (2) is true, or, equivalently,

$(\nabla_{Y}\,\varphi)Z+(\nabla_{Z}\,\varphi)Y=2\,g(Y,Z)\,\xi-\eta(Z)Y-\eta(Y)Z,\qquad
Y,Z\in\mathfrak{X}_{M}.$ (4)

In addition to (4), the following two conditions, which are automatically
satisfied by almost contact metric manifolds, play a significant role in this
paper:

$\displaystyle(\nabla_{X}\,\widetilde{Q})\,Y=0,\quad X\in\mathfrak{X}_{M},\
Y\in\ker\eta,$ (5) $\displaystyle R_{\widetilde{Q}X,Y}Z\in\ker\eta,\quad
X,Y,Z\in\ker\eta,$ (6)

where $\widetilde{Q}=Q-{\rm id}_{\,TM}$ is a “small” tensor, and
$R_{{X},{Y}}=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]}$ for all
$X,Y\in\mathfrak{X}_{M}$, is the curvature tensor, e.g., [5]. From (6) and the
Bianchi identity we get

$R_{\,X,Y}\,\widetilde{Q}Z\in\ker\eta,\quad X,Y,Z\in\ker\eta.$

Using the equalities $(\nabla_{\xi}\,\varphi)\,\xi=0$ and $\varphi\,\xi=0$ for
a weak nearly Sasakian manifold, we conclude that $\xi$ is a geodesic field
($\nabla_{\xi}\,\xi=0$). From (5) and $\nabla_{\xi}\,\xi=0$, we get

$\nabla_{\xi}\,\widetilde{Q}=0.$

The $\xi$-curves form a Riemannian geodesic foliation, and $-\nabla\xi$ is its
splitting operator (co-nullity tensor), see [19, Section 1.3.1]. Taking
derivative of the equality $g(\varphi V,Z)=-g(V,\varphi Z)$, we see that the
tensor $\nabla_{Y}\,\varphi$ of a weak nearly Sasakian manifold is skew-
symmetric:

$g((\nabla_{Y}\,\varphi)V,Z)=-g((\nabla_{Y}\,\varphi)Z,V).$ (7)

The Ricci identity (commutation formula) for the (1,1)-tensor $\varphi$ can be
written as (e.g., [5])

$g((\nabla^{2}_{X,Y}\,\varphi)V,Z)-g((\nabla^{2}_{Y,X}\,\varphi)V,Z)=g(R_{\,X,Y}\,\varphi
V,Z)+g(R_{\,X,Y}V,\varphi Z).$ (8)

where the second covariant derivative is given by
$\nabla^{2}_{X,Y}=\nabla_{X}\nabla_{Y}-\nabla_{\nabla_{X}Y}$. Taking covariant
derivative of (7), we see that $\nabla^{2}_{X,Y}\,\varphi$ of a weak nearly
Sasakian manifold is skew-symmetric:

$g((\nabla^{2}_{X,Y}\,\varphi)V,Z)=-g((\nabla^{2}_{X,Y}\,\varphi)Z,V).$

In the rest of the section, we survey some properties of weak nearly Sasakian
manifolds, see [17]. Define a (1,1)-tensor field $h$ on $M$ (as in the
classical case, e.g., [10]),

$h=\nabla\xi+\varphi.$ (9)

We get $\eta\circ h=0$ and $h(\ker\eta)\subset\ker\eta$. Since $\xi$ is a
geodesic vector field, we also get $h\,\xi=0$. Since $\xi$ is a Killing vector
field, see [16], and $\varphi$ is skew-symmetric, the tensor $h$ is skew-
symmetric. The distribution $\ker\eta$ is integrable, $[X,Y]\in\ker\eta\
(X,Y\in\ker\eta)$, if and only if $h=\varphi$, and in this case, our manifold
splits along $\xi$ and $\ker\eta$, i.e., is locally the metric product.

######  Lemma 1 (see [17]).

For a weak nearly Sasakian manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ we
obtain

$\displaystyle(\nabla_{X}\,h)\,\xi=-h(h-\varphi)X,$ (10)
$\displaystyle(\nabla_{X}\,\varphi)\,\xi=-\varphi(h-\varphi)X,$ (11)
$\displaystyle h\,\varphi+\varphi\,h=-2\,\widetilde{Q},$ (12) $\displaystyle
h\,Q=Q\,h\quad(h\ {\rm commutes\ with}\ Q).$ (13)

Moreover, $h^{2}\varphi=\varphi h^{2}$, $h\varphi^{2}=\varphi^{2}h$,
$h^{2}\varphi^{2}=\varphi^{2}h^{2}$, etc.

If a weak nearly Sasakian manifold satisfies the condition (6), then its
contact distribution is curvature invariant, see [17], i.e.,

$R_{\,X,Y}Z\in\ker\eta,\quad X,Y,Z\in\ker\eta.$ (14)

For example, the tangent bundle of a totally geodesic submanifold in a
Riemannian manifold and the distribution $\ker\eta$ of any 1-form $\eta$ on a
real space form satisfy (14).

######  Lemma 2 (see [17]).

For a weak nearly Sasakian manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ with
the condition (6), we obtain

$\displaystyle
R_{\,X,\xi}\,Y=(\nabla_{X}\,(h-\varphi)\,)\,Y=(\nabla_{X}\,(h-\varphi))Y=g((h-\varphi)^{2}X,Y)\,\xi-\eta(Y)\,(h-\varphi)^{2}X.$
(15)

In particular, $\nabla_{\xi}\,h=\nabla_{\xi}\,\varphi=\varphi h+\widetilde{Q}$
and

$\displaystyle
g(R_{\,\xi,X}Y,Z)=\eta(Y)\,g((h-\varphi)^{2}X,Z)-\eta(Z)\,g((h-\varphi)^{2}X,Y).$
(16)

A Sasakian manifold is defined as a contact metric manifold such that the
tensor $[\varphi,\varphi]+d\eta\otimes\xi$ vanishes identically, where the
Nijenhuis torsion $[\varphi,\varphi]$ of $\varphi$ is given by (for example,
[1])

$\displaystyle[\varphi,\varphi](X,Y)=\varphi^{2}[X,Y]+[\varphi X,\varphi
Y]-\varphi[\varphi X,Y]-\varphi[X,\varphi Y].$

The following statement, generalizing [12, Lemma 2.1], will be used in the
proof of Theorem 2.

######  Proposition 1 (see also [17]).

For a weak nearly Sasakian manifold with the property (5), the equality $h=0$
holds if and only if the manifold is Sasakian.

For a weak nearly Sasakian manifold satisfying (5) and (6), the eigenvalues
(and their multiplicities) of the self-adjoint operator $h^{2}$ are constant,
see [17]. Since $h$ is skew-symmetric, the nonzero eigenvalues of $h^{2}$ are
negative, and the spectrum of $h^{2}$ has the form

$Spec(h^{2})=\\{0,-\lambda_{1}^{2},\ldots-\lambda_{r}^{2}\\},$ (17)

where $\lambda_{i}$ is a positive real number and $\lambda_{i}\neq\lambda_{j}$
for $i\neq j$. If $X$ is a unit eigenvector of $h^{2}$ with eigenvalue
$-\lambda^{2}_{i}$, then by (12) and (13), $X,\varphi X,hX$ and $h\,\varphi X$
are nonzero eigenvectors of $h^{2}$ with eigenvalue $-\lambda^{2}_{i}$. Denote
by $[\xi]$ the 1-dimensional distribution generated by $\xi$, and by $D_{0}$ a
distribution of the eigenvectors with eigenvalue 0 orthogonal to $\xi$. Denote
by $D_{i}$ a distribution of the eigenvectors with eigenvalue
$-\lambda^{2}_{i}$. Note that $D_{0}$ and $D_{i}\ (i=1,\ldots,r)$ belong to
$\ker\eta$ and are $\varphi$\- and $h$\- invariant. The tensors $h$ and
$\widetilde{Q}$ of a weak nearly Sasakian manifold vanish on $D_{0}$, [17].

The following statement generalizes [3, Theorems 3.3 and 3.5].

######  Theorem 1 (see [17]).

Let $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ be a weak nearly Sasakian manifold with
conditions (5) and (6), and let the spectrum of the self-adjoint operator
$h^{2}$ have the form (17). Then, the distribution $[\xi]\oplus D_{0}$ and
each distribution $[\xi]\oplus D_{i}\ (i=1,\ldots,r)$ are integrable with
totally geodesic leaves. In particular, the eigenvalue $0$ has multiplicity
$2p+1$ for some integer $p\geq 0$. If $p>0$, then

$(a)$ the distribution $[\xi]\oplus D_{1}\oplus\ldots\oplus D_{r}$ is
integrable and defines a Riemannian foliation with totally geodesic leaves;

$(b)$ the leaves of $[\xi]\oplus D_{0}$ are $(2p+1)$-dimensional Sasakian
manifolds.

##  3 Main results

Here, we prove some properties of the curvature tensor and covariant
derivatives of the tensors $\varphi$ and $h$, and prove two theorems
characterizing Sasakian manifolds in the class of weak almost contact metric
manifolds.

First, we consider weak almost contact metric manifolds with the condition (5)
and characterize Sasakian manifolds in this class using the property (1).

######  Theorem 2.

Let $M(\varphi,Q,\xi,\eta,g)$ be a weak almost contact metric manifold with
conditions (1) and (5). Then $Q={\rm id}_{\,TM}$ and the structure
$(\varphi,\xi,\eta,g)$ is Sasakian.

###### Proof.

Differentiating the equality $g(\varphi\,Y,\xi)=0$ yields

$0=Xg(\varphi\,Y,\xi)=g((\nabla_{X}\,\varphi)Y,\xi)+g(\varphi\,Y,(h-\varphi)X).$

Applying to this (1) and then (3) gives $h\varphi=-\widetilde{Q}$. Since
$h,\varphi$ are skew-symmetric and $\widetilde{Q}$ is self-adjoint, we also
get $\varphi h=-\widetilde{Q}$. Therefore,

$h\varphi=\varphi h=-\widetilde{Q}.$ (18)

Differentiating (3) and using (5), (18) and the skew-symmetry of
$\nabla_{X}\,\varphi$, we get

$-\eta(Y)\,g(hX,Z)-\eta(Z)\,g(\varphi X+Q\,(h-\varphi)X,Y)=0.$

Taking $Y=\xi$ in the above equality, we get $g(hX,Z)=0$ for all
$X,Z\in\mathfrak{X}_{M}$. Therefore, $h=\widetilde{Q}=0$ and by Proposition 1,
the structure is Sasakian. ∎

The following statement generalizes Lemma 2.4 in [11].

######  Lemma 3.

For covariant derivatives of the tensor $\varphi$ on a weak nearly Sasakian
manifold with the condition (5) we obtain

$\displaystyle g((\nabla_{X}\,\varphi)\varphi
Y,Z)=g((\nabla_{X}\,\varphi)Y,\varphi
Z)+\eta(Y)\,g((h-\varphi)X,Z)+\eta(Z)\,g((h-\varphi)X,QY),$ (19)
$\displaystyle g((\nabla_{\varphi
X}\,\varphi)Y,Z)=g((\nabla_{X}\,\varphi)\,Y,\varphi
Z)-\eta(X)\,g(hY,Z)-2\,\eta(Y)\,g(\varphi X,Z)$
$\displaystyle\quad\qquad+\,2\,\eta(Z)\,g(\varphi
X,Y)-\eta(Z)\,g(QX,(h-\varphi)Y).$ (20) $\displaystyle g((\nabla_{\varphi
X}\,\varphi)\,\varphi Y,Z)=-g((\nabla_{X}\,\varphi)Y,QZ)+\eta(X)\,g(Y,h\varphi
Z)+\eta(Y)\,g(h\varphi X+\varphi^{2}X,Z)$
$\displaystyle\quad\qquad+\,\eta(Z)\,g(\varphi(h-\varphi)X,Y)+\eta(Z)\,g((h-\varphi)\varphi
X,QY).$ (21)

Proof. Differentiating (3) and using (12), (5) and the skew-symmetry of
$\nabla_{X}\,\varphi$, we get (19). We obtain (3) from (19) by the condition
(4). Replacing $X$ by $\varphi X$ in (19) gives

$g((\nabla_{\varphi X}\,\varphi)\,\varphi Y,Z)=g((\nabla_{\varphi
X}\,\varphi)\,Y,\varphi Z)+\eta(Y)\,g((h-\varphi)\varphi
X,Z)+\eta(Z)\,g((h-\varphi)\varphi X,QY),$

from which, using (3) and (3), we get (3). $\square$

######  Lemma 4.

The curvature tensor of a weak nearly Sasakian manifold satisfies

$\displaystyle g(R_{\,\varphi X,Y}Z,V)+g(R_{\,X,\varphi
Y}Z,V)+g(R_{\,X,Y}\,\varphi Z,V)+g(R_{\,X,Y}Z,\varphi V)$
$\displaystyle=g(Y,V)\,g((h-\varphi)X,Z)-g(X,Y)\,g(Z,(h-\varphi)V)+g(Y,Z)\,g(X,(h-\varphi)V)$
$\displaystyle-(1/2)\,g(Z,V)\,g((h-\varphi)X,Y)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)V)\quad(X,Y,Z,V\in\mathfrak{X}_{M}).$
(22)

The curvature tensor of a weak nearly Sasakian manifold with the condition (6)
satisfies

$\displaystyle g(R_{\,\varphi X,\varphi Y}Z,V)=g(R_{\,X,Y}\,\varphi Z,\varphi
V)-(1/2)\,\delta(X,Y,Z,V)-g(X,Y)\,g(Z,\widetilde{Q}V)$ $\displaystyle\
+(1/2)\,g(Y,V)\,g(\varphi(h-\varphi)X,Z)-(1/2)\,g(Y,\varphi
Z)\,g(X,(h-\varphi)V)$ $\displaystyle\ -(1/2)\,g(Y,\varphi
V)\,g((h-\varphi)X,Z)-(1/2)\,g(Y,Z)\,g(X,(h-\varphi)\varphi V)$
$\displaystyle\ -(1/4)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)-(1/4)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V),$ (23)
$\displaystyle g(R_{\,\varphi X,\varphi Y}\,\varphi Z,\varphi
V)=g(R_{\,QX,QY}Z,V)-\eta(X)\,g(R_{\,\xi,QY}Z,V)+\eta(Y)\,g(R_{\,\xi,QX}Z,V)$
$\displaystyle\ +(1/2)\,g(\varphi Y,V)\,g((h-\varphi)\varphi X,\varphi
Z)+(1/2)\,g(\varphi^{2}Y,Z)\,g(X,\varphi(h-\varphi)V)$ $\displaystyle\
-(1/2)g(\varphi^{2}Y,V)g((h-\varphi)\varphi X,Z)+(1/2)g(\varphi Y,Z)g(\varphi
X,(h-\varphi)\varphi V)$ $\displaystyle\
+(1/4)\,g(\varphi^{2}X,Z)\,g(Y,\varphi(h-\varphi)V)+(1/4)\,g(\varphi
X,Z)\,g(\varphi Y,(h-\varphi)\varphi V)$ $\displaystyle\
-g(\varphi^{2}X,Y)g(Z,\widetilde{Q}V)+(1/2)\,\delta(\varphi X,\varphi Y,Z,V),$
(24)

where the “small” tensor $\delta$ is defined by

$\delta(X,Y,Z,V)=g(R_{\,X,Y}\widetilde{Q}Z,V)+g(R_{\,X,Y}Z,\widetilde{Q}V)-g(R_{\widetilde{Q}X,Y}Z,V)-g(R_{\,X,\widetilde{Q}Y}Z,V).$

######  Remark 1.

The tensor $\delta$ of a weak nearly Sasakian manifold has the following
symmetries:

$\displaystyle\delta(Y,X,Z,V)=\delta(X,Y,V,Z)=\delta(Z,V,X,Y)=-\delta(X,Y,Z,V).$

If (6) is true, then by (16), we get

$\displaystyle\delta(X,Y,Z,\xi)=g(R_{\,X,Y}\,\widetilde{Q}Z,\xi)-g(R_{\widetilde{Q}X,Y}Z,\xi)-g(R_{\,X,\widetilde{Q}Y}Z,\xi)$
$\displaystyle\qquad=g(R_{\,\xi,\,\widetilde{Q}Z}Y,X)+g(R_{\,\xi,Z}\,\widetilde{Q}X,Y)+g(R_{\,\xi,Z}\,X,\widetilde{Q}Y)=0,$

hence,
$\delta(\xi,Y,Z,V)=\delta(X,\xi,Z,V)=\delta(X,Y,\xi,V)=\delta(X,Y,Z,\xi)=0$.

The following lemma generalizes the result obtained in [11], see also equation
(6) in [10].

######  Lemma 5.

For a weak nearly Sasakian manifold with the conditions (5) and (6), we obtain

$\displaystyle g((\nabla_{X}\,\varphi)Y,hZ)=-\eta(X)\,g((\varphi
h^{2}+\widetilde{Q}h)Z,Y)+\eta(Y)\,g((\varphi h^{2}-h+\widetilde{Q}h)Z,X).$
(25)

The proofs of Lemmas 4–5 are given in Section 4. Next, we generalize [10,
Proposition 3.1].

######  Proposition 2.

For a weak nearly Sasakian manifold with conditions (5) and (6), we obtain

$\displaystyle(\nabla_{X}\,\varphi)Y=\eta(X)\,(\varphi
hY+\widetilde{Q}Y)-\eta(Y)\,(\varphi hX+QX)+g(\varphi hX+QX,Y)\,\xi,$ (26)
$\displaystyle(\nabla_{X}\,h)Y=\eta(X)\,(\varphi
hY+\widetilde{Q}Y)-\eta(Y)\,h(h-\varphi)X+g(h(h-\varphi)X,Y)\,\xi,$ (27)
$\displaystyle(\nabla_{X}\,\varphi h)Y=\eta(X)\,(\varphi
h^{2}Y-hY+\widetilde{Q}\varphi Y)-\eta(Y)\,g(\varphi
h^{2}X-QhX+2\,\widetilde{Q}\varphi X)$ $\displaystyle\quad+g(\varphi
h^{2}X-hX+\widetilde{Q}hX,Y)\,\xi.$ (28)

###### Proof.

From (25) we have

$\displaystyle g((\nabla_{X}\,\varphi)Y,hZ)=\eta(X)\,g(\varphi
hY+\widetilde{Q}Y,hZ)-\eta(Y)\,g(\varphi hX+X+\widetilde{Q}X,hZ),$

which is compatible with (26). On the other hand,

$\displaystyle
g((\nabla_{X}\,\varphi)Y,\xi)=-g(Y,(\nabla_{X}\,\varphi)\xi)=g(Y,\varphi(h-\varphi)X)=g(QX+\varphi
hX,Y)-\eta(X)\,\eta(Y).$

By the above, to prove (26), we need only to show (since
$\widetilde{Q}|_{D_{0}}=0$)

$\displaystyle g((\nabla_{X}\,\varphi)Y,V)=-\eta(Y)\,g(X,V),\quad
X,Y\in\mathfrak{X}_{M},\ V\in{D}_{0}.$ (29)

By Theorem 1, the distribution $\bar{D}=[\xi]\oplus D_{1}\oplus\ldots\oplus
D_{r}$ is integrable with totally geodesic leaves. If $X,Y\in\bar{D}$, then
both sides of (29) vanish. If $X\in D_{0}$ and $Y\in\bar{D}$, then

$\displaystyle
g((\nabla_{X}\,\varphi)Y,V)=-g(Y,(\nabla_{X}\,\varphi)V)=-\eta(Y)\,g(X,V),$

since $[\xi]\oplus D_{0}$ is integrable with totally geodesic leaves with the
induced Sasakian structure on each leaf, so
$(\nabla_{X}\,\varphi)V=g(X,V)\,\xi-\eta(V)\,X$. If $X\in\bar{D}$ and $Y\in
D_{0}$, then $g((\nabla_{X}\,\varphi)Y,V)=-\eta(X)\,g(Y,V)$, and applying (4),
we get

$\displaystyle
g((\nabla_{X}\,\varphi)Y,V)=-g((\nabla_{Y}\,\varphi)X+\eta(X)Y,V)=0.$

If we take $X,Y\in D_{0}$, then (26) is true because of (1) and the
orthogonality of vector fields $X,Y,V$ to $\xi$. This completes the proof of
(26). From (26), using (15), we obtain (27). Finally, using (26) and (27) in
the equality $(\nabla_{X}\,\varphi
h)Y=(\nabla_{X}\,\varphi)\,hY+\varphi(\nabla_{X}\,h)Y$ gives

$\displaystyle(\nabla_{X}\,\varphi h)Y=\eta(X)\,(\varphi
h^{2}Y{-}hY+\widetilde{Q}\varphi Y+\eta(Y)\,\xi)$
$\displaystyle-\eta(Y)\,g(\varphi
h^{2}X-hX+(2\,\varphi-h)\widetilde{Q}X+\eta(X)\,\xi)+g(\varphi
h^{2}X{-}hX+\widetilde{Q}hX,Y)\,\xi,$

where (3) and (12) were also used, hence (2) is true. ∎

Recall, see [5, pp. 15–16], that for any 2-form $\beta$ and 1-form $\eta$ we
have

$\displaystyle
3\,(\eta\wedge\beta)(X,Y,Z)=\eta(X)\,\beta(Y,Z)+\eta(Y)\,\beta(Z,X)+\eta(Z)\,\beta(X,Y),$
$\displaystyle
3\,d\beta(X,Y,Z)=X\beta(Y,Z){+}Y\beta(Z,X){+}Z\beta(X,Y){-}\beta([X,Y],Z){-}\beta([Z,X],Y){-}\beta([Y,Z],X)$
$\displaystyle\qquad=(\nabla_{X}\,\beta)(Y,Z)+(\nabla_{Y}\,\beta)(Z,X)+(\nabla_{Z}\,h)(X,Y).$

The following proposition will be used in the proof of our main result,
Theorem 3.

######  Proposition 3 (see Proposition 3.2 in [10]).

Let $\eta$ be a contact 1-form on a smooth manifold $M$ of dimension $2n+1>5$.
Then, the following operator is injective:

$\Upsilon_{d\eta}:\beta\in\Lambda^{2}(M)\to
d\eta\wedge\beta\in\Lambda^{4}(M),$

where $\Lambda^{p}(M)$ is the vector bundle of differential $p$-forms on $M$.

We will generalize Theorem 3.3 in [10] on characterization of Sasakian
manifolds.

######  Theorem 3.

Let $M(\varphi,Q,\xi,\eta,g)$ be a weak nearly Sasakian manifold of dimension
greater than $5$ with conditions (5) and (6). Then $Q={\rm id}_{\,TM}$ and the
structure $(\varphi,\xi,\eta,g)$ is Sasakian.

###### Proof.

We consider the following 2-forms:

$\displaystyle\Phi_{0}(X,Y)=g(hX,Y),\quad\Phi_{1}(X,Y)=g(\varphi
hX,Y),\quad\Phi_{2}(X,Y)=g(\varphi h^{2}X,Y),$
$\displaystyle\Psi_{0}(X,Y)=g(\widetilde{Q}X,Y),\quad\Psi_{1}(X,Y)=g(\widetilde{Q}\varphi
X,Y),\quad\Psi_{2}(X,Y)=g(\widetilde{Q}hX,Y).$

It is easy to calculate, see [13]:

$3\,d\Phi_{0}(X,Y,Z)=g((\nabla_{X}\,h)Z,Y)+g((\nabla_{Y}\,h)X,Z)+g((\nabla_{Z}\,h)Y,X).$

Applying (27) to the above, gives

$d\Phi_{0}=\eta\wedge(\Phi_{1}+\Psi_{0}).$ (30)

We also have

$3\,d\Phi_{1}(X,Y,Z)=g((\nabla_{X}\,\varphi h)Z,Y)+g((\nabla_{Y}\,\varphi
h)X,Z)+g((\nabla_{Z}\,\varphi h)Y,X).$

Similarly to (30), applying (2) to the above, gives

$d\Phi_{1}=\eta\wedge(\Phi_{2}-\Phi_{0}+\Psi_{1}).$ (31)

Also, using (5), we find that $d\Psi_{0}=-\eta\wedge(\Psi_{2}-\Psi_{1})$:

$\displaystyle
3\,d\Psi_{0}=g((\nabla_{X}\,\widetilde{Q})Z,Y)+g((\nabla_{Y}\,\widetilde{Q})X,Z)+g((\nabla_{Z}\,\widetilde{Q})Y,X)$
$\displaystyle=\eta(Z)\,g((\nabla_{X}\,\widetilde{Q})\,\xi,Y)+\eta(X)\,g((\nabla_{Y}\,\widetilde{Q})\,\xi,Z)+\eta(Y)\,g((\nabla_{Z}\,\widetilde{Q})\,\xi,X)$
$\displaystyle=-\eta(Z)\,g(\widetilde{Q}(h{-}\varphi)X,Y)-\eta(X)\,g(\widetilde{Q}(h{-}\varphi)Y,Z)-\eta(Y)\,g(\widetilde{Q}(h{-}\varphi)Z,X)=-3\,\eta\wedge(\Psi_{2}-\Psi_{1}).$

Using this and (31), from (30) we obtain

$\displaystyle
0=d^{2}\Phi_{0}=d\eta\wedge(\Phi_{1}+\Psi_{0})-\eta\wedge(d\Phi_{1}+d\Psi_{0})$
$\displaystyle=d\eta\wedge(\Phi_{1}+\Psi_{0})-\eta\wedge\eta\wedge(\Phi_{2}-\Phi_{0}+2\,\Psi_{1}-\Psi_{2})=d\eta\wedge(\Phi_{1}+\Psi_{0}).$
(32)

If $\dim M>5$, $\eta$ being a contact form, the fact that
$d\eta\wedge(\Phi_{1}+\Psi_{0})=0$, see (3) and Proposition 3, imply
$\Phi_{1}+\Psi_{0}=0$, i.e., $\varphi h=-\widetilde{Q}$. By (12), we get
$h\varphi=\varphi h=-\widetilde{Q}$, see (18). Using this in (26) gives the
equality (1) well known for Sasakian structure. Using the equality (1) in
(19), we get

$-\eta(Y)\,g(hX,Z)+\eta(Z)\,g((\widetilde{Q}\,\varphi-Q\,h)X,Y)=0.$

Taking $Y=\xi$ in the above equality, we get $g(hX,Z)=0$ for all
$X,Z\in\mathfrak{X}_{M}$. We come to the conclusion that $h=\widetilde{Q}=0$,
therefore, by Proposition 1, the structure is Sasakian. ∎

##  4 Proofs of Lemmas 4 and 5

Here, in the proofs of Lemmas 4–5 we use the approach [11].

Proof of Lemma 4. This consists of two steps.

Step 1. Some formulas appearing in the proof of (4) are also used in the proof
of (4). Differentiating (4), we find

$\displaystyle(\nabla^{2}_{X,Y}\,\varphi)Z+(\nabla^{2}_{X,Z}\,\varphi)Y=2\,g(Y,Z)\,(h{-}\varphi)X-g((h{-}\varphi)X,Z)Y-g((h{-}\varphi)X,Y)Z.$
(33)

Applying the Ricci identity (8), from (33) and the skew-symmetry of
$\nabla^{2}_{X,Y}\,\varphi$ we get

$\displaystyle g(R_{X,Y}Z,\varphi V)-g(R_{X,Y}V,\varphi
Z)+g((\nabla^{2}_{X,Z}\,\varphi)Y,V)-g((\nabla^{2}_{Y,Z}\,\varphi)X,V)$
$\displaystyle=2\,g(Y,Z)g((h{-}\varphi)X,V)-2\,g(X,Z)g((h{-}\varphi)Y,V)$
$\displaystyle-g(Y,V)g((h{-}\varphi)X,Z)+g(X,V)g((h{-}\varphi)Y,Z)+2\,g(Z,V)g((h{-}\varphi)Y,X).$
(34)

By Bianchi and Ricci identities, we find

$\displaystyle g(R_{X,Y}Z,\varphi V)=-g(R_{Y,Z}X,\varphi V)-g(R_{Z,X}Y,\varphi
V)$
$\displaystyle=g((\nabla^{2}_{Y,Z}\,\varphi)V,X)-g((\nabla^{2}_{Z,Y}\,\varphi)V,X)-g(R_{Y,Z}V,\varphi
X)-g(R_{Z,X}Y,\varphi V).$ (35)

Substituting (4) into (4), it follows that

$\displaystyle g(R_{X,Z}Y,\varphi V)-g(R_{X,Y}V,\varphi Z)-g(R_{Y,Z}V,\varphi
X)$
$\displaystyle-g((\nabla^{2}_{Z,Y}\,\varphi)V,X)-g((\nabla^{2}_{X,Z}\,\varphi)V,Y)=2\,g((\nabla^{2}_{Y,Z}\,\varphi)X,V)$
$\displaystyle+2\,g(Y,Z)g((X-\varphi)X,V)-2\,g(X,Z)g((h-\varphi)Y,V)$
$\displaystyle+2\,g(Z,V)g((h{-}\varphi)Y,X)-g(Y,V)g((h{-}\varphi)X,Z)+g(X,V)g((h{-}\varphi)Y,Z).$
(36)

On the other hand, using (33) and the Ricci identity (8), we see that

$\displaystyle g(R_{X,Z}Y,\varphi V)-g(R_{X,Z}V,\varphi
Y)-g((\nabla^{2}_{X,Z}\,\varphi)Y,V)+g((\nabla^{2}_{Z,X}\,\varphi)Y,V)=0.$
(37)

Adding (37) to (4), we get

$\displaystyle 2\,g(R_{X,Z}Y,\varphi V)-g(R_{X,Y}V,\varphi
Z)-g(R_{Y,Z}V,\varphi X)-g(R_{X,Z}V,\varphi Y)$
$\displaystyle=2\,g((\nabla^{2}_{Y,V}\,\varphi)Z,X)+2\,g(Y,Z)g((h-\varphi)X,V)-2\,g(X,Z)g((h-\varphi)Y,V)$
$\displaystyle+2\,g(Z,V)g((h-\varphi)Y,X)-g(Y,V)g((h-\varphi)X,Z)+g(X,V)g((h-\varphi)Y,Z).$
(38)

Swapping $Y$ and $V$ in (4), we find

$\displaystyle 2\,g(R_{X,Z}V,\varphi Y)-g(R_{X,V}Y,\varphi
Z)-g(R_{V,Z}Y,\varphi X)-g(R_{X,Z}Y,\varphi V)$
$\displaystyle=2\,g((\nabla^{2}_{V,Y}\,\varphi)Z,X)+2\,g(Z,V)g((h-\varphi)X,Y)-2\,g(X,Z)g((h-\varphi)V,Y)$
$\displaystyle+2\,g(Y,Z)g((h-\varphi)V,X)-g(Y,V)g((h-\varphi)X,Z)+g(X,Y)g((h-\varphi)V,Z).$
(39)

Subtracting (4) from (4), and using the Bianchi and Ricci identities, we get

$\displaystyle g(R_{\varphi X,Z}Y,V)+g(R_{X,\varphi Z}Y,V)+g(R_{X,Z}\varphi
Y,V)+g(R_{X,Z}Y,\varphi V)$
$\displaystyle=g(Z,V)g((h{-}\varphi)X,Y)-g(X,Z)g((h{-}\varphi)V,Y)+g(Y,Z)g((h{-}\varphi)V,X)$
$\displaystyle-(1/2)\,g(Y,V)g((h-\varphi)X,Z)+(1/2)\,g(X,Y)g((h-\varphi)V,Z),$

which (by replacing $Z$ and $Y$) gives (4).

Step 2. Replacing $X$ by $\varphi X$ in (4) and using (3), we have

$\displaystyle-g(R_{\,QX,Y}Z,V){+}\eta(X)g(R_{\,\xi,Y}Z,V){+}g(R_{\,\varphi
X,\varphi Y}Z,V){+}g(R_{\,\varphi X,Y}\,\varphi Z,V){+}g(R_{\,\varphi
X,Y}Z,\varphi V)$ $\displaystyle=g(Y,V)\,g((h-\varphi)\varphi X,Z)-g(\varphi
X,Y)\,g(Z,(h-\varphi)V)+g(Y,Z)\,g(\varphi X,(h-\varphi)V)$
$\displaystyle-(1/2)\,g(Z,V)\,g((h-\varphi)\varphi X,Y)+(1/2)\,g(\varphi
X,Z)\,g(Y,(h-\varphi)V).$ (40)

Exchanging $X$ and $Y$ in (4), we find

$\displaystyle g(R_{\,X,QY}Z,V){+}\eta(Y)g(R_{\,\xi,X}Z,V){-}g(R_{\,\varphi
X,\varphi Y}Z,V){+}g(R_{\,\varphi Y,X}\,\varphi Z,V){+}g(R_{\,\varphi
Y,X}Z,\varphi V)$ $\displaystyle=g(X,V)\,g((h-\varphi)\varphi Y,Z)-g(\varphi
X,Y)\,g(Z,(h-\varphi)V)-g(X,Z)\,g(\varphi(Y,h-\varphi)V)$
$\displaystyle-(1/2)\,g(Z,V)\,g(X,(h-\varphi)\varphi Y)+(1/2)\,g(\varphi
Y,Z)\,g(X,(h-\varphi)V).$ (41)

Subtracting (4) from (4), we obtain

$\displaystyle 2\,g(R_{\,\varphi X,\varphi
Y}Z,V)-2\,g(R_{\,X,Y}Z,V)+\eta(X)\,g(R_{\,\xi,Y}Z,V)-\eta(Y)\,g(R_{\,\xi,X}Z,V)$
$\displaystyle\ +g(R_{\,\varphi X,Y}\,\varphi Z,V)-g(R_{\,\varphi
Y,X}\,\varphi Z,V)+g(R_{\,\varphi X,Y}Z,\varphi V)-g(R_{\,\varphi
Y,X}Z,\varphi V)$ $\displaystyle\
-g(R_{\widetilde{Q}X,Y}Z,V)-g(R_{\,X,\widetilde{Q}Y}Z,V)$
$\displaystyle=g(Y,V)\,g((h-\varphi)\varphi
X,Z)+g(Y,Z)\,g((h-\varphi)V,\varphi X)+g(Z,V)\,g(\widetilde{Q}X,Y)$
$\displaystyle\ +(1/2)\,g(\varphi X,Z)\,g(Y,(h-\varphi)V)-(1/2)\,g(\varphi
Y,Z)\,g(X,(h-\varphi)V)$ $\displaystyle\ -g(X,V)\,g((h-\varphi)\varphi
Y,Z)+g(X,Z)\,g(Y,\varphi(h-\varphi)V).$ (42)

Then, replacing $Z$ by $\varphi Z$ and also $V$ by $\varphi V$ in (4) and
using (3), we get two equations

$\displaystyle-g(R_{\,X,Y}QZ,V)={-}\eta(Z)\,g(R_{\,\xi,V}\,X,Y){-}g(R_{\,X,Y}\,\varphi
Z,\varphi V){-}g(R_{\,X,\varphi Y}\,\varphi Z,V){-}g(R_{\,\varphi
X,Y}\,\varphi Z,V)$ $\displaystyle\
-g(Y,V)\,g(\varphi(h-\varphi)X,Z)-g(X,Y)\,g(\varphi
Z,(h-\varphi)V)+g(Y,\varphi Z)\,g(X,(h-\varphi)V)$ $\displaystyle\
-(1/2)\,g(\varphi Z,V)\,g((h-\varphi)X,Y)+(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V),$ (43)
$\displaystyle-g(R_{\,X,Y}Z,QV)=\eta(V)\,g(R_{\,\xi,Z}\,X,Y){-}g(R_{\,X,Y}\,\varphi
Z,\varphi V){-}g(R_{\,\varphi X,Y}Z,\varphi V){-}g(R_{\,X,\varphi Y}Z,\varphi
V)$ $\displaystyle\ +g(Y,\varphi
V)\,g((h-\varphi)X,Z)-g(X,Y)\,g(Z,(h-\varphi)\varphi
V)+g(Y,Z)\,g(X,(h-\varphi)\varphi V)$ $\displaystyle\ -(1/2)\,g(Z,\varphi
V)\,g((h-\varphi)X,Y)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V).$ (44)

Adding (4) to (4), we get

$\displaystyle-2\,g(R_{\,X,Y}Z,V)=-2\,g(R_{\,X,Y}\,\varphi Z,\varphi
V)+g(R_{\,X,Y}\widetilde{Q}Z,V)+g(R_{\,X,Y}Z,\widetilde{Q}V)$ $\displaystyle\
-\,\eta(Z)\,g(R_{\,\xi,V}\,X,Y)-g(R_{\,X,\varphi Y}\,\varphi
Z,V)-g(R_{\,\varphi X,Y}\,\varphi Z,V)$ $\displaystyle\
+\,\eta(V)\,g(R_{\,\xi,Z}\,X,Y)-g(R_{\,\varphi X,Y}Z,\varphi
V)-g(R_{\,X,\varphi Y}Z,\varphi V)$ $\displaystyle\
-g(Y,V)\,g(\varphi(h-\varphi)X,Z)+g(Y,\varphi Z)\,g(X,(h-\varphi)V)$
$\displaystyle\ +g(Y,\varphi
V)\,g((h-\varphi)X,Z)+2\,g(X,Y)\,g(Z,\widetilde{Q}V)+g(Y,Z)\,g(X,(h-\varphi)\varphi
V)$ $\displaystyle\ +(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V).$

Substituting the above equation into (4), we have

$\displaystyle 2\,g(R_{\,\varphi X,\varphi Y}Z,V)-2\,g(R_{\,X,Y}\,\varphi
Z,\varphi V)-\eta(Z)\,g(R_{\,\xi,V}X,Y)+\eta(V)\,g(R_{\,\xi,Z}\,X,Y)$
$\displaystyle+\,\eta(X)\,g(R_{\,\xi,Y}Z,V)-\eta(Y)\,g(R_{\,\xi,X}Z,V)+\delta(X,Y,Z,V)$
$\displaystyle-g(Y,V)\,g(\varphi(h-\varphi)X,Z)+g(Y,\varphi
Z)\,g(X,(h-\varphi)V)$ $\displaystyle+g(Y,\varphi
V)\,g((h-\varphi)X,Z)+g(Y,Z)\,g(X,(h-\varphi)\varphi
V)+2\,g(X,Y)\,g(Z,\widetilde{Q}V)$ $\displaystyle+(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V)=0.$ (45)

Using (16) that is supporting by (6), (4) and symmetry of $h^{2}$ and $Q$, we
obtain (4):

$\displaystyle 2\,g(R_{\,\varphi X,\varphi Y}Z,V)-2\,g(R_{\,X,Y}\,\varphi
Z,\varphi V)+\delta(X,Y,Z,V)+2\,g(X,Y)\,g(Z,\widetilde{Q}V)$
$\displaystyle-g(Y,V)\,g(\varphi(h-\varphi)X,Z)+g(Y,\varphi
Z)\,g(X,(h-\varphi)V)$ $\displaystyle+g(Y,\varphi
V)\,g((h-\varphi)X,Z)+g(Y,Z)\,g(X,(h-\varphi)\varphi V)$
$\displaystyle+(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V)=0.$

Replacing $X$ by $\varphi X$ and $Y$ by $\varphi Y$ in (4) and using (3), we
get (4). $\square$

Proof of Lemma 5. In Step 1, our formulas depend on four vectors
$X,Y,Z,V\in\mathfrak{X}_{M}$ and contain many terms. In Step 2, the formulas
depend only on three vectors from $TM$ and contain a relatively small number
of terms.

Step 1. Differentiating (19) and using
$g((\nabla_{X}\varphi)(\nabla_{V}\varphi)Y,Z)=-g((\nabla_{V}\varphi)Y,(\nabla_{X}\varphi)Z)$
gives

$\displaystyle
g((\nabla_{V}\,\varphi)Y,(\nabla_{X}\,\varphi)Z)+g((\nabla_{X}\,\varphi)Y,(\nabla_{V}\,\varphi)Z)$
$\displaystyle=g((\nabla^{2}_{V,X}\,\varphi)\,\varphi
Y,Z)+g((\nabla^{2}_{V,X}\,\varphi)\,\varphi
Z,Y)-g(Y,(h-\varphi)V)\,g((h-\varphi)X,Z)$ $\displaystyle\
-\,\eta(Y)\,g((\nabla_{V}(h-\varphi))X,Z)-\nabla_{V}\big{(}\eta(Z)\,g((h-\varphi)X,\,QY)\big{)}.$
(46)

On the other hand, using (33), (4) and (3), we find $\nabla^{2}$-terms in (4):

$\displaystyle g((\nabla^{2}_{V,X}\,\varphi)\varphi
Y,Z)=g((\nabla^{2}_{V,\varphi Y}\,\varphi)Z,X)$ $\displaystyle\
+2\,g(X,\varphi
Y)\,g(Z,(h-\varphi)V)+g(X,Z)\,g(Y,\varphi(h-\varphi)V)-g(\varphi
Y,Z)\,g(X,(h-\varphi)V)$
$\displaystyle=-g(R_{\,X,Z}V,QY)-\eta(Y)\,g(R_{\,\xi,V}\,X,Z)+(3/2)\,g(X,\varphi
Y)\,g(Z,(h-\varphi)V)$ $\displaystyle\
+g(Z,V)\,g(\varphi(h-\varphi)X,Y)+(1/2)\,g(\varphi Y,V)\,g((h-\varphi)X,Z)$
$\displaystyle\ -(1/2)\,g(R_{\,X,V}\,\varphi Y,\varphi
Z)-(1/2)\,g(R_{\,V,Z}\,\varphi Y,\varphi X)-(1/2)\,g(R_{\,X,Z}\,\varphi
Y,\varphi V),$ (47) $\displaystyle g((\nabla^{2}_{V,X}\,\varphi)\varphi
Z,Y)=g((\nabla^{2}_{V,\varphi Z}\,\varphi)Y,X)$ $\displaystyle\
+2\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)+g(X,Y)\,g(Z,\varphi(h-\varphi)V)-g(Y,\varphi
Z)\,g(X,(h-\varphi)V)$
$\displaystyle=-g(R_{\,X,Y}V,QZ)-\eta(Z)\,g(R_{\,\xi,V}\,X,Y)+(3/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)$ $\displaystyle\
+g(Y,V)\,g(\varphi(h-\varphi)X,Z)+(1/2)\,g(\varphi Z,V)\,g((h-\varphi)X,Y)$
$\displaystyle\ -(1/2)\,g(R_{\,X,V}\,\varphi Z,\varphi
Y)-(1/2)\,g(R_{\,V,Y}\,\varphi Z,\varphi X)-(1/2)\,g(R_{\,X,Y}\,\varphi
Z,\varphi V).$ (48)

Using (16) and (4)–(4), we get from (4) the equality

$\displaystyle
g((\nabla_{V}\,\varphi)Y,(\nabla_{X}\,\varphi)Z)+g((\nabla_{X}\,\varphi)Y,(\nabla_{V}\,\varphi)Z)=-g(R_{\,X,Z}V,QY)-\eta(Y)\,g(R_{\,\xi,V}\,X,Z)$
$\displaystyle\ +(1/2)g(R_{\,Y,V}\,\varphi Z,\varphi
X){+}(1/2)g(R_{\,Y,X}\,\varphi Z,\varphi V){-}(1/2)g(R_{\,V,Z}\,\varphi
Y,\varphi X){-}(1/2)g(R_{\,X,Z}\,\varphi Y,\varphi V)$
$\displaystyle-g(R_{\,X,Y}V,QZ)-\eta(Z)\,g(R_{\,\xi,V}\,X,Y)-g((h-\varphi)V,Y)\,g((h-\varphi)X,Z)$
$\displaystyle\ +(3/2)\,g(X,\varphi
Y)\,g(Z,(h-\varphi)V){+}g(Z,V)\,g(\varphi(h-\varphi)X,Y){+}(1/2)\,g(\varphi
Y,V)\,g((h-\varphi)X,Z)$ $\displaystyle\ +(3/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V){+}g(Y,V)\,g(\varphi(h-\varphi)X,Z){+}(1/2)\,g(\varphi
Z,V)\,g((h-\varphi)X,Y)$
$\displaystyle-\,\eta(Y)\,g((\nabla_{V}(h-\varphi))X,Z)-\nabla_{V}\big{(}\eta(Z)\,g((h-\varphi)X,\,QY)\big{)}.$
(49)

Using (4) for $g(R_{\,Y,V}\,\varphi Z,\varphi X)=g(R_{\,\varphi X,\varphi
Z}V,Y)$ and $g(R_{\,Y,X}\,\varphi Z,\varphi V)=g(R_{\,\varphi V,\varphi
Z}X,Y)$ in (4), and replacing $(\nabla_{V}(h-\varphi))X$ by (15) and using
$g(R_{\,\varphi X,\,\varphi Y}\,Z,\,\xi)=0$, see (14), we get

$\displaystyle
g((\nabla_{V}\,\varphi)Y,(\nabla_{X}\,\varphi)Z)+g((\nabla_{X}\,\varphi)Y,(\nabla_{V}\,\varphi)Z)=-g(R_{\,X,Z}V,QY)-g(R_{\,X,Y}V,QZ)$
$\displaystyle\ +g(R_{\,V,Z}\,\varphi X,\varphi Y)+g(R_{\,X,Z}\,\varphi
V,\varphi Y)-g(Y,(h-\varphi)V)\,g((h-\varphi)X,Z)$ $\displaystyle\
-\eta(X)\eta(Z)\,\,g(Y,(h-\varphi)^{2}V)+\eta(Y)\,\eta(Z)\,g(X,(h-\varphi)^{2}V)$
$\displaystyle\
-(1/2)\,g(Y,Z)\,g(\widetilde{Q}X+\varphi^{2}X,V)+g(Y,V)\,g(\varphi(h-\varphi)X,Z)$
$\displaystyle\
+(3/4)\,g(Z,V)\,g(\varphi(h-\varphi)X,Y)-(1/4)\,g(X,Z)\,g((h-\varphi)\varphi
Y,V)$ $\displaystyle\ -(1/4)\,g(X,V)\,g((h-\varphi)\varphi
Y,Z)-g(Z,V)\,g(\widetilde{Q}Y,X)-(1/2)g(X,Z)\,g(\widetilde{Q}Y,V)$
$\displaystyle\ -(5/4)\,g(X,\varphi Z)\,g((h-\varphi)Y,V)-(1/4)\,g(Z,\varphi
V)\,g((h-\varphi)X,Y)$ $\displaystyle\ +(3/2)\,g(X,\varphi
Y)\,g((h-\varphi)V,Z){-}(1/2)\,g(Y,\varphi V)\,g((h-\varphi)X,Z)$
$\displaystyle\
-(1/4)\,\delta(V,Z,X,Y)-(1/4)\,\delta(X,Z,V,Y)-\nabla_{V}\big{(}\eta(Z)\,g((h-\varphi)X,\,QY)\big{)}.$
(50)

Replacing $Z$ and $V$ by $\varphi Z$ and $\varphi V$ in (4), we find

$\displaystyle g((\nabla_{\varphi V}\,\varphi)Y,(\nabla_{X}\,\varphi)\varphi
Z){+}g((\nabla_{X}\,\varphi)Y,(\nabla_{\varphi V}\,\varphi)\varphi
Z)=g(R_{\,X,\varphi Z}\,\varphi^{2}V,\varphi Y){-}g(R_{\,X,\varphi Z}\,\varphi
V,QY)$ $\displaystyle\ -g(R_{\,X,Y}\,Q\varphi Z,\varphi V)+g(R_{\,\varphi
X,\varphi Y}\,\varphi Z,\varphi V)-g(Y,(h-\varphi)\varphi
V)\,g((h-\varphi)X,\varphi Z)$ $\displaystyle\ -(1/2)\,g(Y,\varphi
Z)\,g(\widetilde{Q}X+\varphi^{2}X,\varphi V)+g(Y,\varphi
V)\,g(\varphi(h-\varphi)X,\varphi Z)$ $\displaystyle\
-(3/4)\,g(\varphi^{2}Z,V)\,g(\varphi(h-\varphi)X,Y)-(1/4)\,g(X,\varphi
Z)\,g((h-\varphi)\varphi Y,\varphi V)$ $\displaystyle\ -(1/4)\,g(X,\varphi
V)\,g((h{-}\varphi)\varphi Y,\varphi
Z){+}g(\varphi^{2}Z,V)\,g(X,\widetilde{Q}Y){-}(1/2)g(X,\varphi
Z)\,g(\widetilde{Q}Y,\varphi V)$ $\displaystyle\
-(5/4)\,g(X,\varphi^{2}Z)\,g((h-\varphi)Y,\varphi V)+(1/4)\,g(\varphi
Z,QV)\,g((h-\varphi)X,Y)$ $\displaystyle\ +(3/2)\,g(X,\varphi
Y)\,g((h-\varphi)\varphi V,\varphi
Z){-}(1/2)\,g(Y,\varphi^{2}V)\,g((h-\varphi)X,\varphi Z)$ $\displaystyle\
-(1/4)\,\delta(\varphi V,\varphi Z,X,Y)-(1/4)\,\delta(X,\varphi Z,\varphi
V,Y).$ (51)

Using (19), (3), (3), (3) and Lemma 1, we rewrite terms in the lhs of (4):

$\displaystyle g((\nabla_{\varphi V}\,\varphi)Y,(\nabla_{X}\,\varphi)\varphi
Z)=g(Q(\nabla_{X}\,\varphi)Z,(\nabla_{V}\,\varphi)Y)-\eta(V)g((\nabla_{X}\,\varphi)Z,\varphi
hY)$ $\displaystyle\
-2\,\eta(Y)g((\nabla_{X}\,\varphi)Z,\varphi^{2}V)+\eta(Z)g((\nabla_{V}\,\varphi)Y,\varphi(h-\varphi)X)$
$\displaystyle\
-\eta(Z)\eta(V)\,g((h-\varphi)X,(h-\varphi)Y)-2\,\eta(Z)\eta(Y)\,g((h-\varphi)X,\varphi
V)$ $\displaystyle\
-\,g(\varphi(h-\varphi)X,Z)g(Y,\varphi(h-\varphi)V)+g(Y,(h-\varphi)\varphi
V)\,g((h-\varphi)X,QZ),$ (52) $\displaystyle
g((\nabla_{X}\,\varphi)Y,(\nabla_{\varphi V}\,\varphi)\varphi
Z)=\eta(Z)\,g((\nabla_{X}\,\varphi)Y,(h{-}\varphi)\varphi
V){-}g((\nabla_{V}\,\varphi)Z,Q(\nabla_{X}\,\varphi)Y)$ $\displaystyle\
+\,\eta(V)\,g((\nabla_{X}\,\varphi)Y,\varphi
hZ)-g(\varphi(h-\varphi)X,Y)\,g(\varphi(h-\varphi)V,Z)$ $\displaystyle\
+\,g(\varphi(h-\varphi)X,Y)\,g((h-\varphi)\varphi V,QZ).$ (53)

From (16) and Lemma 1, we have

$\displaystyle g(R_{\,X,\varphi Z}\,\varphi V,Y)-g(R_{\,X,\varphi
Z}\,\varphi^{2}V,\varphi Y)=g(R_{\,X,\varphi Z}\,\varphi V,Y)$
$\displaystyle+g(R_{\,X,\varphi Z}QV,\varphi
Y)-\eta(X)\,\eta(V)\,g((h-\varphi)^{2}\varphi Y,\varphi Z).$ (54)

On the other hand, from (4) and (4) it follows that

$\displaystyle g(R_{\,X,Z}\,\varphi V,\varphi
Y)+g(R_{\,X,Z}\,\varphi^{2}V,Y)+g(R_{\,X,\varphi Z}\,\varphi
V,Y)+g(R_{\,\varphi X,Z}\,\varphi V,Y)$ $\displaystyle\
=-g(Y,Z)\,g(\varphi(h-\varphi)X,V)+g(X,Z)\,g(\varphi(h-\varphi)Y,V)+g(Z,\varphi
V)\,g(X,hY{-}\varphi Y)$ $\displaystyle\ -(1/2)\,g(Y,\varphi
V)\,g((h-\varphi)X,Z)+(1/2)\,g(X,\varphi V)\,g((h-\varphi)Y,Z),$ (55)
$\displaystyle g(R_{\,\varphi X,Z}\,\varphi
V,\varphi^{2}Y)=g(R_{\,\varphi^{2}X,\varphi Z}V,\varphi
Y){+}(1/2)\delta(\varphi X,Z,V,\varphi Y){+}g(\varphi
X,Z)\,g(\widetilde{Q}\varphi Y,V)$ $\displaystyle\ +(1/2)\,g(\varphi
Y,Z)\,g((h-\varphi)\varphi X,\varphi V)+(1/2)\,g(Z,\varphi V)\,g(\varphi
X,(h-\varphi)\varphi Y)$ $\displaystyle\
+(1/2)\,g(\varphi^{2}Y,Z)\,g((h-\varphi)\varphi
X,V)+(1/2)\,g(Z,V)\,g((h-\varphi)\varphi X,QY)$ $\displaystyle\
-(1/4)\,g(\varphi^{2}X,V)\,g((h-\varphi)\varphi Y,Z)+(1/4)\,g(\varphi
X,V)\,g(QY,(h-\varphi)Z).$ (56)

From (4), using (3), we find

$\displaystyle-g(R_{\,\varphi X,Z}\,\varphi V,QY)+\eta(Y)\,g(R_{\,\varphi
X,Z}\,\varphi V,\xi)=-g(R_{\,QX,\varphi Z}V,\varphi Y)$ $\displaystyle\
+\eta(X)\,g(R_{\,\xi,\varphi Z}V,\varphi Y)+(1/2)\,\delta(\varphi
X,Z,V,\varphi Y){+}g(\varphi X,Z)\,g(\widetilde{Q}\varphi Y,V)$
$\displaystyle\ +(1/2)\,g(\varphi Y,Z)\,g((h-\varphi)\varphi X,\varphi
V)+(1/2)\,g(Z,\varphi V)\,g(\varphi X,(h-\varphi)\varphi Y)$ $\displaystyle\
+(1/2)\,g(\varphi^{2}Y,Z)\,g((h-\varphi)\varphi
X,V)+(1/2)\,g(Z,V)\,g((h-\varphi)\varphi X,QY)$ $\displaystyle\
-(1/4)\,g(\varphi^{2}X,V)\,g((h-\varphi)\varphi Y,Z)+(1/4)\,g(\varphi
X,V)\,g(QY,(h-\varphi)Z).$ (57)

Summing up the formulas (4) and (4) (and using (4), (4), (16) and Lemma 1), we
get

$\displaystyle g(R_{\,\varphi X,Z}\,\varphi V,Y)+g(R_{\,QX,\varphi Z}V,\varphi
Y)=-g(R_{\,X,Z}\,\varphi V,\varphi Y)-g(R_{\,X,Z}\,\varphi^{2}V,Y)$
$\displaystyle-g(R_{\,X,\varphi Z}\,\varphi V,Y)+g(R_{\,\varphi X,Z}\,\varphi
V,QY)-\eta(Y)\,\eta(Z)\,g(\varphi X,(h-\varphi)^{2}\varphi V)$ $\displaystyle\
-g(Y,Z)\,g(\varphi(h-\varphi)X,V)+g(X,Z)\,g(\varphi(h-\varphi)Y,V)+g(Z,\varphi
V)\,g(X,(h-\varphi)Y)$ $\displaystyle\ -(1/2)\,g(Y,\varphi
V)\,g((h-\varphi)X,Z)+(1/2)\,g(X,\varphi V)\,g((h-\varphi)Y,Z)$
$\displaystyle\ +\eta(X)\,\eta(V)\,g(\varphi Y,(h-\varphi)^{2}\varphi
Z){+}(1/2)\,\delta(\varphi X,Z,V,\varphi Y){+}g(\varphi
X,Z)\,g(\widetilde{Q}\varphi Y,V)$ $\displaystyle\ +(1/2)\,g(\varphi
Y,Z)\,g((h-\varphi)\varphi X,\varphi V)+(1/2)\,g(Z,\varphi V)\,g(\varphi
X,(h-\varphi)\varphi Y)$ $\displaystyle\
+(1/2)\,g(\varphi^{2}Y,Z)\,g((h-\varphi)\varphi
X,V)+(1/2)\,g(Z,V)\,g((h-\varphi)\varphi X,QY)$ $\displaystyle\
-(1/4)\,g(\varphi^{2}X,V)\,g((h-\varphi)\varphi Y,Z)+(1/4)\,g(\varphi
X,V)\,g(QY,(h-\varphi)Z).$ (58)

Substituting (4) into (4), we get

$\displaystyle g(R_{\,X,\varphi Z}\,\varphi V,Y)-g(R_{\,X,\varphi
Z}\,\varphi^{2}V,\varphi Y)=-g(R_{\,QX,\varphi Z}V,\varphi
Y)-g(R_{\,X,Z}\,\varphi V,\varphi Y)$ $\displaystyle\
-g(R_{\,X,Z}\,\varphi^{2}V,Y)-g(R_{\,X,\varphi Z}\,\varphi V,Y)+g(R_{\,\varphi
X,Z}\,\varphi V,QY)+g(R_{\,X,\varphi Z}QV,\varphi Y)$ $\displaystyle\
+g(Y,Z)\,g((h-\varphi)X,\varphi
V)+g(X,Z)\,g(\varphi(h-\varphi)Y,V)+g(Z,\varphi V)\,g(X,(h-\varphi)Y)$
$\displaystyle\ -(1/2)\,g(Y,\varphi V)\,g((h-\varphi)X,Z)+(1/2)\,g(X,\varphi
V)\,g((h-\varphi)Y,Z)$ $\displaystyle\ +(1/2)\,\delta(\varphi X,Z,V,\varphi
Y){+}g(\varphi X,Z)\,g(\widetilde{Q}\varphi
Y,V)-\eta(Y)\,\eta(Z)\,g((h-\varphi)^{2}\varphi V,\varphi X)$ $\displaystyle\
+(1/2)\,g(\varphi Y,Z)\,g((h-\varphi)\varphi X,\varphi V)+(1/2)\,g(Z,\varphi
V)\,g(\varphi X,h\varphi Y{+}QY)$ $\displaystyle\
+(1/2)\,g(\varphi^{2}Y,Z)\,g((h-\varphi)\varphi
X,V)+(1/2)\,g(Z,V)\,g((h-\varphi)\varphi X,QY)$ $\displaystyle\
-(1/4)\,g(\varphi^{2}X,V)\,g((h-\varphi)\varphi Y,Z)+(1/4)\,g(\varphi
X,V)\,g(QY,(h-\varphi)Z).$ (59)

By means of (16), from (4) and (4), we have

$\displaystyle g(R_{\,\varphi X,\varphi Y}\,\varphi Z,\varphi
V)-g(R_{\,X,Y}\,\varphi Z,\varphi V)=-g(R_{\,\varphi X,\varphi
Y}Z,V)+g(R_{\,QX,QY}Z,V)$
$\displaystyle{-}\eta(X)\,g(R_{\,\xi,QY}Z,V)+\eta(Y)\,g(R_{\,\xi,QX}Z,V)$
$\displaystyle\ +(1/2)\,g(\varphi Y,V)\,g((h-\varphi)\varphi X,\varphi
Z)+(1/2)\,g(\varphi^{2}Y,Z)\,g(X,\varphi(h-\varphi)V)$ $\displaystyle\
-(1/2)g(\varphi^{2}Y,V)g((h-\varphi)\varphi X,Z)+(1/2)g(\varphi Y,Z)g(\varphi
X,(h-\varphi)\varphi V)$ $\displaystyle\
+(1/4)\,g(\varphi^{2}X,Z)\,g(Y,\varphi(h-\varphi)V)+(1/4)\,g(\varphi
X,Z)\,g(\varphi Y,(h-\varphi)\varphi V)$ $\displaystyle\
+(1/2)\,g(Y,V)\,g(\varphi(h-\varphi)X,Z)-(1/2)\,g(Y,\varphi
Z)\,g(X,(h-\varphi)V)$ $\displaystyle\ -(1/2)\,g(Y,\varphi
V)\,g((h-\varphi)X,Z)-(1/2)\,g(Y,Z)\,g(X,(h-\varphi)\varphi V)$
$\displaystyle\ -(1/4)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)-(1/4)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V)$
$\displaystyle\
-g(\varphi^{2}X+X,Y)\,g(\widetilde{Q}V,Z)+(1/2)\,\delta(\varphi X,\varphi
Y,Z,V)-(1/2)\,\delta(X,Y,Z,V).$ (60)

Step 2. Substituting (4), (4), (4) and (4) into (4) and adding the result to
(4), we obtain a multi-term equation (which we do not place here). Then
putting $\xi$ on $V$, we get the following equation with two
$\nabla\varphi$-terms and a relatively small number of other terms:

$\displaystyle g((\nabla_{X}\,\varphi)Z,\widetilde{Q}Y+Q(\varphi
h+\widetilde{Q})Y)+g((\nabla_{X}\,\varphi)Y,(2\,\varphi
h+\widetilde{Q})Z-Q(\varphi h+\widetilde{Q})Z)$ $\displaystyle\
-\eta(Z)\,g((h-\varphi)X,\varphi(\varphi
h+\widetilde{Q})Y+(h-\varphi)Y)=-g(R_{\,X,Z}\,\xi,QY)$ $\displaystyle\
-g(R_{\,X,Y}\,\xi,QZ)+g(R_{\,\xi,Z}\,\varphi X,\varphi
Y)-\eta(X)\,g(R_{\,\xi,QY}Z,\xi)+\eta(Y)\,g(R_{\,\xi,QX}Z,\xi)$
$\displaystyle\ +g(R_{\,QX,\varphi Z}\,\xi,\varphi Y)-g(R_{\,X,\varphi
Z}\,\xi,\varphi Y)-g(R_{\,\varphi X,\varphi Y}Z,\xi)+g(R_{\,QX,QY}Z,\xi)$
$\displaystyle\
+\eta(Y)\,g(\varphi(h-\varphi)X,Z)+(3/4)\,\eta(Z)\,g(\varphi(h-\varphi)X,Y)-(1/4)\,\eta(X)\,g((h-\varphi)\varphi
Y,Z)$ $\displaystyle\
-\eta(Z)\,g(\widetilde{Q}Y,X)-(1/2)\,\eta(Z)\,g((h-\varphi)\varphi
X,QY)+(1/2)\,\eta(Y)\,g(\varphi(h-\varphi)X,Z).$

After applying (16) the above equation is reduced to the following form:

$\displaystyle
g((\nabla_{X}\,\varphi)Z,Q\varphi(h{-}\varphi)Y-Y)+g((\nabla_{Y}\,\varphi)X,\widetilde{Q}\varphi(h-\varphi)Z-\widetilde{Q}Z-\varphi
hZ)$ $\displaystyle\
+\eta(X)\,g(p_{1}(Z),Y)+\eta(Y)\,g(p_{2}(Z),X)+\eta(Z)\,g(p_{3}(X),Y)=0,$ (61)

where $p_{1},p_{2},p_{3}$ are linear operators (polynomials in
$h,\varphi,\widetilde{Q}$) on $TM$ and $p_{1},p_{2}$ vanish on $\xi$.

Swapping $X\leftrightarrow Z$ and $V\leftrightarrow Y$ in the multi-term
equation (mentioned above), then putting $\xi$ on $V$, we get another equation
with two $\nabla\varphi$-terms and a relatively small number of other terms:

$\displaystyle-g((\nabla_{Z}\,\varphi)X,2\,\varphi
hY+\widetilde{Q}\varphi(h-\varphi)Y)+g((\nabla_{Y}\,\varphi)X,\,\widetilde{Q}\varphi(h-\varphi)Z)$
$\displaystyle\
+\eta(X)\,g(2\,\varphi^{2}Y+2\,QY,\varphi(h-\varphi)Z)-\eta(Y)\,g(\varphi(h-\varphi)Z,\varphi
hX)=-g(R_{\,Z,X}Y,\xi)$ $\displaystyle\
+g(R_{\,\xi,Z}Y,QX)+g(R_{\,Z,X}\,\varphi^{2}Y,\xi)+g(R_{\,Z,\varphi
X}\,\varphi Y,\xi)-g(R_{\,\varphi Z,X}\,\varphi Y,\xi)$ $\displaystyle\
-g(R_{\,\xi,Z}\,\varphi Y,\widetilde{Q}\varphi
X)+\eta(X)\,g(Z,(h-\varphi)^{2}Y)-(1/2)\,\eta(X)\,g(\widetilde{Q}Z+\varphi^{2}Z,Y)$
$\displaystyle\ -\eta(X)\,g((h-\varphi)Z,\varphi
Y)+\eta(X)\,g((h-\varphi)^{2}\varphi Y,\varphi
Z)+\eta(Y)\,g(\varphi(h-\varphi)Z,X)$ $\displaystyle\
+(1/2)\,\eta(Y)\,g(\varphi(h-\varphi)Z,X)-(1/2)\,\eta(X)\,g(Z,(h-\varphi)\varphi
Y)-\eta(Z)\,g(\widetilde{Q}Y,X).$

After applying (16) and (4) the above equation is reduced to the following
form:

$\displaystyle g((\nabla_{X}\,\varphi)Z,2\,\varphi
hY+\widetilde{Q}\varphi(h-\varphi)Y)+g((\nabla_{Y}\,\varphi)X,\widetilde{Q}\varphi(h-\varphi)Z)$
$\displaystyle\
+\eta(X)\,g(q_{1}(Z),Y)+\eta(Y)\,g(q_{2}(Z),X)+\eta(Z)\,g(q_{3}(X),Y)=0,$ (62)

where $q_{1},q_{2},q_{3}$ are linear operators (polynomials in
$h,\varphi,\widetilde{Q}$) on $TM$ and $q_{1},q_{2}$ vanish on $\xi$.

Deducing (4) and (4), we also used Lemma 1, (14), Remark 1 (that all
$\delta$-terms vanish) and $\nabla_{\xi}\big{(}\eta(Z)\,g(hX-\varphi
X,\widetilde{Q}Y)\big{)}=0$. Subtracting (4) from (4), we get

$\displaystyle-g((\nabla_{Y}\,\varphi)X,\ \varphi
hZ+\widetilde{Q}Z)-g((\nabla_{X}\,\varphi)Z,\ \varphi hY-\widetilde{Q}Y)$
$\displaystyle\
+\eta(X)\,g(r_{1}(Z),Y)+\eta(Y)\,g(r_{2}(Z),X)+\eta(Z)\,g(r_{3}(X),Y)=0,$ (63)

where $r_{1},r_{2},r_{3}$ are linear operators on $TM$ and $r_{1},r_{2}$
vanish on $\xi$.

Swapping $X$ and $Y$ in (4) we get

$\displaystyle-g((\nabla_{X}\,\varphi)Y,\ \varphi
hZ+\widetilde{Q}Z)-g((\nabla_{Y}\,\varphi)Z,\ \varphi hX-\widetilde{Q}X)$
$\displaystyle\
+\eta(Y)\,g(r_{1}(Z),X)+\eta(X)\,g(r_{2}(Z),Y)+\eta(Z)\,g(r_{3}(Y),X)=0.$ (64)

Adding (4) to the resulting equation (4) with the use of (4) and (16), we get

$\displaystyle-g((\nabla_{X}\,\varphi)Z,\ \varphi
hY-\widetilde{Q}Y)-g((\nabla_{Y}\,\varphi)Z,\ \varphi hX-\widetilde{Q}X)$
$\displaystyle\
+\eta(X)\,g(a_{1}(Z),Y)+\eta(Y)\,g(a_{2}(Z),X)+\eta(Z)\,g(a_{3}(X),Y)=0,$ (65)

where $a_{1},a_{2},a_{3}$ are linear operators on $TM$ and $a_{1},a_{2}$
vanish on $\xi$.

Swapping $X$ and $Z$ in (4) we get

$\displaystyle-g((\nabla_{Z}\,\varphi)X,\ \varphi
hY-\widetilde{Q}Y)-g((\nabla_{Y}\,\varphi)X,\ \varphi hZ-\widetilde{Q}Z)$
$\displaystyle\
+\eta(Z)\,g(a_{1}(X),Y)+\eta(Y)\,g(a_{2}(X),Z)+\eta(X)\,g(a_{3}(Z),Y)=0.$ (66)

Adding the resulting equation (4) to (4), then applying (15), we get equation
of the form

$\displaystyle-2\,g((\nabla_{Y}\,\varphi)X,\ \varphi
hZ)+\eta(X)\,g(b_{1}(Z),Y)+\eta(Y)\,g(b_{2}(X),Z)+\eta(Z)\,g(b_{3}(X),Y)=0,$
(67)

where $b_{1},b_{2},b_{3}$ are linear operators (polynomials in
$h,\varphi,\widetilde{Q}$) on $TM$ and $b_{1},b_{2}$ vanish on $\xi$. Taking
$Z=\xi$ in (67), we get $g(b_{3}(X),Y)=0$ for all $X,Y$. Hence $b_{3}=0$ and
(67) is reduced to

$\displaystyle 2\,g((\nabla_{X}\,\varphi)Y,\varphi
hZ)=\eta(X)\,g(b_{1}(Z),Y)+\eta(Y)\,g(b_{2}(Z),X).$ (68)

Replacing $Y$ by $\varphi Y$ in (68) and using (19) and (3), we obtain

$\displaystyle
g((\nabla_{X}\,\varphi)Y,hQZ)=\eta(X)\,g(c_{1}(Z),Y)+\eta(Y)\,g(c_{2}(Z),X)+\eta(Z)\,g(c_{3}(X),Y),$
(69)

where $c_{1},c_{2},c_{3}$ are linear operators, and $c_{1},c_{2}$ vanish on
$\xi$. Taking $Z=\xi$ in (69), we get $c_{3}=0$.

Taking $X=\xi$ in (69), and using $(\nabla_{\xi}\,\varphi)X=\varphi
hX+\widetilde{Q}X$, we find $c_{1}=-(\varphi h^{2}+\widetilde{Q}h)Q$. Taking
$Y=\xi$ in (69), and using (11), we find $c_{2}=(\varphi
h^{2}-h+\widetilde{Q}h)Q$. Thus, (69) reads as

$\displaystyle g((\nabla_{X}\,\varphi)Y,hQZ)=-\eta(X)\,g((\varphi
h^{2}+\widetilde{Q}h)QZ,Y)+\eta(Y)\,g((\varphi h^{2}-h+\widetilde{Q}h)QZ,X).$
(70)

Finally, replacing $QZ$ by $Z$ in (70), since $Q$ is nonsingular, we get (25).
$\square$

## Conclusions

Some properties of the curvature tensor and the tensors $\varphi$ and $h$ of
nearly Sasakian manifolds (see [3, 10]) were extended to weak nearly Sasakian
manifolds with the conditions (5) and (6). Our results answer the question in
the introduction and show that the weak nearly Sasakian structure allows us to
expand the theory of contact metric manifolds. Our Theorem 1 characterizes
Sasakian manifolds in the class of weak almost contact metric manifolds with
the condition (5) using the property (1). According to our Theorem 3, the
properties (5) and (6) are sufficient for a weak nearly Sasakian manifold of
dimension greater than five to be Sasakian, this generalizes Theorem 3.3 from
[10], see also [4, Theorem 4.9]. After [3, 4, 9] we are looking forward to
studying five-dimensional weak nearly Sasakian manifolds. A more difficult
task is to extend a theory such as twistor spinors, see [8], by considering
non-singular skew-symmetric tensors instead of linear complex structures.
Based on applications of the nearly Sasakian structure, we expect that certain
weak structure will be useful for geometry and theoretical physics.

## References

  * [1] Blair, D.E. _Riemannian geometry of contact and symplectic manifolds_ , Second edition, Springer-Verlag, New York, 2010
  * [2] Blair, D.E., Showers, D.K., Komatu, Y. Nearly Sasakian manifolds. Kodai Math. Sem. Rep. 1976, 27, 175–180.
  * [3] Cappelletti-Montano, B. and Dileo, G. Nearly Sasakian geometry and $SU(2)$-structures. Annali di Matematica 195, 897–922 (2016)
  * [4] Cappelletti-Montano, B., Nicola, A.D., Dileo, G. and Yudin, I. Nearly Sasakian manifolds revisited. Complex manifolds, 6, 320–334 (2019)
  * [5] Chow, B., Lu, P., Ni, L. Hamilton’s Ricci flow. Grad. Stud. in Math., 77, AMS, Providence, RI (2006).
  * [6] Gray, A.: Nearly Kähler manifolds. J. Differ. Geom. 4, 283–309 (1970)
  * [7] Herrera, A.C. Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras. Preprint, 2022, ArXiv[Math.DG]:2012.09356. Accepted in: Revista de la Unión Matemática Argentina.
  * [8] Kim, E.C. The first positive and negative Dirac eigenvalues on Sasakian manifolds. J. Korean Math. Soc. 60 (2023), No. 5, 999–1021
  * [9] Massamba, F. and Nzunogera, A. A note on nearly Sasakian manifolds. Mathematics 2023, 11, 2634. DOI: 10.3390/math11122634
  * [10] Nicola, A.D., Dileo, G. and Yudin, I. On nearly Sasakian and nearly cosymplectic manifolds, Annali di Matematica, 197 (2018), 127–138, DOI: 10.1007/s10231-017-0671-2
  * [11] Olszak, Z. Nearly Sasakian manifolds. Tensor (N.S.) 33 (3), 277–286 (1979)
  * [12] Olszak, Z. Five-dimensional nearly Sasakian manifolds. Tensor New Ser. 1980, 34, 273–276
  * [13] Patra, D.S. and Rovenski, V. On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds. Differential Geometry and its Applications, 90 (2023) 102043
  * [14] Rovenski, V. and Wolak, R. New metric structures on $\mathfrak{g}$-foliations, Indagationes Mathematicae, 33 (2022), 518–532
  * [15] Rovenski, V. Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds. Communications in Analysis and Mechanics, 2023, Volume 15, Issue 2: 177–188. DOI: 10.3934/cam.2023010
  * [16] Rovenski, V. Weak nearly Sasakian and weak nearly cosymplectic manifolds. Mathematics 2023, 11(20), 4377. DOI: 10.3390/math11204377
  * [17] Rovenski, V. Foliated structure of weak nearly Sasakian manifolds. Preprint, 10 pp. arXiv:2312.11411
  * [18] Rovenski, V. On the splitting of weak nearly cosymplectic manifolds. Preprint, 16 pp. arXiv:2310.19518v2
  * [19] Rovenski, V. and Walczak, P. G. _Extrinsic geometry of foliations_ , Progress in Mathematics, vol. 339, Birkhäuser, Cham, 2021