LACUNARY STATISTICAL CONVERGENCE OF ORDER $\beta$

IN DIFFERENCE SEQUENCES OF FUZZY NUMBERS

Hıfsı ALTINOK Damla YAĞDIRAN

Department of Mathematics, Fırat University, 23119, Elazığ, TURKEY.

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

> Abstract. In this paper, we define the spaces
> $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right),$
> $S_{\theta}^{\beta}\left(F,\Delta^{m}\right),$
> $w_{p}^{\beta}\left(F,\Delta^{m}\right)$ for sequences of fuzzy numbers
> using generalized difference operator $\Delta^{m}$ and a lacunary sequence
> $\theta$ and give some relations between them, where
> $\beta\in\left(0,1\right]$ and $p>0$. Furthermore, in the last section of
> paper, some inclusion theorems are presented related to the spaces
> $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ and
> $w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$ according to modulus
> function $f$.

Key words and phrases: Fuzzy number, sequence of fuzzy numbers, statistical
convergence, lacunary sequence, Cesàro summability, modulus function

Mathematics Subject Classification: 40A05; 40A25; 40A30; 40C05; 03E72.

1 Introduction

The idea of statistical convergence was given by Zygmund [26] in the first
edition of his monograph published in Warsaw in 1935. The concept of
statistical convergence was introduced by Steinhaus [22] and Fast [11] and
later reintroduced by Schoenberg [21] independently. Over the years and under
different names statistical convergence has been discussed in the theory of
Fourier analysis, ergodic theory, number theory, measure theory, trigonometric
series, turnpike theory and Banach spaces. Later on it was further
investigated from the sequence space point of view and linked with summability
theory by Connor [7], Et [8], Fridy [13], Mursaleen [17] and many others. In
recent years, generalizations of statistical convergence have appeared in the
study of strong integral summability and the structure of ideals of bounded
continuous functions on locally compact spaces. Statistical convergence and
its generalizations are also connected with subsets of the Stone-Čech
compactification of the natural numbers. Moreover, statistical convergence is
closely related to the concept of convergence in probability.

Now, we recall some basic concepts which we shall use throughout the paper.

By a lacunary sequence we mean an increasing integer sequence
$\theta=\left(k_{r}\right)$ such that $k_{0}=0$ and
$h_{r}=\left(k_{r}-k_{r-1}\right)\rightarrow\infty$ as $r\rightarrow\infty.$
Throught this paper the intervals determined by $\theta$ will be denoted by
$I_{r}=\left(k_{r-1},k_{r}\right]$ and the ratio $\frac{k_{r}}{k_{r-1}}$ will
be abbreviated by $q_{r}.$ Recently lacunary sequence spaces have been studied
in ([12],[14],[4])

The order of statistical convergence of a sequence of numbers was given by
Gadjiev and Orhan [15] and later on statistical convergence of order $\alpha$
and strong $p-$Cesàro summability of order $\alpha$ was studied by Çolak [5].
Lacunary statistical convergence of order $\alpha$ in real number sequences
was defined by Şengül and Et [23].

The difference spaces $\ell_{\infty}\left(\Delta\right)$,
$c\left(\Delta\right)$ and $c_{0}\left(\Delta\right)$, consisting of all real
valued sequences $x=\left(x_{k}\right)$ such that $\Delta
x=\Delta^{1}x=\left(x_{k}-x_{k+1}\right)$ in the sequence spaces
$\ell_{\infty}$, $c$ and $c_{0}$, were defined by Kızmaz [16]. The idea of
difference sequences was generalized by Et and Çolak [10], Altinok [2], Çolak
et al. [6], Tripathy and Baruah [25] and many others.

A fuzzy number is a mapping $X:\mathbb{R}\longrightarrow[0,1]$ which satisfies
the following properties:

$i)$ $X$ is normal, i.e., there exists an $t_{0}\in\mathbb{R}$ such that
$X(t_{0})=1;$

$ii)$ $X$ is fuzzy convex, i.e., for all $\lambda\in[0,1]$ and all
$s,t\in\mathbb{R}$ and $0\leq\lambda\leq 1,X(\lambda
s+(1-\lambda)t)\geq\min[X(s),X(t)];$

$iii)$ $X$ is upper semicontinuous on real numbers set;

$iv)$ The set $\overline{\\{t\in\mathbb{R}:X(t)>0\\}},$ denoted by $[X]^{0},$
is compact, where $\overline{\\{t\in\mathbb{R}:X(t)>0\\}}$ is closure of
$\\{t\in\mathbb{R}:X(t)>0\\}$ with usual topology of $\mathbb{R}$.

$L(\mathbb{R})$ denotes the set of all fuzzy numbers on real numbers set
$\mathbb{R}$ and is said to be a fuzzy number space.

For $\alpha\in\left(0,1\right]$, the $\alpha$-level set $[X]^{\alpha}$ of
fuzzy number $X$ is defined by.

$[X]^{\alpha}=\left\\{\begin{array}[]{cc}\\{t\in\mathbb{R}:X(t)\geq\alpha\\},&\text{for
}\alpha\in\left(0,1\right]\\\
\overline{\\{t\in\mathbb{R}:X(t)>\alpha\\}},&\text{for
}\alpha=0\end{array}\right.$

The aritmetic operations for $\alpha$-level sets are defined as follows:

For $X,Y\in L(\mathbb{R}),$ now and what follows we will denote $\alpha-$level
sets by $\left[X\right]^{\alpha}=\left[u_{1}^{\alpha},v_{1}^{\alpha}\right],$
$\left[Y\right]^{\alpha}=\left[u_{2}^{\alpha},v_{2}^{\alpha}\right],$
$\alpha\in\left[0,1\right].$ Then we have

$\displaystyle\left[X+Y\right]^{\alpha}$
$\displaystyle=\left[u_{1}^{\alpha}+u_{2}^{\alpha},v_{1}^{\alpha}+v_{2}^{\alpha}\right]$
$\displaystyle\left[X-Y\right]^{\alpha}$
$\displaystyle=\left[u_{1}^{\alpha}-v_{2}^{\alpha},v_{1}^{\alpha}-u_{2}^{\alpha}\right]$
$\displaystyle\left[X.Y\right]^{\alpha}$
$\displaystyle=\left[\min_{i,j\in\left[1,2\right]}u_{i}^{\alpha}v_{j}^{\alpha},\max_{i,j\in\left[1,2\right]}u_{i}^{\alpha}v_{j}^{\alpha}\right].$

A sequence $X=(X_{k})$ of fuzzy numbers is a function $X$ from the set
$\mathbb{N}$ of all natural numbers into $L(\mathbb{R})$ [18].

Let $\beta\in\left(0,1\right]$ and $X=\left(X_{k}\right)$ be a sequence of
fuzzy numbers. Then the sequence $X=\left(X_{k}\right)$ of fuzzy numbers is
said to be statistically convergent of order $\beta,$ to fuzzy number $X_{0}$
if for every $\varepsilon>0,$

$\lim_{n\rightarrow\infty}\frac{1}{n^{\beta}}\left|\left\\{k\leq
n:d\left(X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|=0,$

where the vertical bars indicate the number of elements in the enclosed set.
In this case we write $S^{\beta}\left(F\right)-\lim X_{k}=X_{0}.$ We denote
the set of all statistically convergent sequences of order $\beta$ by
$S^{\beta}\left(F\right)$ [3].

Let $w\left(F\right)$ be the set of all sequences of fuzzy numbers. The
operator $\Delta^{m}:w\left(F\right)\rightarrow w\left(F\right)$ is defined by

$\displaystyle\left(\Delta^{0}X\right)_{k}$ $\displaystyle=$ $\displaystyle
X_{k},\left(\Delta^{1}X\right)_{k}=\Delta^{1}X_{k}=X_{k}-X_{k+1},$
$\displaystyle\left(\Delta^{m}X\right)_{k}$ $\displaystyle=$
$\displaystyle\Delta^{1}\left(\Delta^{m-1}X\right)_{k},\left(m\geq
2\right),\text{ for all }k\in\mathbb{N}.$

2 Main Results

In this section, we define the spaces
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ and
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$ and examine some inclusion
relations between them and space $w_{p}^{\beta}\left(F,\Delta^{m}\right).$

Definition 2.1 Let $\theta=\left(k_{r}\right)$ be a lacunary sequence,
$X=\left(X_{k}\right)$ be a sequence of fuzzy numbers and
$\beta\in\left(0,1\right]$ be given. The sequence $X=\left(X_{k}\right)\in
w\left(F\right)$ is said to be
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-$statistically convergent (or
lacunary statistically convergent sequence of order $\beta)$ if there is a
fuzzy number $X_{0}$ such that

$\lim_{r\rightarrow\infty}\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|=0,$

where $I_{r}=\left(k_{r-1},k_{r}\right]$ and $h_{r}^{\beta}$ denote the
$\beta$th power $\left(h_{r}\right)^{\beta}$ of $h_{r},$ that is
$h^{\beta}=(h_{r}^{\beta})=(h_{1}^{\beta},h_{2}^{\beta},...,h_{r}^{\beta},...).$
In this case we write $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim
X_{k}=X_{0}$. The set of all
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-$statistically convergent
sequences will be denoted by $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$.
For $\theta=\left(2^{r}\right),$ we shall write
$S^{\beta}\left(F,\Delta^{m}\right)$ instead of
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ and we shall write
$S\left(F,\Delta^{m}\right)$ instead of
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ in the special case $\beta=1$
and $\theta=\left(2^{r}\right).$

Lemma 2.2. Let $\theta=\left(k_{r}\right)$ be a lacunary sequence and
$K\subset\mathbb{N}.$ Then $S_{\theta}^{\beta}\left(K\right)\leq
S_{\theta}^{\alpha}\left(K\right)$ for $0<\alpha\leq\beta\leq 1$.

Proof. Let $0<\alpha\leq\beta\leq 1$. We can write

$\dfrac{1}{h_{r}^{\beta}}\left|\left\\{k_{r-1}<k\leq k_{r}:k\in
K\right\\}\right|\leq\dfrac{1}{h_{r}^{\alpha}}\left|\left\\{k_{r-1}<k\leq
k_{r}:k\in K\right\\}\right|$

since $h_{r}^{\alpha}\leq h_{r}^{\beta}$. Hence
$S_{\theta}^{\beta}\left(K\right)\leq$ $S_{\theta}^{\alpha}\left(K\right)$.

Let $\beta\in\left(0,1\right].$ If a sequence of fuzzy numbers is lacunary
statistically convergent of order $\beta$, then it is lacunary statistically
convergent. Really, we obtain

$\dfrac{1}{h_{r}^{\alpha}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|\geq\dfrac{1}{h_{r}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|$

since $h_{r}^{\alpha}\leq h_{r}$ for $\beta\in\left(0,1\right].$ Taking limit
as $r\rightarrow\infty$ we get $X_{k}\rightarrow
X_{0}\left(S_{\theta}^{\alpha},\Delta^{m}\right)$ and so $X_{k}\rightarrow
X_{0}\left(S_{\theta},\Delta^{m}\right).$

Note that the lacunary statistical convergence of order $\beta$ is well
defined for $\beta\in\left(0,1\right]$, but not well defined for $\beta>1$. To
show this, consider the sequence $X=\left(X_{k}\right)$ of fuzzy numbers
following:

$X_{k}\left(t\right)=\left\\{\begin{array}[]{cc}\left.\begin{array}[]{cc}t,&\text{for
}0\leq t\leq 1\\\ -t+2,&\text{for }1\leq t\leq 2\\\
0,&\text{otherwise}\end{array}\right\\}:=X^{\prime},&\begin{array}[]{c}\text{if
}k=2r\\\ (r=1,2,3,...)\end{array}\\\ \left.\begin{array}[]{cc}t-3,&\text{for
}3\leq t\leq 4\\\ -t+5,&\text{for }4\leq t\leq 5\\\
0,&\text{otherwise}\end{array}\right\\}:=X^{\prime\prime},&\text{if }k\neq
2r\text{ }\end{array}\right.$

We can find the $\alpha-$level sets of sequences $\left(X_{k}\right)$ and
$\left(\Delta^{m}X_{k}\right)$ after some arithmetic operations as follows

$\left[X_{k}\right]^{\alpha}=\left\\{\begin{array}[]{cc}\left[\alpha,2-\alpha\right],&\text{if
}k=2r\\\ \left[3+\alpha,5-\alpha\right],&\text{if }k\neq 2r\text{
}\end{array}\right..$

and

$\left[\Delta^{m}X_{k}\right]^{\alpha}=\left\\{\begin{array}[]{cc}\left[2^{m}\alpha-2^{m-1}.5,-2^{m}\alpha-2^{m-1}\right]:=X^{\prime},&\text{if
}k=2r\\\
\left[2^{m}\alpha+2^{m-1},-2^{m}\alpha+2^{m-1}.5\right]:=X^{\prime\prime},&\text{if
}k\neq 2r\text{ }\end{array}\right.,\left(m=1,2,...\right)$

then we can write

$\lim_{r\rightarrow\infty}\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X^{\prime}\right)\geq\varepsilon\right\\}\right|\leq\lim_{r\rightarrow\infty}\frac{k_{r}-k_{r-1}}{2h_{r}^{\beta}}=\lim_{r\rightarrow\infty}\frac{h_{r}}{2h_{r}^{\beta}}=0$

and

$\lim_{r\rightarrow\infty}\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X^{\prime\prime}\right)\geq\varepsilon\right\\}\right|\leq\lim_{r\rightarrow\infty}\frac{k_{r}-k_{r-1}}{2h_{r}^{\beta}}=\lim_{r\rightarrow\infty}\frac{h_{r}}{2h_{r}^{\beta}}=0$

for $\beta>1.$ Hence sequence $\left(X_{k}\right)$ is $\Delta^{m}-$lacunary
statistically convergent of order $\beta,$ both to $X^{\prime}$ and
$X^{\prime\prime},$ i.e., $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim
X_{k}$ $=X^{\prime}$ and $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim
X_{k}$ $=X^{\prime\prime}.$ But this is impossible.

Theorem 2.3 Let $0<\beta\leq 1$ and $X=\left(X_{k}\right),$
$Y=\left(Y_{k}\right)$ be sequences of fuzzy numbers, then

$\left(i\right)$ If $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim
X_{k}=X_{0}$ and $c\in\mathbb{C},$ then
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim\left(cx_{k}\right)=cx_{0},$

$\left(ii\right)$ If $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim
X_{k}=X_{0}$ and $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim
Y_{k}=Y_{0},$ then
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-\lim\left(X_{k}+Y_{k}\right)=X_{0}+Y_{0}.$

Proof. $(i)$ It is clear for the case $c=0.$ Suppose that $c\neq 0,$ then the
proof of $\left(i\right)$ follows from

$\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(c\Delta^{m}X_{k},cX_{0}\right)\geq\varepsilon\right\\}\right|=\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\frac{\varepsilon}{\left|c\right|}\right\\}\right|$

and that of $\left(ii\right)$ follows from

$\displaystyle\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}\left(X_{k}+Y_{k}\right),\left(X_{0}+Y_{0}\right)\right)\geq\varepsilon\right\\}\right|$
$\displaystyle\leq$ $\displaystyle\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\frac{\varepsilon}{2}\right\\}\right|$
$\displaystyle+\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}Y_{k},Y_{0}\right)\geq\frac{\varepsilon}{2}\right\\}\right|.$

Definition 2.4 Let $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers,
$\theta=\left(k_{r}\right)$ be a lacunary sequence, $\beta\in\left(0,1\right]$
be any real number and let $p$ be a positive real number. A sequence $X$ of
fuzzy numbers is said to be strongly
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)-$summable (or strongly
$N_{\theta}\left(p,F,\Delta^{m}\right)-$summable of order $\beta)$ if there is
a fuzzy number $X_{0}$ such that

$\underset{r\rightarrow\infty}{\lim}\frac{1}{h_{r}^{\beta}}\mathop{\textstyle\sum}\limits_{k\in
I_{r}}d\left(\Delta^{m}X_{k},X_{0}\right)^{p}=0\text{.}$

In this case we write $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)-\lim
X_{k}=X_{0}$. The strong
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)-$summability reduces to the
strong $N_{\theta}\left(p,F,\Delta^{m}\right)-$summability for $\beta=1$. The
set of all strongly $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)-$summable
sequences will be denoted by $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$.

If we take $\theta=\left(2^{r}\right)$ in space
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right),$ then we obtain strongly
$p-$Cesaro summable sequences set of order $\beta$ following

$w_{p}^{\beta}\left(F,\Delta^{m}\right)=\left\\{X=\left(X_{k}\right):\exists
X_{0},\underset{n}{\lim}\dfrac{1}{n^{\beta}}\sum_{k=1}^{n}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}=0\right\\}$

This set will be denoted by $w_{p,0}^{\beta}$ in case of $X_{0}=\bar{0}$ [2].

Theorem 2.5 If $0<\beta<\gamma\leq 1$ then
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)\subset
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$ and the inclusion is strict.

Proof. Proof can be obtained from the inequality

$\frac{1}{h_{r}^{\gamma}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|\leq\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|.$

Taking $\theta=\left(2^{r}\right)$ we show the strictness of the inclusion
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)\subset
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$ for a special case. For this,
consider the sequence $X=\left(X_{k}\right)$ of fuzzy numbers defined by

$X_{k}\left(t\right)=\left\\{\begin{array}[]{cc}\left.\begin{array}[]{cc}2t-\left(2k-1\right),&\text{for
}k-\frac{1}{2}\leq t\leq k\\\ -2t+\left(2k+1\right),&\text{for }k\leq t\leq
k+\frac{1}{2}\\\ 0,&\text{otherwise}\end{array}\right\\},&\text{if }k=n^{3}\\\
\left.\begin{array}[]{cc}2t-1,&\text{for }\frac{1}{2}\leq t\leq 1\\\
-2t+3,&\text{for }1\leq t\leq\frac{3}{2}\\\
0,&\text{otherwise}\end{array}\right\\},&\text{if }k\neq
n^{3}\end{array}\right.$

After calculating $\alpha-$level sets of sequences $\left(X_{k}\right)$ and
$\left(\Delta^{m}X_{k}\right),$ we obtain
$S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)-\lim
X_{k}=\left[2^{m-1}\left(\alpha-1\right),2^{m-1}\left(1-\alpha\right)\right],$
i.e. $X\in S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$ for
$\frac{1}{3}<\gamma\leq 1,$ but $X\notin
S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ for $0<\beta\leq\frac{1}{3}.$
This implies that the inclusion
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)\subset
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$ is strict for
$\beta,\gamma\in\left(0,1\right]$ such that
$\beta\in\left(0,\frac{1}{3}\right]$ and
$\gamma\in\left(\frac{1}{3},1\right].$

Corollary 2.6 If a sequence of fuzzy numbers is
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)-$statistically convergent to
fuzzy number $X_{0}$, then it is
$S_{\theta}\left(F,\Delta^{m}\right)-$statistically convergent to fuzzy number
$X_{0}.$

Theorem 2.7 Let $\beta$ and $\gamma$ be fixed real numbers such that
$0<\beta\leq\gamma\leq 1,$ $X=\left(X_{k}\right)$ be a sequence of fuzzy
numbers and $\theta=\left(k_{r}\right)$ be a lacunary sequence. For
$0<p<\infty,$ $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)\subset
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$ and the inclusion is strict for
some $\beta$’s and $\gamma$’s.

Proof. For any sequence $X=\left(X_{k}\right)$ of fuzzy numbers and
$\varepsilon>0$, we have

$\displaystyle\mathop{\textstyle\sum}\limits_{k\in
I_{r}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}$ $\displaystyle=$
$\displaystyle\mathop{\textstyle\sum}\limits_{\begin{subarray}{c}k\in I_{r}\\\
d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\end{subarray}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}+\mathop{\textstyle\sum}\limits_{\begin{subarray}{c}k\in
I_{r}\\\
d\left(\Delta^{m}X_{k},X_{0}\right)<\varepsilon\end{subarray}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}$
$\displaystyle\geq$
$\displaystyle\mathop{\textstyle\sum}\limits_{\begin{subarray}{c}k\in I_{r}\\\
d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\end{subarray}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}$
$\displaystyle\geq$ $\displaystyle\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|\varepsilon^{p}$

and so that

$\displaystyle\frac{1}{h_{r}^{\beta}}\mathop{\textstyle\sum}\limits_{k\in
I_{r}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}$
$\displaystyle\geq$ $\displaystyle\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|\varepsilon^{p}$
$\displaystyle\geq$ $\displaystyle\frac{1}{h_{r}^{\gamma}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|\varepsilon^{p}.$

It follows that if $X=\left(X_{k}\right)$ is strongly
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)-$summable to fuzzy number
$X_{0}$, then it is
$S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)-$statistically convergent to
$X_{0}$.

Taking $\beta=\gamma,$ $\theta=\left(2^{r}\right)$ and $p=1$ we show the
strictness of the inclusion
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)\subset
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$ for a special case. For this we
can select a sequence of fuzzy numbers as follows:

$X_{k}\left(t\right)=\left\\{\begin{array}[]{cc}\left.\begin{array}[]{cc}\frac{t}{k}+1,&\text{for
}-k\leq t\leq 0\\\ -\frac{t}{k}+1,&\text{for }0\leq t\leq k\\\
0,&\text{otherwise}\end{array}\right\\},&\begin{array}[]{c}\text{if
}k=n^{2}\\\ \left(n=1,2,...\right)\end{array}\\\
\left.\begin{array}[]{cc}\frac{t}{2}-1,&\text{for }2\leq t\leq 4\\\
-\frac{t}{2}+3,&\text{for }4\leq t\leq 6\\\
0,&\text{otherwise}\end{array}\right\\}:=X_{0}&\text{if }k\neq
n^{2}\end{array}\right.$

After calculating $\alpha-$level sets of sequences $\left(X_{k}\right)$ and
$\left(\Delta^{m}X_{k}\right),$ we see that $\left(X_{k}\right)$ is
$\Delta^{m}-$statistically convergent of order $\gamma$ to $X_{0}$ for
$\frac{1}{2}<\beta\leq 1,$ where
$\left[X_{0}\right]^{\alpha}=\left[2^{m+1}\left(\alpha-1\right),2^{m}\left(1-\alpha\right)\right],$
but it is not strongly
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$-summable to $X_{0}.$ Therefore
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)\subset
S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ for $\frac{1}{2}<\beta\leq 1.$

Corollary 2.8 Let $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers,
$\beta\in\left(0,1\right]$ and $p$ is a positive real number. Then

$i)$ If $X\in N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right),$ then $X\in
S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$ and the limits are same.

$ii)$ If $X\in N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right),$ then $X\in
S_{\theta}\left(F,\Delta^{m}\right)$ and the limits are same.

Theorem 2.9 Let $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers,
$\beta\in\left(0,1\right]$ and $\theta=\left(k_{r}\right)$ be a lacunary
sequence. If $\lim\inf_{r}q_{r}>1,$ then
$S^{\beta}\left(F,\Delta^{m}\right)\subset
S_{\theta}^{\beta}\left(F,\Delta^{m}\right).$

Proof. Suppose that $\lim\inf_{r}q_{r}>1,$ then there exists a number
$\delta>0$ such that $q_{r}\geq 1+\delta$ for sufficiently large $r\ $and so

$\frac{h_{r}}{k_{r}}\geq\frac{\delta}{1+\delta}\Longrightarrow\left(\frac{h_{r}}{k_{r}}\right)^{\beta}\geq\left(\frac{\delta}{1+\delta}\right)^{\beta}\Longrightarrow\frac{1}{k_{r}^{\beta}}\geq\frac{\delta^{\beta}}{\left(1+\delta\right)^{\beta}}\frac{1}{h_{r}^{\beta}}.$

If $X_{k}\rightarrow X_{0}\left(S^{\beta}\left(F,\Delta^{m}\right)\right),$
then we prove the suffıciency from following inequalities for every
$\varepsilon>0$ and for sufficiently large $r$

$\displaystyle\frac{1}{k_{r}^{\beta}}\left|\left\\{k\leq
k_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|$
$\displaystyle\geq$ $\displaystyle\frac{1}{k_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|$
$\displaystyle\geq$
$\displaystyle\frac{\delta^{\beta}}{\left(1+\delta\right)^{\beta}}\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|;$

Theorem 2.10 If $\lim_{r\rightarrow\infty}\inf\frac{h_{r}^{\beta}}{k_{r}}>0,$
then $S\left(F,\Delta^{m}\right)\subset
S_{\theta}^{\beta}\left(F,\Delta^{m}\right).$

Proof. For a given $\varepsilon>0,$ we can write

$\left\\{k\leq
k_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\supset\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}.$

From here we obtain

$\displaystyle\frac{1}{k_{r}}\left|\left\\{k\leq
k_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|$
$\displaystyle\geq$ $\displaystyle\frac{1}{k_{r}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|$
$\displaystyle=$
$\displaystyle\frac{h_{r}^{\beta}}{k_{r}}\frac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|.$

Taking limit according to $r$ and using
$\lim_{r\rightarrow\infty}\inf\frac{h_{r}^{\beta}}{k_{r}}>0,$ we get

$X_{k}\rightarrow X_{0}\left(S,F,\Delta^{m}\right)\Longrightarrow
X_{k}\rightarrow X_{0}\left(S_{\theta}^{\beta},F,\Delta^{m}\right).$

Theorem 2.11 Let $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers,
$0<\beta\leq 1,$and $\theta=\left(k_{r}\right)$ be a lacunary sequence. If
$\lim\sup_{r}q_{r}<\infty,$ then
$S_{\theta}^{\beta}\left(F,\Delta^{m}\right)\subset
S\left(F,\Delta^{m}\right).$

Proof. Omitted.

Theorem 2.12 Let $0<\beta<\gamma\leq 1,$ $X=\left(X_{k}\right)$ be a sequence
of fuzzy numbers and $p>0$, then there exists inclusion
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)\subset
N_{\theta}^{\gamma}\left(p,F,\Delta^{m}\right)$ and this is strict.

Proof. Let $X=\left(X_{k}\right)\in
N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$. Then given $\beta$ and
$\gamma$ such that $0<\beta<\gamma\leq 1$ and a $p>0,$ we can write inequality

$\frac{1}{h_{r}^{\gamma}}\mathop{\textstyle\sum}\limits_{k\in
I_{r}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}\leq\frac{1}{h_{r}^{\beta}}\mathop{\textstyle\sum}\limits_{k\in
I_{r}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p}$

and this gives that $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)\subset
N_{\theta}^{\gamma}\left(p,F,\Delta^{m}\right).$

To show that the inclusion is strict take lacunary sequence
$\theta=\left(2^{r}\right)$ and $p=1.$ Consider the sequence
$X=\left(X_{k}\right)$ of fuzzy numbers as follows

$X_{k}\left(t\right)=\left\\{\begin{array}[]{cc}\left.\begin{array}[]{cc}t-2,&\text{for
}2\leq t\leq 3\\\ -t+4,&\text{for }3\leq t\leq 4\\\
0,&\text{otherwise}\end{array}\right\\},&\begin{array}[]{c}\text{if
}k=n^{3}\\\ \left(n=1,2,...\right)\end{array}\\\
\left.\begin{array}[]{cc}\frac{t}{3}-\frac{5}{3},&\text{for }5\leq t\leq 8\\\
-\frac{t}{3}+\frac{11}{3},&\text{for }8\leq t\leq 11\\\
0,&\text{otherwise}\end{array}\right\\}:=X_{0}&\text{if }k\neq
n^{3}\end{array}\right.$

After calculating $\alpha-$level sets of sequences $\left(X_{k}\right)$ and
$\left(\Delta^{m}X_{k}\right),$ we obtain $X\in
N_{\theta}^{\gamma}\left(p,F,\Delta^{m}\right)$ since $\left(X_{k}\right)$ is
strongly $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$-summable to $X_{0}$
for $\frac{1}{3}<\gamma\leq 1,$ where
$\left[X_{0}\right]^{\alpha}=\left[2^{m}\left(3\alpha-3\right),2^{m}\left(3-3\alpha\right)\right],$
but $X\notin N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$ for
$0<\beta\leq\frac{1}{3},$ i.e. it is not strongly
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$-summable to fuzzy number
$X_{0}.$ So, this shows that inclusion is strict. We show the limits of
sequence $\Delta^{m}X_{k}$ in Figure 1.

Corollary 2.13 Let $0<\beta\leq\gamma\leq 1,$ $X=\left(X_{k}\right)$ be a
sequence of fuzzy numbers and $p>0$. Then

$(i)$ If $\beta=\gamma,$ then
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)=N_{\theta}^{\gamma}\left(p,F,\Delta^{m}\right),$

$(ii)$ For each $\beta\in(0,1]$ and $0<p<\infty,$
$N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)\subseteq
N_{\theta}\left(p,F,\Delta^{m}\right).$

Theorem 2.14. Let $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers and
$\theta=\left(k_{r}\right)$ be a lacunary sequence. If $\lim\inf_{r}q_{r}>1,$
then $w_{p}^{\beta}\left(F,\Delta^{m}\right)\subseteq
N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$ for $0<\beta\leq 1$ and
$0<p<\infty.$

Proof. If $\lim\inf_{r}q_{r}>1,$ then there exists a number $\delta>0$ such
that $q_{r}\geq 1+\delta$ for all $r\geq 1.$ For $0<\beta\leq 1$ and $X\in
w_{p,0}^{\beta}$ we can write

$\displaystyle\tau_{r}^{\beta}$ $\displaystyle=$
$\displaystyle\dfrac{1}{h_{r}^{\beta}}\sum_{i=1}^{k_{r}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}-\dfrac{1}{h_{r}^{\beta}}\sum_{i=1}^{k_{r-1}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}$
$\displaystyle=$
$\displaystyle\dfrac{k_{r}^{\beta}}{h_{r}^{\beta}}\left(\dfrac{1}{k_{r}^{\beta}}\sum_{i=1}^{k_{r}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}\right)-\dfrac{k_{r-1}^{\beta}}{h_{r}^{\beta}}\left(\dfrac{1}{k_{r-1}^{\beta}}\sum_{i=1}^{k_{r-1}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}\right)$

and so

$\dfrac{k_{r}^{\beta}}{h_{r}^{\beta}}\leq\dfrac{\left(1+\delta\right)^{\beta}}{\delta^{\beta}}\text{
and }\dfrac{k_{r-1}^{\beta}}{h_{r}^{\beta}}\leq\dfrac{1}{\delta^{\beta}}$

for $h_{r}=k_{r}-k_{r-1}$. It can be seen that both of
terms$\dfrac{1}{k_{r}^{\beta}}\sum_{i=1}^{k_{r}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}$
and
$\dfrac{1}{k_{r-1}^{\beta}}\sum_{i=1}^{k_{r-1}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}$
converge to $0$ and so $\tau_{r}^{\beta}$ does. Hence $X\in
N_{\theta,0}^{\beta}\left(p,F,\Delta^{m}\right)$, that is
$w_{p}^{\beta}\left(F,\Delta^{m}\right)\subseteq
N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$.

Theorem 2.15. If $\lim\sup_{r}\dfrac{k_{r}}{k_{r-1}^{\beta}}<\infty$, then
$N_{\theta}\left(p,F,\Delta^{m}\right)\subseteq
w_{p}^{\beta}\left(F,\Delta^{m}\right)$ for $0<\beta\leq 1.$

Proof. If $\lim\sup_{r}\dfrac{k_{r}}{k_{r-1}^{\beta}}<\infty,$ then there
exists a number $M>0$ such that $\dfrac{k_{r}}{k_{r-1}^{\beta}}<M$ for all
$r\geq 1.$ Let $X\in N_{\theta,0}\left(p,F,\Delta^{m}\right)$ and
$\varepsilon>0.$ Then we can find numbers $R>0$ and $K>0$ such that
$\sup_{i\geq R}\tau_{i}<\varepsilon$ and $\tau_{i}<K$ for $i=1,2,3,...$ .
Therefore we can write

$\displaystyle\dfrac{1}{t^{\beta}}\sum_{i=1}^{t}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}\begin{array}[]{c}\leq\end{array}\dfrac{1}{k_{r-1}^{\beta}}\sum_{i=1}^{k_{r-1}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}$
$\displaystyle\qquad\qquad\begin{array}[]{c}=\end{array}\dfrac{1}{k_{r-1}^{\beta}}\left(\sum_{I_{1}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}+\sum_{I_{2}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}+...+\sum_{I_{r}}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}\right)$
$\displaystyle\qquad\qquad\begin{array}[]{c}=\end{array}\dfrac{k_{1}}{k_{r-1}^{\beta}}\tau_{1}+\dfrac{k_{2}-k_{1}}{k_{r-1}^{\beta}}\tau_{2}+...+\dfrac{k_{R}-k_{R-1}}{k_{r-1}^{\beta}}\tau_{R}+\dfrac{k_{R+1}-k_{R}}{k_{r-1}^{\beta}}\tau_{R+1}+...+\dfrac{kr-
k_{r-1}}{k_{r-1}^{\beta}}\tau_{r}$
$\displaystyle\qquad\qquad\begin{array}[]{c}\leq\end{array}\left(\sup_{i\geq
1}\tau_{i}\right)\dfrac{k_{R}}{k_{r-1}^{\beta}}+\left(\sup_{i\geq
R}\tau_{i}\right)\dfrac{k_{r}-k_{R}}{k_{r-1}^{\beta}}\begin{array}[]{c}<\end{array}K\dfrac{k_{R}}{k_{r-1}^{\beta}}+\epsilon
M$

where $k_{r-1}<t\leq k_{r}$ and $r>R.$ We get
$\dfrac{1}{t^{\beta}}\sum_{i=1}^{t}d\left(\Delta^{m}X_{i},\bar{0}\right)^{p}\rightarrow
0$ since $t\rightarrow\infty$ for $k_{r-1}\rightarrow\infty,$ hence $X\in
w_{p,0}^{\beta}\left(F,\Delta^{m}\right).$

Theorem 2.16. Let $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers,
$\theta=\left(k_{r}\right)$ be a lacunary sequence and
$\beta\in\left(0,1\right]$. If $X\in w^{\beta}\left(F,\Delta^{m}\right)\cap
N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$ and
$\lim\sup_{r}\dfrac{k_{r}}{k_{r-1}^{\beta}}<\infty,$ then the limits of
$\left(X_{k}\right)$ are same.

Proof. Suppose that $w_{p}^{\beta}\left(F,\Delta^{m}\right)-\lim
X_{k}=X^{\prime},$ $N_{\theta,p}^{\beta}\left(p,F,\Delta^{m}\right)-\lim
X_{k}=X^{\prime\prime}$ and $X^{\prime}\neq X^{\prime\prime}$. We get
$N_{\theta,0}\left(p,F,\Delta^{m}\right)\subseteq
w_{p,0}^{\beta}\left(F,\Delta^{m}\right)$ since
$\lim\sup_{r}\dfrac{k_{r}}{k_{r-1}^{\beta}}<\infty$ from Theorem 2.15. Because
of $d\left(\Delta^{m}X_{i},X^{\prime\prime}\right)\in
N_{\theta,0}\left(p,F,\Delta^{m}\right)$, so
$d\left(\Delta^{m}X_{i},X^{\prime\prime}\right)\in
w_{p,0}^{\beta}\left(F,\Delta^{m}\right)$. Therefore for $p=1$

$\dfrac{1}{t^{\beta}}\sum_{i=1}^{t}d\left(\Delta^{m}X_{i},X^{\prime\prime}\right)\rightarrow
0$

In the following inequality, both of terms at left converges to $0,$ but this
is a contradiction and so $X^{\prime}=X^{\prime\prime}.$

$\dfrac{1}{t^{\beta}}\sum_{i=1}^{t}d\left(\Delta^{m}X_{i},X^{\prime\prime}\right)+\dfrac{1}{t^{\beta}}\sum_{i=1}^{t}d\left(\Delta^{m}X_{i},X^{\prime}\right)\geq\dfrac{1}{t^{\beta}}d\left(X^{\prime},X^{\prime\prime}\right)>0$

3\. Results Related to Modulus Function

We give relation between sets $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$
and $w_{p}^{\beta}\left[\theta,f,F,\Delta^{m}\right]$ according to modulus
function $f$.

We first quote the definition of a modulus function:

The concept of modulus function was formally introduced by Nakano [19]. A
mapping $f:[0,\infty)$ $\rightarrow$ $[0,\infty)$ is said to be a modulus if

$i)$ $f\left(x\right)=0$ iff $x=0$,

$ii)$ $f\left(x+y\right)\leq f\left(x\right)+f\left(y\right)$ for $x,y\geq 0,$

$iii)$ $f$ is increasing,

$iv)$ $f$ is right-continuous at $0.$

The continuity of $f$ everywhere on $[0,\infty)$ follows from above
definition. A modulus function can be bounded or unbounded. For example
$f\left(x\right)=x^{p},\left(0<p\leq 1\right)$ is bounded and
$f\left(x\right)=\frac{x}{1+x}$ is bounded. The concept of modulus function
for sequences of fuzzy numbers was first investigated by Sarma [20]. Later on
it was explotied by Talo and Başar [24].

Definition 3.1. Let $f$ be a modulus function, $X=\left(X_{k}\right)$ be a
sequence of fuzzy numbers and $p=\left(p_{k}\right)$ be a sequence of positive
real numbers and $\beta\in\left(0,1\right]$. We define space
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$ as follows

$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)=\left\\{x=\left(x_{k}\right):\lim_{r\rightarrow\infty}\dfrac{1}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[d\left(\Delta^{m}X_{k},X_{0}\right)\right]^{p_{k}}=0,\text{ for
some }X_{0}\right\\}$

We shall write $N_{\theta}^{\beta}\left(p,F,\Delta^{m}\right)$ instead of
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$ in the special cases
$p_{k}=1$ and $f\left(x\right)=x$ for all $k\in\mathbb{N}$.

We consider that $p=\left(p_{k}\right)$ is bounded and $0<h=\inf_{k}p_{k}\leq
p_{k}\leq\sup_{k}p_{k}=H<\infty$ in the following theorems.

Theorem 3.2. Let $\beta,\gamma\in\left(0,1\right],$ $\beta\leq\gamma,$ $f$ be
a modulus function, $X=\left(X_{k}\right)$ be a sequence of fuzzy numbers and
$\theta=\left(k_{r}\right)$ be a lacunary sequence. Then,
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)\subset
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right).$

Proof. Take $x\in w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$,
$\varepsilon>0$ and let $\sum\limits_{1}$ and $\sum\limits_{2}$ be summations
on $k\in I_{r},$ $d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon$ and
$k\in I_{r},$ $d\left(\Delta^{m}X_{k},X_{0}\right)<\varepsilon,$ respectively.
Since $h_{r}^{\beta}<h_{r}^{\gamma}$ for each $r,$ we get

$\displaystyle\dfrac{1}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}$
$\displaystyle=$
$\displaystyle\dfrac{1}{h_{r}^{\beta}}\left[\sum_{1}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}+\sum_{2}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}\right]$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{h_{r}^{\gamma}}\left[\sum_{1}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}+\sum_{2}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}\right]$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{h_{r}^{\gamma}}\sum_{1}\left[f\left(\varepsilon\right)\right]^{p_{k}}$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{h_{r}^{\gamma}}\sum_{1}\min\left(\left[f\left(\varepsilon\right)\right]^{h},\left[f\left(\varepsilon\right)\right]^{H}\right)$
$\displaystyle\geq$ $\displaystyle\dfrac{1}{h_{r}^{\gamma}}\left|\left\\{k\in
I_{r}:d\left(\Delta^{m}X_{k},X_{0}\right)\geq\varepsilon\right\\}\right|\min\left(\left[f\left(\varepsilon\right)\right]^{h},\left[f\left(\varepsilon\right)\right]^{H}\right)$

Left side of above inequality converges to $0$ as $r\rightarrow\infty$ because
of $X\in w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$, so right one
converges to zero and hence $X\in
S_{\theta}^{\gamma}\left(F,\Delta^{m}\right)$.

Theorem 3.3. Let $f$ be a bounded modulus function, $X=\left(X_{k}\right)$ be
a sequence of fuzzy numbers and $\theta=\left(k_{r}\right)$ be a lacunary
sequence. If $\lim\limits_{r\rightarrow\infty}\dfrac{h_{r}}{h_{r}^{\beta}}=1,$
then $S_{\theta}^{\beta}\left(F,\Delta^{m}\right)\subset
w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$.

Proof. Suppose that $f$ is bounded and $X\in
S_{\theta}^{\beta}\left(F,\Delta^{m}\right)$. Given $\varepsilon>0.$ There is
an integer $K$ such that $\left(x\right)\leq K$ since $f$ is bounded. Now we
can write

$\displaystyle\dfrac{1}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}$
$\displaystyle=$
$\displaystyle\dfrac{1}{h_{r}^{\beta}}\sum_{1}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}+\dfrac{1}{h_{r}^{\beta}}\sum_{2}\left[f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\right]^{p_{k}}$
$\displaystyle\leq$
$\displaystyle\dfrac{1}{h_{r}^{\beta}}\sum_{1}\max\left(K^{h},K^{H}\right)+\dfrac{1}{h_{r}^{\beta}}\sum_{2}\left[f\left(\varepsilon\right)\right]^{p_{k}}$
$\displaystyle\leq$
$\displaystyle\max\left(K^{h},K^{H}\right)\dfrac{1}{h_{r}^{\beta}}\left|\left\\{k\in
I_{r}:f\left(d\left(\Delta^{m}X_{k},X_{0}\right)\right)\geq\varepsilon\right\\}\right|$
$\displaystyle+\dfrac{h_{r}}{h_{r}^{\beta}}\max\left(f\left(\varepsilon\right)^{h},\text{
}f\left(\varepsilon\right)^{H}\right).$

Hence we get $X\in w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)$ for
$r\in\mathbb{N}$ from above inequalities.

Theorem 3.4. If $\lim p_{k}>0$ and sequence $X=\left(X_{k}\right)$ is strongly
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)-$summable to $X_{0},$ then
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)-\lim X_{k}=X_{0}$ is unique.

Proof. Let $\lim p_{k}=s>0,$
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)-\lim X_{k}=X^{\prime}$ and
$w_{p}^{\beta}\left(\theta,f,F,\Delta^{m}\right)-\lim X_{k}=X^{\prime\prime}$.
Then we obtain

$\lim_{r\rightarrow\infty}\dfrac{1}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(\Delta^{m}X_{k},X^{\prime}\right)\right)\right]^{p_{k}}=0$

and

$\lim_{r\rightarrow\infty}\dfrac{1}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(\Delta^{m}X_{k},X^{\prime\prime}\right)\right)\right]^{p_{k}}=0$

On the other hand, since $f$ is increasing and $f\left(x+y\right)\leq
f\left(x\right)+f\left(y\right)$ we can write

$\displaystyle\dfrac{1}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(X^{\prime},X^{\prime\prime}\right)\right)\right]^{p_{k}}$
$\displaystyle\leq$ $\displaystyle\dfrac{D}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left(\left[f\left(d\left(\Delta^{m}X_{k},X^{\prime}\right)\right)\right]^{p_{k}}+\left[f\left(d\left(\Delta^{m}X_{k},X^{\prime\prime}\right)\right)\right]^{p_{k}}\right)$
$\displaystyle\leq$ $\displaystyle\dfrac{D}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(\Delta^{m}X_{k},X^{\prime}\right)\right)\right]^{p_{k}}+\dfrac{D}{h_{r}^{\beta}}\sum_{k\in
I_{r}}\left[f\left(d\left(\Delta^{m}X_{k},X^{\prime\prime}\right)\right)\right]^{p_{k}}\rightarrow
0,$

where $p_{k}=H$ and $D=\left(1,2^{H-1}\right)$. Left right of above
inequalities tends to zero as $r\rightarrow\infty$ and because of $\lim
p_{k}=s$ we obtain

$\lim_{r\rightarrow\infty}\left[f\left(d\left(X^{\prime},X^{\prime\prime}\right)\right)\right]^{p_{k}}=\left[f\left(d\left(X^{\prime},X^{\prime\prime}\right)\right)\right]^{s}$

Hence $X^{\prime}=X^{\prime\prime}$ and the limit is unique.

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The $k$-almost Ricci solitons and contact geometry
Amalendu Ghosh 1 and Dhriti Sundar Patra 2

Abstract: The aim of this article is to study the $k$-almost Ricci soliton and
$k$-almost gradient Ricci soliton on contact metric manifold. First, we prove
that if a compact $K$-contact metric is a $k$-almost gradient Ricci soliton
then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result
on a compact $k$-almost Ricci soliton when the flow vector field $X$ is
contact. Finally, we study some special types of $k$-almost Ricci solitons
where the potential vector field $X$ is point wise collinear with the Reeb
vector field $\xi$ of the contact metric structure.

Mathematics Subject Classification 2010: 53C25, 53C20, 53C15

Keywords: Contact metric manifold, $k$-almost Ricci soliton, $k$-almost
gradient Ricci soliton, $K$-contact manifold, Sasakian manifold, Einstein
manifold.

##  1 Introduction

A Riemannian manifold $(M^{n},g)$ is said to be a Ricci soliton if there
exists a vector field $X$ on $M^{n}$ and a constant $\lambda$ satisfying the
equation $S+\frac{1}{2}\pounds_{X}g=\lambda g$, where $\pounds_{X}g$ denotes
the Lie-derivative of $g$ along the vector field $X$ on $M^{n}$ and $S$ is the
Ricci tensor of $g$. In general, $X$ and $\lambda$ are known as the potential
vector field and the soliton constant, respectively. Ricci solitons are the
fixed points of Hamilton’s Ricci flow: $\frac{\partial}{\partial
t}g(t)=-2S(g(t))$ (where $g(t)$ a one-parameter family of metrics on $M^{n}$)
viewed as a dynamical system on the space of Riemannian metrics modulo
diffeomorphisms and scalings (cf. [13]). Recently, the notion of Ricci soliton
was generalized by Pigoli-Rigoli-Rimoldi-Setti [18] to almost Ricci soliton by
allowing the soliton constant $\lambda$ to be a smooth function.

Recently, Wang-Gomes-Xia [20] extended the notion of almost Ricci soliton to
$k$-almost Ricci soliton which is defined as

######  Definition 1.1

A complete Riemannian manifold $(M^{n},g)$ is said to be a $k$-almost Ricci
soliton, denoted by $(M^{n},g,X,k,\lambda)$, if there exists a smooth vector
field $X$ on $M^{n}$, a soliton function $\lambda\in C^{\infty}(M^{n})$ and a
positive real valued function $k$ on $M^{n}$ such that

$\displaystyle S+\frac{k}{2}\pounds_{X}g=\lambda g.$ (1.1)

This notion has been justified as follows. Suppose $(M^{n},g_{0})$ be a
complete Riemannian manifold of dimension $n$ and let $g(t)$ be a solution of
the Ricci flow equation defined on $[0,\epsilon)$, $\epsilon>0$, such that
$\psi_{t}$ is a one-parameter family of diffeomorphisms of $M^{n}$, with
$\psi_{0}=id_{M}$ and $g(t)(x)=\rho(x,t)\psi_{t}^{*}g_{0}(x)$ for every $x\in
M^{n}$, where $\rho(x,t)$ is a positive smooth function on
$M^{n}\times[0,\epsilon)$. Then one can deduce

$\displaystyle\frac{\partial}{\partial t}g(t)(x)=\frac{\partial}{\partial
t}\rho(x,t)\psi_{t}^{*}g_{0}(x)+\rho(x,t)\psi_{t}^{*}\pounds_{\frac{\partial}{\partial
t}\psi(x,t)}g_{0}(x).$

When $t=0$, the foregoing equation reduces to

$\displaystyle S_{g_{0}}+\frac{k}{2}\pounds_{X}g_{0}=\lambda g_{0},$

where $X=\frac{\partial}{\partial t}\psi(x,0)$,
$\lambda(x)=-\frac{1}{2}\frac{\partial}{\partial t}\rho(x,0)$ and
$k(x)=\rho(x,0)$.

A $k$-almost Ricci soliton is said to be $shrinking$, $steady$ or $expanding$
accordingly as $\lambda$ is positive, zero or negative, respectively. It is
trivial (Einstein) if the flow vector field $X$ is homothetic, i.e.
$\pounds_{X}g=cg$, for some constant $c$. Otherwise, it is non-trivial. A
$k$-almost Ricci soliton is said to be a $k$-almost gradient Ricci soliton if
the potential vector field $X$ can be expressed as a gradient of a smooth
function $u$ on $M^{n}$, i.e. $X=Du$, where $D$ is the gradient operator of
$g$ on $M^{n}$. In this case, we denotes $(M^{n},g,Du,k,\lambda)$ as a
$k$-almost gradient Ricci soliton with potential function $u$. Further, the
fundamental equation (1.1) takes the form

$\displaystyle S+k\nabla^{2}u=\lambda g,$ (1.2)

where $\nabla^{2}u$ denotes the Hessian of $u$.

In particular, a Ricci soliton is the $1$-almost Ricci soliton with constant
$\lambda$, and an almost Ricci soliton is just the $1$-almost Ricci soliton.
Barros and Ribeiro Jr. proved (cf. [2]) that a compact almost Ricci soliton
with constant scalar curvature is isometric to a Euclidean sphere. An
analogous result has also been proved by Wang-Gomes-Xia [20] for the case of
$k$-almost Ricci soliton.

Theorem [WGX]: Let $(M^{n},g,X,k,\lambda)$, $n\geq 3$ be a non-trivial
$k$-almost Ricci soliton with constant scalar curvature $r$. If $M^{n}$ is
compact, then it is isometric to a standard sphere $S^{n}(c)$ of radius
$c=\sqrt{\frac{2n(2n+1)}{r}}$.

Recall that a smooth manifold $M^{n}$ together with a Riemannian metric $g$ is
said to be a generalized quasi-Einstein manifold if there exist smooth
functions $f$, $\mu$ and $\lambda$ such that (cf. [8])

$\displaystyle S+\nabla^{2}f-\mu df\otimes df=\lambda g.$

For $\mu=\frac{1}{m}$, the generalized quasi-Einstein manifold is known as
generalized $m$-quasi-Einstein manifold (cf. [1, 3]), and when $\lambda$ is
constant the generalized quasi-Einstein manifold is simply known as $m$-quasi-
Einstein manifold. Case-Shu-Wei [7] proved that any complete quasi-Einstein-
metric with constant scalar curvature is trivial (Einstein). Subsequently,
this has been extended by Barros-Gomes [1]. In fact, they proved that any
compact generalized $m$-quasi-Einstein metric with constant scalar curvature
is isometric to a standard Euclidean sphere $S^{n}$. Particularly, Barros-
Ribeiro [3] construct a family of nontrivial generalized $m$-quasi-Einstein
metric on the unit sphere $S^{n}(1)$ that are rigid in the class of constant
scalar curvature. It is interesting to note that by a suitable choice of the
function $f$ it is possible to reduce any generalized $m$-quasi-Einstein
metric to a $k$-almost Ricci soliton. For instance, if we take
$u=e^{\frac{f}{m}}$ and $k=-\frac{m}{u}$, then (1.2) reduces to

$\displaystyle S+\nabla^{2}f-\frac{1}{m}df\otimes df=\lambda g.$

Thus, in one hand $k$-almost Ricci soliton generalizes generalized $m$-quasi-
Einstein metric, on the other it covers gradient Ricci soliton and gradient
almost Ricci soliton. For details we refer to [20]. Recently, Yun-Co-Hwang
[23] studied Bach-flat $k$-almost gradient Ricci solitons.

During the last few years Ricci soliton and almost Ricci soliton have been
studied by several authors (cf. [19], [11], [9], [10] and [12]) within the
frame-work of contact geometry. In [19], Sharma initiated the study of
gradient Ricci soliton within the frame-work of $K$-contact manifold and prove
that “any complete $K$-contact metric admitting a gradient Ricci soliton is
Einstein and Sasakian”. Later on, this has been generalized by the second
author [12] who proved that “if a complete $K$-contact metric (in particular,
Sasakian) represents a gradient almost Ricci soliton, then is it isometric to
the unit sphere $S^{2n+1}$”. Inspired by these results, here we consider
contact metric manifolds whose metric is a $k$-almost Ricci soliton. Following
[3], one can construct a family of non-trivial examples of generalized
$m$-quasi-Einstein metrics on the odd dimensional unit sphere $S^{2n+1}$.
Another motivation arises from the fact that any odd dimensional unit sphere
satisfies the generalized $m$-quasi-Einstein condition and hence it satisfies
the gradient $k$-almost Ricci soliton equation (1.2). Since any odd
dimensional unit sphere $S^{2n+1}$ admits standard $K$-contact (Sasakian)
structure we are interested in studying $K$-contact metric as a gradient
$k$-almost Ricci soliton. We address this issue in Section 3 and prove that if
a compact $K$-contact manifold admits a $k$-almost gradient Ricci soliton then
it is isometric to a unit sphere $S^{2n+1}$. Next, we study $k$-almost Ricci
soliton in the frame-work of compact $K$-contact manifold when the potential
vector field is contact. Finally, a couple of results on contact metric
manifolds admitting $k$-almost Ricci soliton are presented under the
assumption that the potential vector field $X$ is point wise collinear with
the Reeb vector field $\xi$ of the contact metric structure.

##  2 Preliminaries

First, we recall some basic definitions and formulas on a contact metric
manifold. By a contact manifold we mean a Riemannian manifold $M^{2n+1}$ of
dimension $(2n+1)$ which carries a global $1$-form $\eta$ such that
$\eta\wedge(d\eta)^{n}\neq 0$ everywhere on $M^{2n+1}$. The form $\eta$ is
usually known as the contact form on $M^{2n+1}$. It is well known that a
contact manifold admits an almost contact metric structure on
$(\varphi,\xi,\eta,g)$, where $\varphi$ is a tensor field of type $(1,1)$,
$\xi$ a global vector field known as the characteristic vector field (or the
Reeb vector field) and $g$ is Riemannian metric, such that

$\displaystyle\varphi^{2}Y=-Y+\eta(Y)\xi,$ (2.1)
$\displaystyle\eta(Y)=g(Y,\xi),$ (2.2) $\displaystyle g(\varphi Y,\varphi
Z)=g(Y,Z)-\eta(Y)\eta(Z),$ (2.3)

for all vector fields $Y$, $Z$ on $M$. It follows from the above equations
that $\varphi\xi=0$ and $\eta\circ\varphi=0$ (see [5], p.43). A Riemannian
manifold $M^{2n+1}$ together with the almost contact metric structure
$(\varphi,\xi,\eta,g)$ is said to be a contact metric if it satisfies ( [5],
p.47)

$\displaystyle d\eta(Y,Z)=g(Y,\varphi Z),$ (2.4)

for all vector fields $Y$, $Z$ on $M$. In this case, we say that $g$ is an
associated metric of the contact metric structure. On a contact metric
manifold $M^{2n+1}(\varphi,\xi,\eta,g),$ we consider two self-adjoint
operators $h=\frac{1}{2}\pounds_{\xi}\varphi$ and $l=R(.,\xi)\xi$, where
$\pounds_{\xi}$ is the Lie-derivative along $\xi$ and $R$ is the Riemann
curvature tensor of $g$. The two operators $h$ and $l$ satisfy (e.g., see [5],
p.84, p.85)

$Tr~{}h=0,\hskip 5.69046ptTr~{}(h\varphi)=0,\hskip 5.69046pth\xi=0,\hskip
5.69046ptl\xi=0,\hskip 5.69046pth\varphi=-\varphi h.$

We now recall the following

###### Lemma 2.1

([5], p.84; p.112; p.111) On a contact metric manifold
$M^{2n+1}(\varphi,\xi,\eta,g)$ we have

$\displaystyle\nabla_{Y}\xi=-\varphi Y-\varphi hY,$ (2.5) $\displaystyle
Ric_{g}(\xi,\xi)=g(Q\xi,\xi)=Tr~{}l=2n-Tr~{}(h^{2}),$ (2.6)
$\displaystyle(\nabla_{Z}\varphi)Y+(\nabla_{\varphi Z}\varphi)\varphi
Y=2g(Y,Z)\xi-\eta(Y)(Z+hZ+\eta(Z)\xi),$ (2.7)

for all vector fields $Y$, $Z$ on $M$; where $\nabla$ is the operator of
covariant differentiation of $g$ and $Q$ the Ricci operator associated with
the $(0,2)$ Ricci tensor given by $S(Y,Z)=g(QY,Z)$ for all vector fields $Y$,
$Z$ on $M$.

A contact metric manifold is said to be $K$-contact if the vector field $\xi$
is Killing, equivalently if $h=0$ ([5], p.87). Hence on a $K$-contact manifold
Eq. (2.5) becomes

$\displaystyle\nabla_{Y}\xi=-\varphi Y,$ (2.8)

for any vector field $Y$ on $M$. Moreover, on a $K$-contact manifold the
following formulas are also valid.

###### Lemma 2.2

(see Blair [5], p.113; p.116) On a $K$-contact manifold
$M^{2n+1}(\varphi,\xi,\eta,g)$ we have

$\displaystyle Q\xi=2n\xi,$ (2.9) $\displaystyle
R(\xi,Y)Z=(\nabla_{Y}\varphi)Z,$ (2.10)

for all vector fields $Y$, $Z$ on $M$.

An almost contact metric structure on $M$ is said to be normal if the almost
complex structure $J$ on $M\times\mathbb{R}$ defined by (e.g., see Blair [5],
p.80)

$\displaystyle J(X,fd/dt)=(\varphi X-f\xi,\eta(X)d/dt),$

where $f$ is a real function on $M\times R$, is integrable. A normal contact
metric manifold is said to be Sasakian. On a Sasakian manifold (e.g., [5],
p.86)

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

for all vector fields $X$, $Y$ on $M$. Further, a contact metric manifold is
Sasakian if and only if the curvature tensor $R$ satisfies (e.g., [5], p.114)

$\displaystyle R(X,Y)\xi=\eta(Y)X-\eta(X)Y,$ (2.11)

for all vector fields $X$, $Y$ on $M$. A Sasakian manifold is $K$-contact but
the converse is true only in dimension $3$ (e.g., [5], p.87).

A contact metric manifold is said to be $\eta$-Einstein if the Ricci tensor
$S$ satisfies $S(Y,Z)=ag(Y,Z)+b\eta(Y)\eta(Z)$ for any vector fields $Y$, $Z$
on $M$ and are arbitrary functions $a$, $b$ on $M$. The functions $a$ and $b$
are constant for a $K$-contact manifold of dimension $>3$ (cf. [22]).

Let $T^{1}M$ be the unit tangent bundle of a compact orientable Riemannian
manifold $(M,g)$ equipped with the Sasaki metric $g_{s}$. Any unit vector
field $U$ determines a smooth map between $(M,g)$ and $(T^{1}M,g_{s})$. The
energy $E(U)$ of the unit vector field $U$ is defined by

$\displaystyle E(U)=\frac{1}{2}\int\parallel
dU\parallel^{2}dM=\frac{n}{2}vol(M,g)+\frac{1}{2}\int_{M}\parallel\nabla
U\parallel^{2}dM,$

where $dU$ denotes the differential of the map $U$ and $dM$ denotes the volume
element of $M$. $U$ is said to be a harmonic vector field if it is a critical
point of the energy functional $E$ defined on the space $\chi^{1}$ of all unit
vector fields on $(M,g)$. A contact metric manifold is said to be an
$H$-contact manifold if the Reeb vector field $\xi$ is harmonic. In [17],
Perrone proved that “A contact metric manifold is an $H$-contact manifold,
that is $\xi$ is a harmonic vector field, if and only if $\xi$ is an
eigenvector of the Ricci operator.” On a contact metric manifold, $\xi$ is an
eigenvector of the Ricci operator implies that $Q\xi=(Tr~{}l)\xi$. This is
true for many contact metric manifolds. Such as, $\eta$-Einstein contact
metric manifolds, $K$-contact (in particular Sasakian) manifolds,
$(k,\mu)$-contact manifolds and the tangent sphere bundle of a Riemannian
manifold of constant curvature. In particular, this condition holds on the
unit sphere $S^{2n+1}$ with standard contact metric structure.

######  Definition 2.1

A vector field $X$ on a contact manifold is said to be a contact vector field
if it preserve the contact form $\eta$, i.e.

$\displaystyle\pounds_{X}\eta=f\eta,$ (2.12)

for some smooth function $f$ on $M$. When $f=0$ on $M$, the vector field $X$
is called a strict contact vector field.

Example 2.2:  It is well know [5] that any odd dimensional unit sphere
$S^{2n+1}$ admits a standard $K$-contact (Sasakian) structure
$(\varphi,\xi,\eta,g)$ and hence the Reeb vector field satisfies (2.8), for
any vector field $Y$ on $S^{2n+1}$. We now recall the theorem of Obata [15]
that a complete connected Riemannian manifold $(M,g)$ of dimension $>2$ is
isometric to a sphere of radius $\frac{1}{c}$ if and only if it admits a non-
trivial solution $k$ of the equation $\nabla\nabla k=-c^{2}kg$. For unit
sphere this transforms to $\nabla\nabla k=-kg$, where $k$ is the eigenfunction
of the Laplacian on $S^{2n+1}$. Let $X$ be a vector field on $S^{2n+1}$ such
that $X=-Dk+\mu\xi$, where $\mu$ is a constant. Differentiating this along an
arbitrary vector field $Y$ on $S^{2n+1}$ and using (2.8) we obtain
$\nabla_{Y}X=-\nabla_{Y}Dk-\mu\varphi Y$. Then by Obata’s theorem and (2.8) we
see that

$\displaystyle\frac{k}{2}(\pounds_{X}g)(Y,Z)+S(Y,Z)=(k^{2}+2n)g(Y,Z),$ (2.13)

for all vector fields $Y$, $Z$ on $S^{2n+1}$. This shows that
$(S^{2n+1},g,X,\lambda)$ is a almost Ricci soliton with $\lambda=k^{2}+2n$.
Moreover, if we take $X=Du$, for some smooth non constant function $u$ on
$S^{2n+1}$, then from (2.13) it follows that $S^{2n+1}$ also admits $k$-almost
gradient Ricci soliton.

##  3  $K$-contact metric as $k$-almost gradient Ricci soliton and $k$-almost
Ricci soliton

We assume that a $K$-contact metric $g$ is a $k$-almost gradient Ricci soliton
with the potential function $u$. Then the $k$-almost gradient Ricci soliton
Eq. (1.2) can be exhibited as

$\displaystyle k\nabla_{Y}Du+QY=\lambda Y,$ (3.1)

for any vector field $Y$ on $M$; where $D$ is the gradient operator of $g$ on
$M$. Taking the covariant derivative of (3.1) along an arbitrary vector field
$Z$ on $M$ yields

$\displaystyle k\nabla_{Z}\nabla_{Y}Du=$
$\displaystyle\frac{1}{k}(Zk)(QY-\lambda Y)-(\nabla_{Z}Q)Y-Q(\nabla_{Z}Y)$
$\displaystyle+(Z\lambda)Y+\lambda\nabla_{Z}Y,$

for any vector field $Y$ on $M$. Using this and (3.1) in the well known
expression of the curvature tensor
$R(Y,Z)=[\nabla_{Y},\nabla_{Z}]-\nabla_{[Y,Z]},$ we can easily find out the
curvature tensor which is given by

$\displaystyle kR(Y,Z)Du=\frac{1}{k}(Yk)(QZ-\lambda
Z)-\frac{1}{k}(Zk)(QY-\lambda Y)$
$\displaystyle+(\nabla_{Z}Q)Y-(\nabla_{Y}Q)Z+(Y\lambda)Z-(Z\lambda)Y,$ (3.2)

for all vector fields $Y$, $Z$ on $M$.

Before entering into our main results we prove the following.

###### Lemma 3.1

On a $K$-contact manifold $M^{2n+1}(\varphi,\xi,\eta,g)$, we have

$\displaystyle\nabla_{\xi}Q=Q\varphi-\varphi Q.$ (3.3)

Proof: Since $\xi$ is Killing on a $K$-contact manifold, we have
$(\pounds_{\xi}Q)Y=0$ for any vector field $Y$ on $M$. Taking into account
(2.8) it follows that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\pounds_{\xi}(QY)-Q(\pounds_{\xi}Y)$ $\displaystyle=$
$\displaystyle\nabla_{\xi}QY-\nabla_{QY}\xi-Q(\nabla_{\xi}Y)+Q(\nabla_{Y}\xi)$
$\displaystyle=$ $\displaystyle(\nabla_{\xi}Q)Y+\varphi QY-Q\varphi Y,$

for any vector field $Y$ on $M$. This completes the proof. $\square$

###### Theorem 3.1

Let $(M^{2n+1},g,Du,k,\lambda)$ be a $k$-almost gradient Ricci soliton with
the potential function $u$. If $(M,g)$ is a compact $K$-contact manifold, then
it is isometric to a unit sphere $S^{2n+1}$.

Proof: Firstly, taking covariant differentiation of (2.9) along an arbitrary
vector field $Y$ on $M$ and using (2.8), we get

$(\nabla_{Y}Q)\xi=Q\varphi Y-2n\varphi Y.$ (3.4)

Now, replacing $\xi$ instead of $Y$ in (3) and making use of the $K$-contact
condition (2.9), (3.4) and (3.3), we get

$\displaystyle kR(\xi,Z)Du=(\frac{\lambda-2n}{k})(Zk)\xi+\frac{1}{k}(\xi
k)(QZ-\lambda Z)-2n\varphi Z$ $\displaystyle+\varphi
QZ+(\xi\lambda)Z-(Z\lambda)\xi,$

for any vector field $Z$ on $M$. Scalar product of the last equation with an
arbitrary vector field $Y$ on $M$ and using (2.10), we obtain

$\displaystyle
kg((\nabla_{Z}\varphi)Y,Du)+(\frac{\lambda-2n}{k})(Zk)\eta(Y)+(\xi\lambda-\frac{\lambda}{k}(\xi
k))g(Y,Z)$ $\displaystyle+\frac{1}{k}(\xi k)g(QY,Z)+2ng(\varphi
Y,Z)-g(Q\varphi Y,Z)-(Z\lambda)\eta(Y)=0,$ (3.5)

for any vector field $Z$ on $M$. Next, substituting $Y$ by $\varphi Y$ and $Z$
by $\varphi Z$ in (3) and using (2.1), $\eta o\varphi=0$ and $\varphi\xi=0$
provides

$\displaystyle kg((\nabla_{\varphi Z}\varphi)\varphi
Y,Du)+(\xi\lambda-\frac{\lambda}{k}(\xi k))\\{g(Y,Z)-\eta(Y)\eta(Z)\\}$
$\displaystyle+\frac{1}{k}(\xi k)g(Q\varphi Y,\varphi Z)+2ng(\varphi
Y,Z)-g(\varphi QY,Z)=0,$

for all vector fields $Y$, $Z$ on $M$. Adding the preceding Eq. with (3) and
using (2.7) (where $h=0$, as $M$ is $K$-contact) yields

$\displaystyle 2\\{k(\xi u)+(\xi\lambda)-\frac{\lambda}{k}(\xi
k)\\}g(Y,Z)+\\{\frac{\lambda}{k}(\xi k)-(\xi\lambda)-k(\xi
u)\\}\eta(Y)\eta(Z)$
$\displaystyle+\\{(\frac{\lambda-2n}{k})(Zk)-k(Zu)-(Z\lambda)\\}\eta(Y)+\frac{1}{k}(\xi
k)g(QY,Z)$ $\displaystyle+4ng(\varphi Y,Z)-g(Q\varphi Y+\varphi
QY,Z)+\frac{1}{k}(\xi k)g(\varphi QY,\varphi Z)=0,$

for all vector fields $Y$, $Z$ on $M$. Anti-symmetrizing the foregoing
equation provides

$\displaystyle\\{(\frac{\lambda-2n}{k})(Zk)-k(Zu)-(Z\lambda)\\}\eta(Y)-2g(Q\varphi
Y+\varphi QY,Z)$
$\displaystyle-\\{(\frac{\lambda-2n}{k})(Yk)-k(Yu)-(Y\lambda)\\}\eta(Z)+8ng(\varphi
Y,Z)=0,$

for all vector fields $Y$, $Z$ on $M$. Moreover, substituting $Y$ by $\varphi
Y$ and $Z$ by $\varphi Z$ in the last equation and using the $K$-contact
condition (2.9), (2.1), $\eta o\varphi=0$ and $\varphi\xi=0$ gives

$\displaystyle g(Q\varphi Y+\varphi QY,Z)=4ng(\varphi Y,Z),$

for all vector fields $Y$, $Z$ on $M$. It follows from last Eq. that

$\displaystyle Q\varphi Y+\varphi QY=4n\varphi Y,$ (3.6)

for any vector field $Y$ on $M$. Let $\\{e_{i},\varphi
e_{i},\xi\\},i=1,2,3,.....,n$, be an orthonormal $\varphi$-basis of $M$ such
that $Qe_{i}=\sigma_{i}e_{i}$. Thus, we have $\varphi Qe_{i}=\sigma_{i}\varphi
e_{i}.$ Substituting $e_{i}$ for $Y$ in (3.6) and using the foregoing
equation, we obtain $Q\varphi e_{i}=(4n-\sigma_{i})\varphi e_{i}$. Using the
$\varphi$-basis and (2.9), the scalar curvature $r$ is given by

$\displaystyle r$ $\displaystyle=$ $\displaystyle
g(Q\xi,\xi)+\sum_{i=1}^{n}[g(Qe_{i},e_{i})+g(Q\varphi e_{i},\varphi e_{i})]$
$\displaystyle=$ $\displaystyle
g(Q\xi,\xi)+\sum_{i=1}^{n}[\sigma_{i}g(e_{i},e_{i})+(4n-\sigma_{i})g(\varphi
e_{i},\varphi e_{i})]$ $\displaystyle=$ $\displaystyle 2n(2n+1).$

Therefore, the scalar curvature $r$ is constant. As $M$ is compact, Theorem
[WGX] shows that $M$ is isometric to $S^{2n+1}(c)$, where
$c=\sqrt[]{\frac{2n(2n+1)}{r}}$ is the radius of the sphere. Since
$r=2n(2n+1)$, we have $c=1$. Hence, $M$ is isometric to a unit sphere
$S^{2n+1}$. This completes the proof. $\square$

###### Remark 3.1

From the last theorem we see that any compact $K$-contact manifold $M$
admitting a gradient $k$-almost Ricci soliton is isometric to a unit sphere
and hence of constant curvature $1$. Consequently, $M$ is Sasakian. Since
$k$-almost Ricci soliton covers Einstein manifold, we may compare this as an
extension of the odd dimensional Goldberg conjecture which states that any
compact Einstein $K$-contact manifold is Sasakian. For details, we refer to
Boyer-Galicki [6].

In particular, the above result is also true for complete Sasakian manifolds.

######  Corollary 3.1

Let $(M^{2n+1},g,Du,k,\lambda)$ be a $k$-almost gradient Ricci soliton with
the potential function $u$. If $(M,g)$ is a complete Sasakian manifold, then
it is compact and isometric to a unit sphere $S^{2n+1}$.

Proof: On a Sasakian manifold the Ricci operator $Q$ and $\varphi$ commutes,
i.e. $Q\varphi=\varphi Q$ (see [5], p.116). Using this in (3.6) implies
$Q\varphi Y=2n\varphi Y$ for any vector field $Y$ on $M$. Substituting $Y$ by
$\varphi Y$ in the last equation and using (2.9) gives $QY=2nY$ for any vector
field $Y$ on $M$. This shows that $M$ is Einstein with Einstein constant $2n$.
As $(M,g)$ is complete, $M$ is compact by Myers’ Theorem [14]. The rest of the
proof follows from the last theorem. $\square$

Next, we extend the Theorem $3.1$ and consider $K$-contact metric as a
$k$-almost Ricci soliton when its potential vector field is a contact vector
field and prove

###### Theorem 3.2

Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a compact $K$-contact manifold with $X$
as a contact vector field. If $g$ is a $k$-almost Ricci soliton with $X$ as
the potential vector field, then $M$ is isometric to a unit sphere $S^{2n+1}$.

Proof:  Firstly, taking Lie-derivative of (2.4) along $X$ and using (1.1) we
have

$\displaystyle k(\pounds_{X}d\eta)(Y,Z)=2g(-QY+\lambda Y,\varphi
Z)+kg(Y,(\pounds_{X}\varphi)Z),$ (3.7)

for all vector fields $Y$, $Z$ on $M$. As $X$ is a contact vector field, we
deduce from (2.12) that

$\displaystyle\pounds_{X}d\eta=d\pounds_{X}\eta=(df)\wedge\eta+f(d\eta).$
(3.8)

Now, making use of (3.8) in (3.7), we obtain

$\displaystyle 2k(\pounds_{X}\varphi)Z=4Q\varphi Z+2(fk-2\lambda)\varphi
Z+k(\eta(Z)Df-(Zf)\xi),$ (3.9)

for any vector field $Z$ on $M$. Next, replacing $\xi$ instead of $Z$ in the
last equation and using $\varphi\xi=0$ we have

$\displaystyle 2(\pounds_{X}\varphi)\xi=Df-(\xi f)\xi,$ (3.10)

where we use $k$ is positive. Further, tracing (1.1) gives

$\displaystyle kdivX=(2n+1)\lambda-r.$ (3.11)

Let $\omega$ be the volume form of $M$, i.e.,
$\omega=\eta\wedge(d\eta)^{n}\neq 0$. Taking Lie-derivative of this along the
vector field $X$ and using the formula $\pounds_{X}\omega=(divX)\omega$ and
equation (3.8) yields $divX=(n+1)f$. By virtue of this, (3.11) provides

$\displaystyle r=(2n+1)\lambda-(n+1)kf.$ (3.12)

Also, Lie-differentiation of $g(\xi,\xi)=1$ along an arbitrary vector field
$X$ on $M$ and by the use of the equations (1.1), (2.9) yields

$\displaystyle kg(\pounds_{X}\xi,\xi)=2n-\lambda.$ (3.13)

Now, taking Lie-derivative of (2.2) on $X$ and using (1.1), (2.9) and (2.12)
we obtain $k\pounds_{X}\xi=(kf-2\lambda+4n)\xi.$ Making use of this in (3.13)
yields $kf=\lambda-2n.$ and therefore we have
$k\pounds_{X}\xi=(2n-\lambda)\xi$. Next, taking Lie-derivative of
$\varphi\xi=0$ along $X$ and using the foregoing equation we get
$(\pounds_{X}\varphi)\xi=0$. In view of this, the Eq. (3.10) becomes $Df=(\xi
f)\xi$, i.e. $df=(\xi f)\eta$. Exterior derivative of the preceding equation
gives $d^{2}f=d(\xi f)\wedge\eta+(\xi f)d\eta$. Using $d^{2}=0$ in the last
equation and then taking the wedge product with $\eta$ we get $(\xi
f)\eta\wedge d\eta=0$. By the definition of contact structure we know that
$\eta\wedge d\eta$ is non-vanishing everywhere on $M$. Hence the previous
equation provides $\xi f=0$. This implies that $df=0$, and therefore $f$ is
constant on $M$. Integrating both sides of $divX=(n+1)f$ over $M$ and applying
the divergence theorem we get $f=0$. Since $kf=\lambda-2n$, it follows that
$\lambda=2n$. Consequently, equation (3.12) gives $r=2n(2n+1)$. This shows
that the scalar curvature is constant. As $M$ is compact, we may invoke
Theorem [WGX] to conclude that $M$ is isometric to $S^{2n+1}(c)$, where
$c=\sqrt[]{\frac{2n(2n+1)}{r}}$ is the radius of the sphere. Since
$r=2n(2n+1)$, we have $c=1$, and hence $M$ is isometric to a unit sphere
$S^{2n+1}$. This completes the proof. $\square$

Waiving the compactness assumption and imposing a commutativity condition we
have

###### Theorem 3.3

Let $M^{2n+1}(\varphi,\xi,\eta,g)$, $n>1$, be a $K$-contact manifold with
$Q\varphi=\varphi Q$. If $g$ is a $k$-almost Ricci soliton such that $X$ is a
contact vector field, then it is trivial and the soliton vector field is
Killing.

Proof:  As $f$ is constant and $fk=\lambda-2n$, the equation (3.9) becomes

$\displaystyle k(\pounds_{X}\varphi)Z=2Q\varphi Z-(\lambda+2n)\varphi Z,$
(3.14)

for all vector field $Z$ on $M$. Also, from (2.12) we have
$k\pounds_{X}\eta=(\lambda-2n)\eta$. Now, taking Lie-derivative of
$\varphi^{2}Z=-Z+\eta(Z)\xi$ along $X$ and then multiplying $k$ on both sides
and using the forgoing Eq., we obtain
$k\varphi(\pounds_{X}\varphi)Z+k(\pounds_{X}\varphi)\varphi Z=0$ for any
vector field $Z$ on $M$. In view of (3.14), the last Eq. becomes

$\displaystyle\varphi Q\varphi Z+Q\varphi^{2}Z=(\lambda+2n)\varphi^{2}Z,$

for any vector field $Z$ on $M$. Since $Q\varphi=\varphi Q$, the last equation
reduces to

$\displaystyle QZ=(\frac{\lambda+2n}{2})Z+(\frac{2n-\lambda}{2})\eta(Z)\xi,$
(3.15)

for any vector field $Z$ on $M$. This shows that $(M,g)$ is $\eta$-Einstein.
Now, differentiating (3.15) along an arbitrary vector field $Y$ on $M$ and
using (2.8), we get

$\displaystyle(\nabla_{Y}Q)Z=(\frac{Y\lambda}{2})Z-(\frac{Y\lambda}{2})\eta(Z)\xi-(\frac{2n-\lambda}{2})\\{g(Z,\varphi
Y)\xi+\eta(Z)\varphi Y\\},$

for any vector field $Z$ on $M$. Tracing the foregoing equation over $Y$ and
$Z$, respectively, we have $Zr=Z\lambda-(\xi\lambda)\eta(Z)$ and
$Zr=nZ(\lambda)$ for any vector field $Z$ on $M$. Since $\xi$ is Killing, $\xi
r=0$. Hence, the last equation provides $\xi\lambda=0$. Consequently, we have
$Zr=Z\lambda$ and $Zr=nZ(\lambda)$ for any vector field $Z$ on $M$. As $n>1$,
these two equations imply that $\lambda$ and $r$ are constant. By virtue of
(3.15), Eq. (3.14) reduces to $(\pounds_{X}\varphi)Z=0$ for any vector field
$Z$ on $M$. At this point, we recall Lemma $1$ (cf. [10]) “if a vector field
$X$ leaves the structure tensor $\varphi$ of the contact metric manifold $M$
invariant, then there exists a constant $c$ such that
$\pounds_{X}g=c(g+\eta\otimes\eta)$ ” to conclude that

$\displaystyle(\pounds_{X}g)(Y,Z)=c\\{g(Y,Z)+\eta(Y)\eta(Z)\\},$

for all vector fields $Y$, $Z$ on $M$. On the other hand, making use of (3.15)
in the Eq. (1.1), we find

$\displaystyle\frac{k}{2}(\pounds_{X}g)(Y,Z)=\lambda g(Y,Z)-S(Y,Z),$

for all vector fields $Y$, $Z$ on $M$. Comparing the last two equations, we
deduce

$\displaystyle\frac{ck}{2}\\{g(Y,Z)+\eta(Y)\eta(Z)\\}=\lambda g(Y,Z)-S(Y,Z).$
(3.16)

Next, putting $Y=Z=\xi$ in (3.16) and using (2.9), we get $ck=2(\lambda-2n)$.
Further, tracing (3.16) yields $ck(n+1)=(2n+1)\lambda-r$. These two equations
together provides $r=4n(n+1)-\lambda$. Moreover, using $kf=\lambda-2n$ in
(3.12) we have $r=(2n+1)\lambda-(n+1)(\lambda-2n)$. Comparing the last two
equations we see that $\lambda=2n$. Utilizing this in (3.15) provides $QY=2nY$
for any vector fields $Y$ on $M$, i.e. the soliton is trivial. This completes
the proof. $\square$

For a Sasakian manifold the commutativity condition $Q\varphi=\varphi Q$ holds
trivially (e.g., see Blair [5]). Thus, we have the following

###### Corollary 3.1

Let $M^{2n+1}(\varphi,\xi,\eta,g)$, $n>1$, be a Sasakian manifold with $X$ is
a contact vector field. If $g$ is a $k$-almost Ricci soliton, then it is
trivial and the soliton vector field is Killing.

##  4 $k$-almost Ricci soliton where $X=\rho\xi$

In this section, we shall discuss about some special type of $k$-almost Ricci
soliton where the potential vector field $X$ is point wise collinear with the
Reeb vector field $\xi$ of the contact metric manifold.

###### Theorem 4.1

Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a compact $H$-contact manifold. If $g$
represents a non-trivial $k$-almost Ricci soliton with non-zero potential
vector field $X$ collinear with the Reeb vector field $\xi$, then $M$ is
Einstein and Sasakian.

Proof: Since the potential vector field $X$ on $M$ is collinear with the Reeb
vector field $\xi$, we have $X=\rho\xi$, where $\rho$ is a non-zero smooth
function on $M$ (as $X$ is non zero). Taking covariant derivative along an
arbitrary vector field $Y$ on $M$ and using (2.5) and (2.6) we get

$\nabla_{Y}X=(Y\rho)\xi-\rho(\varphi Y+\varphi hY).$ (4.1)

By virtue of this the soliton equation (1.1) becomes

$\displaystyle k(Y\rho)\eta(Z)+k(Z\rho)\eta(Y)-2k\rho g(\varphi
hY,Z)+2S(Y,Z)=2\lambda g(Y,Z),$ (4.2)

for all vector fields $Y$, $Z$ on $M$. Replacing $\xi$ instead of $Z$ in (4.2)
gives

$\displaystyle kD\rho+k(\xi\rho)\xi+2(Q\xi-\lambda\xi)=0.$ (4.3)

At this point, putting $Y=Z=\xi$ in (4.2) and making use of (2.6) yields

$\displaystyle k(\xi\rho)+Trl=\lambda.$ (4.4)

Since $M$ is $H$-contact, the Reeb vector field $\xi$ is an eigenvector of the
Ricci operator at each point of $M$, i.e. $Q\xi=(Trl)\xi$. Substituting this
in (4.2) and then using (4.4), we have $kD\rho=k(\xi\rho)\xi$. Since the
$k$-almost Ricci soliton is non trivial and $k$ is a positive function, we
have $D\rho=(\xi\rho)\xi$. Next, taking covariant derivative along an
arbitrary vector field $Y$ on $M$ and using (2.5) yields
$\nabla_{Y}D\rho=Y(\xi\rho)\xi-(\xi\rho)(\varphi Y+\varphi hY).$ In view of
$g(\nabla_{Y}D\rho,Z)=g(\nabla_{Z}D\rho,Y)$, the foregoing equation yields

$\displaystyle Y(\xi\rho)\eta(Z)-Z(\xi\rho)\eta(Y)+2(\xi\rho)d\eta(Y,Z)=0,$
(4.5)

for all vector fields $Y$, $Z$ on $M$. Choosing $X$, $Y$ orthogonal to $\xi$
and noting that $d\eta\neq 0$, the last equation provides $\xi\rho=0$. Hence
$\rho$ is constant. Thus, the equation (4.2) reduces to

$\displaystyle QZ+(k\rho)h\varphi Z=\lambda Z,$ (4.6)

for any vector field $Z$ on $M$. Taking the trace of (4.6) we obtain
$r=(2n+1)\lambda$. Further, covariant derivative of (4.6) along an arbitrary
vector field $Y$ on $M$ gives

$(\nabla_{Y}Q)Z+(k\rho)(\nabla_{Y}h\varphi)Z+\rho(Yk)h\varphi Z=(Y\lambda)Z.$

Contracting this over $Y$ yields

$\displaystyle\frac{1}{2}(Zr)+\rho((h\varphi
Z)k)+(k\rho)div(h\varphi)Z=(Z\lambda),$ (4.7)

for any vector field $Z$ on $M$. On a contact metric manifold it is known that
$div(h\varphi)Z=g(Q\xi,Z)-2n\eta(Z)$ for all vector field $Z$ on $M$ (see
[5]). Using $Q\xi=(Trl)\xi$ in the previous equation we have
$div(h\varphi)Z=(Trl-2n)\eta(Z)=|h|^{2}\eta(Z)$. Hence, equation (4.7) reduces
to

$\displaystyle\frac{1}{2}(Zr)+\rho((h\varphi
Z)k)+(k\rho)(Trl-2n)\eta(Z)=(Z\lambda),$ (4.8)

for any vector field $Z$ on $M$. Setting $Z=\xi$ and making use of
$r=(2n+1)\lambda$ and (2.6) equation (4.8) reduces to

$\displaystyle\frac{2n-1}{2}(\xi r)-(k\rho)|h|^{2}=0.$ (4.9)

Taking into account (4.1) and $X=\rho\xi$ we see that $div(rX)=\rho(\xi
r)+r(\xi\rho)=\rho(\xi r)$, where we have also used $tr(h\varphi)=0$. Using
this equation in (4.9) gives $\frac{2n-1}{2}div(rX)=(k\rho^{2})|h|^{2}$.
Integrating this over $M$ and using divergence theorem we obtain

$\int{k\rho^{2}\left|h\right|^{2}}dM=0.$

Since the soliton is non-trivial with non-zero potential vector field $X$ and
$k$ being positive, the foregoing equation implies $h=0$ and hence $M$ is
$K$-contact. Therefore, equation (4.6) shows that $QZ=\lambda Z$. Using (2.9)
it follows that $\lambda=2n$. Thus, $M$ is Einstein with Einstein constant
$2n$. So, we can apply the result of Boyer-Galicki [6] which states that “any
compact $K$-contact Einstein manifold is Sasakian” to conclude the proof.
$\square$

For a $K$-contact manifold it is known that $Q\xi=2n\xi$. Thus, from the above
theorem we have the following

###### Corollary 4.1

If a complete $K$-contact metric represents a non-trivial $k$-almost Ricci
soliton with non-zero potential vector field $X$ collinear with the Reeb
vector field $\xi$, then it is Einstein and Sasakian.

Next, replacing the “compact $H$-contact” of the previous theorem by the
commutativity condition $Q\varphi=\varphi Q$ we prove

###### Theorem 4.2

Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a contact metric manifold satisfying
$Q\varphi=\varphi Q$. If $g$ represents a non-trivial $k$-almost Ricci soliton
with nonzero potential vector field $X$ collinear with the Reeb vector field
$\xi$, then $M$ is Einstein and $K$-contact. In addition, if $M$ is complete,
then it is compact Sasakian.

Proof : The commutativity condition $Q\varphi=\varphi Q$ together with (2.6)
and $\varphi\xi=0$ shows that $Q\xi=(Trl)\xi$. Further, since the potential
vector field $X$ is collinear with the Reeb vector field $\xi$, from equations
(4.1) to (4.9) are also valid here. Now we replace $Z$ by $\varphi Z$ in (4.6)
and using $h\xi=0$ we get

$\displaystyle Q\varphi Z-(k\rho)hZ=\lambda\varphi Z.$ (4.10)

On the other hand, operating (4.6) by $\varphi$ and using $h\varphi=-\varphi
h$ we obtain

$\displaystyle\varphi QZ+(k\rho)hZ=\lambda\varphi Z.$ (4.11)

Adding (4.10) and (4.11) along with $Q\varphi=\varphi Q$ gives $Q\varphi
Z=\lambda\varphi Z$. Therefore, replacing $Z$ by $\varphi Z$ in the last
equation and using $Q\xi=(Trl)\xi$, we deduce $QZ=\lambda
Z+(Trl-\lambda)\eta(Z)\xi$. Since $\lambda=Trl$ (follows from (4.4), as
$\xi\rho=0$), the foregoing equation implies $QZ=\lambda Z$ and hence $M$ is
Einstein. Consequently, the scalar curvature $r$ and $\lambda$ are constant.
Thus, from (4.9), it follows that $(k\rho)|h|^{2}=0.$ Since $k$ is positive
and the soliton vector field $X$ is non-zero, we can conclude that $h=0$, and
hence $M$ is $K$-contact. From these, we see that $M$ is $K$-contact and
Einstein with Einstein constant $2n$. Now, if $M$ is complete then applying
Myers’ theorem $M$ becomes compact. Finally, using Boyer-Galicki’s theorem [6]
we conclude the proof. $\square$

Acknowledgments: The authors are very much thankful to the reviewer for
his/her comments. The author D. S. Patra is financially supported by the
Council of Scientific and Industrial Research, India (grant no.
17-06/2012(i)EU-V).

## References

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1 Department of Mathematics,
Chandernagore College
Hooghly: 712 136 (W.B.), INDIA
E-mail<EMAIL_ADDRESS>

2 Department of Mathematics,
Jadavpur University,
188\. Raja S. C. Mullick Road,
Kolkata:700 032, INDIA
E-mail<EMAIL_ADDRESS>


# A stochastic PDE approach to large N problems in quantum field theory: a
survey

Hao Shen  Department of Mathematics, University of Wisconsin - Madison, USA
<EMAIL_ADDRESS>
###### Abstract.

In this survey we review some recent rigorous results on large N problems in
quantum field theory, stochastic quantization and singular stochastic PDEs,
and their mean field limit problems. In particular we discuss the $O(N)$
linear sigma model on two and three dimensional torus. The stochastic
quantization procedure leads to a coupled system of $N$ interacting $\Phi^{4}$
equations. In $d=2$, we show uniform in $N$ bounds for the dynamics and
convergence to a mean-field singular SPDE. For large enough mass or small
enough coupling, the invariant measures (i.e. the $O(N)$ linear sigma model)
converge to the massive Gaussian free field, the unique invariant measure of
the mean-field dynamics, in a Wasserstein distance. We also obtain tightness
for certain $O(N)$ invariant observables as random fields in suitable Besov
spaces as $N\to\infty$, along with exact descriptions of the limiting
correlations. In $d=3$, the estimates become more involved since the equation
is more singular. We discuss in this case how to prove convergence to the
massive Gaussian free field. The proofs of these results build on the recent
progress of singular SPDE theory and combine many new techniques such as
uniform in $N$ estimates and dynamical mean field theory. These are based on
joint papers with Scott Smith, Rongchan Zhu and Xiangchan Zhu.

###### 2010 Mathematics Subject Classification:

60H15; 35R60

##  1\. Introduction and physics background

In quantum field theory (QFT), there are important models where the “field”
takes values in an $N$ dimensional space. A natural question is that what is
the asymptotic behavior of such models as dimensionality of the target space
$N\to\infty$. The perspective of investigating these questions in this article
is called stochastic quantization. Stochastic quantization is a procedure
which formulates a quantum field theory in terms of a stochastic PDE (SPDE).
More precisely, for a quantum field theory where the target space has
dimension $N$, its stochastic quantization yields a system of $N$ coupled
SPDEs. This procedure then brings in many powerful analytical tools, including
the recent developments of SPDE solution theories and their a priori
estimates, and mean field theory techniques. The purpose of this article is to
survey the recent results along this direction.

For most of the article, we will focus on the linear $\sigma$-model in $2$ and
$3$ dimensions. But we will first briefly review the origin of the large N
problems in physics.

###  1.1. Large N problem in QFT

Large $N$ methods (or “$1/N$ expansions”) in theoretical physics are
ubiquitous and are generally applied to models where dimensionality of the
target space is large. It was first noted by Stanley [stanley1968spherical]
for lattice spin models (rather than QFT), that they exhibit considerable
simplification as $N$ (the number of spin components) becomes large: at least
at the level of free energy, as $N\to\infty$, these models become the
“spherical model”, which is a solvable model introduced by Berlin and Kac in
1952 [berlin1952spherical]. See Baxter’s book [baxter2016exactly], Section 5,
for an expository discussion on spherical model. Later, the large $N$
convergence to the spherical model was established at the level of free
energies [kac1971spherical] and then of correlations [MR471851, MR982418].
Again in the context of spin systems, [brezin1973critical] found a systematic
way to expand in powers of $1/N$, and obtained $1/N$ order corrections to the
spherical model.

In 1973 Wilson exploited this idea in quantum field theory
[wilson1973quantum]. In particular Wilson studied both bosonic (linear
$\sigma$-model) and fermionic models. The linear $\sigma$-model is the
$N$-component generalization of the $\Phi^{4}_{d}$ model, given by the
(formal) measure

${\mathord{{\rm d}}}\nu^{N}(\Phi)\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\frac{1}{Z_{N}}\mathop{\mathrm{exp}}\nolimits\bigg{(}-\int_{\mathbb{T}^{d}}\sum_{j=1}^{N}|\nabla\Phi_{j}|^{2}+m\sum_{j=1}^{N}\Phi_{j}^{2}+\frac{\lambda}{2N}\Big{(}\sum_{j=1}^{N}\Phi_{j}^{2}\Big{)}^{2}{\mathord{{\rm
d}}}x\bigg{)}\mathcal{D}\Phi$ (1.1)

over $\mathbb{R}^{N}$ valued fields $\Phi=(\Phi_{1},\Phi_{2},...,\Phi_{N})$,
where $Z_{N}$ is a normalization constant and $\mathcal{D}\Phi$ is the formal
“Lebesgue measure”. We will discuss below the physics predictions and how they
are related with our rigorous results. Soon later, Coleman–Jackiw–Politzer
[coleman1974] calculated the effective potential and studied spontaneous
symmetry breaking of this model for $d\leqslant 4$, see also [abbott1976bound]
for more extensive analysis in $d=4$. 111Remark that in this survey we will
not discuss phase transition or spontaneous symmetry breaking, although it is
an interesting topic.

Around the same time, Gross and Neveu [gross1974dynamical] studied the $1/N$
expansion for a two-dimensional fermionic model, where $N$ Dirac fermions
interact via a quartic term (“four-fermion interactions”): namely the
Lagrangian is given by
$\sum_{i=1}^{N}\bar{\psi}_{i}(i\not{\partial})\psi_{i}+\frac{\lambda}{N}(\sum_{i=1}^{N}\bar{\psi}_{i}\psi_{i})^{2}$.
This model had also been touched in Wilson’s paper [wilson1973quantum]. It was
observed that the model has asymptotic freedom, symmetry breaking (for a
discrete chiral symmetry) and produces a fermion mass.

Another class of models considered from the 70s is the nonlinear
$\sigma$-models in two dimensions. For instance, see [bardeen1976phase] which
considered models where $\Phi$ takes values in a sphere in $\mathbb{R}^{N}$.
It can be viewed as the limit of $\mathbb{R}^{N}$ valued $\sigma$-model with a
potential $K(\Phi\cdot\Phi-a^{2})^{2}$ in a formal limit $K\to\infty$. The
idea was also extended to $CP^{N-1}$ type nonlinear $\sigma$-models
[d1979confinement, d19781n] (where more general homogenous space $G/H$ valued
models were also formulated).

Matrix-valued field theories are considered to be more difficult. In 1974
t’Hooft [tHooft1974planar] gave the solution of the two dimensional Yang-Mills
model with $SU(N)$ gauge group, as $N\to\infty$. After this seminal work by
t’Hooft, the large $N$ behavior of gauge theories was extensively studied by
physicists, see e.g. [makeenko1979exact, polyakov1980gauge, kazakov1980non,
kazakov1981wilson]. See also [gopakumar1995mastering] (and the references
therein) for a broad class of matrix models.

It is certainly impossible to list all the vast literature, instead we refer
to [LargeN1993] for an edited comprehensive collection of articles on large
$N$ as applied to a wide spectrum of problems in quantum field theory and
statistical mechanics; see also the review articles [witten1980] and
[moshe2003] or Coleman’s excellent book [coleman1988, Chapter 8] for summaries
of the progress.

###  1.2. Perturbation theory heuristics

A standard approach in quantum field theory is perturbation theory, namely,
calculating physical quantities of a given model by perturbative expansion in
a coupling constant of the model. In large $N$ problems, terms in perturbation
theory can have different orders in $1/N$, and the “$1/N$ expansion” is a re-
organization of the series in the parameter $1/N$, with each term typically
being a (formal) sum of infinitely many orders of the ordinary perturbation
theory.

To explain this formalism a bit more, we first recall the usual perturbation
theory for $\Phi^{4}$ model, with $N=1$ and $m=0$ (for simplicity) in (1.1).
This could be found in a standard QFT textbook, e.g. Peskin and Schroeder
[MR1402248]. Loosely speaking, the perturbative calculation of for instance a
two-point correlation of $\Phi$ is given by a sum of Feynman graphs with two
external legs and degree-$4$ internal vertices that represents the quartic
interaction $\Phi^{4}$, such as

$(a)\quad$ $\qquad(b)\quad$

For the vector model (1.1) with $N>1$ which was studied by Wilson
[wilson1973quantum], the perturbative calculation of the two-point correlation
of $\Phi_{i}$ (with $i$ fixed) is also given by the sum of Feynman graphs with
two external legs and degree-$4$ internal vertices, but now each vertex
carrying two distinct summation variables and a factor $1/N$ that represents
the interaction $\frac{1}{N}\sum_{j,k}\Phi_{j}^{2}\Phi_{k}^{2}$ term in (1.1),
such as (a) (b) below

$(a)\quad$ $i$$i$$j$$j$ $\qquad(b)\quad$ $i$$i$$j$$j$$i$

Heuristically, graph (a) is of order $\frac{1}{N}\sum_{j}\approx O(1)$ and
graph (b) is of order $\frac{1}{N^{2}}\sum_{j}\approx O(\frac{1}{N})$. The
philosophy of [wilson1973quantum] is that graphs with “self-loops” such as (a)
get cancelled by Wick renormalization, 222We will have more precise discussion
on Wick renormalization in Section LABEL:sec:Phi42. and all other graphs with
internal vertices such as (b) are at least of order $O(1/N)$ and thus vanish,
so the theory would be asymptotically Gaussian free field. (We rigorously
prove this in Section LABEL:sec:GFF2d and Section LABEL:sec:GFF3d in 2D and
3D.)

Since the model (1.1) is invariant under any $O(N)$ rotation of $\Phi$, it is
more natural to study the $O(N)$-invariant observables than the field itself,
such as $\sum_{j=1}^{N}\Phi_{j}^{2}$. One needs to find a suitable scaling for
such observables to see interesting fluctuations. In the “law of large numbers
behavior” scaling $\frac{1}{N}\sum_{j=1}^{N}\Phi_{j}^{2}$ one would expect a
deterministic limit. Interesting fluctuation then arises in the next scale.

Following the above heuristic, we assume that the model is Wick renormalized
and so the perturbation series does not contain self-loops, and we consider
$\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\,\colon\\!\\!\Phi_{j}^{2}\\!\colon$. Note
that here $\,\colon\\!\\!\Phi_{j}^{2}\\!\colon$ is the Wick square. The
bubbled diagrams of the following form are all the $O(1)$ contributions to its
two-point correlation:

$\leavevmode\hbox to140.19pt{\vbox
to22.7pt{\pgfpicture\makeatletter\hbox{\hskip 10.4588pt\lower-11.34805pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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