# On trans-Sasakian $3$-manifolds as $\eta$-Einstein solitons

Dipen Ganguly, Santu Dey and Arindam Bhattacharyya

(†† The first author is the corresponding author. )

> Abstract. The present paper is to deliberate the class of $3$-dimensional
> trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have
> studied $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds
> where the Ricci tensors are Codazzi type and cyclic parallel. We have also
> discussed some curvature conditions admitting $\eta$-Einstein solitons on
> $3$-dimensional trans-Sasakian manifolds and the vector field is torse-
> forming. We have also shown an example of $3$-dimensional trans-Sasakian
> manifold with respect to $\eta$-Einstein soliton to verify our results.

M.S.C. 2010: 53C25, 53C15, 53C21.
Key words: Einstein soliton; $\eta$-Einstein soliton; Trans-Sasakian manifold;
codazzi type Ricci tensor; $C$-Bochner curvature tensor.

##  1 Introduction

In 2016, G. Catino and L. Mazzieri [7] introduced the notion of Einstein
soliton which can be viewed as a self-similar solution to the Einstein flow

(1.1) $\frac{\partial g}{\partial t}=-2(S-\frac{r}{2}g),$

where $g$ is the Riemannian metric, $S$ is the Ricci tensor and $r$ is the
scalar curvature. It can be easily seen that the Einstein soliton is analogous
to the Ricci soliton which is also generated by a self-similar solution to the
very famous geometric revolution equation Ricci flow. It is a well-known fact
now that the study of Ricci soliton has tremendous contribution in solving the
longstanding Thurston’s geometric conjecture. Similarly it is also interesting
to study the Einstein soliton from various directions to solve many physical
and geometrical problems. However in this paper we consider, a slight
perturbation of the Einstein soliton by $\eta\otimes\eta$, called the
$\eta$-Einstein soliton. The mathematical expression for the $\eta$-Einstein
soliton [1] is given by the following equation

(1.2) $\mathcal{L}_{\xi}g+2S+(2\lambda-r)g+2\mu\eta\otimes\eta=0,$

where $\mathcal{L}_{\xi}$ denotes the Lie derivative along the direction of
the vector field $\xi$, $S$ is the Ricci tensor, $r$ is the scalar curvature
and $\lambda$, $\mu$ are real constants. The $\eta$-Einstein soliton is called
shrinking if $\lambda<0$, steady if $\lambda=0$ and expanding if $\lambda>0$.
In particular, if $\mu=0$, the $\eta$-Einstein soliton reduces to the Einstein
soliton $(g,\xi,\lambda)$.

J. T. Cho and M. Kimura [6] introduced the concept of $\eta$-Ricci soliton and
later C. Calin and M. Crasmareanu [5] studied it on Hopf hypersufaces in
complex space forms. A Riemannian manifold $(M,g)$ is said to admit an
$\eta$-Ricci soliton if for a smooth vector field $V$, the metric $g$
satisfies the following equation

(1.3) $\mathcal{L}_{\xi}g+2S+2\lambda g+2\mu\eta\otimes\eta=0,$

where $\mathcal{L}_{\xi}$ is the Lie derivative along the direction of $\xi$,
$S$ is the Ricci tensor and $\lambda$, $\mu$ are real constants. It is to be
noted that if the manifold has constant scalar curvature, then the data
$(g,\xi,\lambda-\frac{r}{2},\mu)$ of the equation $(1.2)$ satisfies the
equation $(1.3)$, i.e; the $\eta$-Einstein soliton reduces to an $\eta$-Ricci
soliton. Hence we can remark that the two notions are different for the
manifolds of non-constant scalar curvature and if the scalar curvature of the
manifold is constant then the concepts of $\eta$-Ricci soliton and
$\eta$-Einstein soliton coincide.

The paper is organised as follows: After a brief introduction, in section $2$,
we recall some basic knowledge on trans-Sasakian manifolds. Section $3$ deals
with $3$-dimensional trans-Sasakian manifolds admitting $\eta$-Einstein
solitons and also the nature of the soliton is dicussed. In this section, we
have constructed an example of a $3$-dimensional trans-Sasakian manifold
satisfying $\eta$-Einstein soliton. In section $4$, we have contrived
$\eta$-Einstein solitons in $3$-dimensional trans-Sasakian manifolds in terms
of Codazzi type and cyclic parallel Ricci tensor and characterized the nature
of the manifold. Sections $5,6,7,8$ are devoted to the study of some curvature
conditions $R\cdot S=0$, $W_{2}\cdot S=0$, $R\cdot E=0$, $B\cdot S=0$, $S\cdot
R=0$ admitting $\eta$-Einstein solitons in $3$-dimensional trans-Sasakian
manifold. In last section we have studied torse forming vector field when
$3$-dimensional trans-Sasakian manifolds admitting $\eta$-Einstein solitons.

##  2 Preliminaries

An $n$-dimensional smooth Riemannian manifold $(M,g)$ is said to be an almost
contact metric manifold [3] if it admits a $(1,1)$ tensor field $\phi$, a
characteristic vector field $\xi$, a global 1-form $\eta$ and an indefinite
metric $g$ on M satisfying the following relations

(2.1) $\displaystyle\phi^{2}$ $\displaystyle=$
$\displaystyle-I+\eta\otimes\xi,$ (2.2) $\displaystyle\eta(\xi)$
$\displaystyle=$ $\displaystyle 1,$ (2.3) $\displaystyle\eta(X)$
$\displaystyle=$ $\displaystyle g(X,\xi),$ (2.4) $\displaystyle g(\phi X,\phi
Y)$ $\displaystyle=$ $\displaystyle g(X,Y)-\eta(X)\eta(Y),$ (2.5)
$\displaystyle g(X,\phi Y)+g(Y,\phi X)$ $\displaystyle=$ $\displaystyle 0,$

for all vector fields $X,Y\in TM$, where $TM$ is the tangent bundle of the
manifold $M$. Also it can be easily seen that $\phi(\xi)=0$, $\eta(\phi X)=0$
and rank of $\phi$ is $(n-1)$.

The geometry of the almost Hermitian manifold $(M\times\mathbb{R},G,J)$ gives
rise to the geometry of the almost contact metric manifold
$(M,g,\phi,\xi,\eta)$, where $G$ is product metric of the product manifold
$M\times\mathbb{R}$ with the complex structure $J$ defined by

(2.6) $J(X,f\frac{d}{dt})=(\phi X-f\xi,\eta(X)\frac{d}{dt}),$

for all vector fields $X$ on the manifold $M$ and smooth function $f$ on the
product manifold $M\times\mathbb{R}$. An almost contact metric manifold
$(M,g,\phi,\xi,\eta)$ is called a trans-Sasakian manifold if the product
manifold $(M\times\mathbb{R},G,J)$ belongs to the class $W_{4}$ [9]. The
notion of trans-Sasakian manifolds was introduced by J. A. Oubina [13] and
later J. C. Marrero [11] completely characterized the local structures of
trans-Sasakian manifolds of dimension $n\geq 5$. The expression for which an
almost contact metric manifold $(M,g,\phi,\xi,\eta)$ becomes a trans-Sasakian
manifold is given by

(2.7) $(\nabla_{X}\phi)(Y)=\alpha[g(X,Y)\xi-\eta(Y)X]+\beta[g(\phi
X,Y)\xi-\eta(Y)\phi X],$

for all $X,Y\in TM$ and for some smooth functions $\alpha,\beta$ on the
manifold $M$. Then such kind of manifold is called a trans-Sasakian manifold
of type $(\alpha,\beta)$. In particular trans-Sasakian manifolds of type
$(0,0)$, $(\alpha,0)$ and $(0,\beta)$ are called cosymplectic,
$\alpha$-Sasakian and $\beta$-Kenmotsu manifolds respectively.

In what follows, by a trans-Sasakian $3$-manifold, we mean a $3$-dimensional
trans-Sasakian manifold $(M,g,\phi,\xi,\eta)$ of type $(\alpha,\beta)$ and we
will use the notation $(M,g)$ to denote it throughout this article. Now from
the above expression $(2.7)$ it can be derived that

(2.8) $\displaystyle\nabla_{X}\xi$ $\displaystyle=$
$\displaystyle-\alpha\phi(X)+\beta(X-\eta(X)\xi),$ (2.9)
$\displaystyle(\nabla_{X}\eta)(Y)$ $\displaystyle=$ $\displaystyle-\alpha
g(\phi(X),Y)+\beta g(\phi(X),\phi(Y)),$

for all vector fields $X,Y$ in $TM$. Again in a trans-Sasakian $3$-manifold
$(M,g)$ the Ricci tensor is given by

(2.10) $\displaystyle S(X,Y)$ $\displaystyle=$
$\displaystyle[\frac{r}{2}+\xi\beta-(\alpha^{2}-\beta^{2})]g(X,Y)-[\frac{r}{2}+\xi\beta-3(\alpha^{2}-\beta^{2})]\eta(X)\eta(Y),$
$\displaystyle-[Y\beta+(\phi(Y)\alpha)]\eta(X)-[X\beta+(\phi(X)\alpha)]\eta(Y).$

Furthermore, if the functions $\alpha$, $\beta$ are constants then, in a
trans-Sasakian $3$-manifold $(M,g)$ the following relations hold,

(2.11) $\displaystyle R(X,Y)\xi$ $\displaystyle=$
$\displaystyle(\alpha^{2}-\beta^{2})[\eta(Y)X-\eta(X)Y],$ (2.12)
$\displaystyle R(\xi,X)Y$ $\displaystyle=$
$\displaystyle(\alpha^{2}-\beta^{2})[g(X,Y)\xi-\eta(Y)X],$ (2.13)
$\displaystyle R(\xi,X)\xi$ $\displaystyle=$
$\displaystyle(\alpha^{2}-\beta^{2})[\eta(X)\xi-X],$ (2.14) $\displaystyle
S(X,Y)$ $\displaystyle=$
$\displaystyle[\frac{r}{2}-(\alpha^{2}-\beta^{2})]g(X,Y)-[\frac{r}{2}-3(\alpha^{2}-\beta^{2})]\eta(X)\eta(Y),$
(2.15) $\displaystyle S(X,\xi)$ $\displaystyle=$ $\displaystyle
2(\alpha^{2}-\beta^{2})\eta(X),$

for all vector fields $X,Y$ in $TM$ and where $R$ is the curvature tensor and
$S$ is the Ricci tensor.

######  Definition 2.1.

A trans-Sasakian $3$-manifold $(M,g)$ is said to be an $\eta$-Einstein
manifold if its Ricci tensor $S$ is of the form

(2.16) $S(X,Y)=ag(X,Y)+b\eta(X)\eta(Y),$

for all $X,Y\in TM$ and smooth functions $a,b$ on the manifold $(M,g)$.

##  3 $\eta$-Einstein solitons on trans-Sasakian $3$-manifolds

Let us consider a trans-Sasakian $3$-manifold $(M,g)$ admitting an
$\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$. Then from equation $(1.2)$ we
can write

(3.1)
$(\mathcal{L}_{\xi}g)(X,Y)+2S(X,Y)+[2\lambda-r]g(X,Y)+2\mu\eta(X)\eta(Y)=0,$

for all $X,Y\in TM$.
Again from the well-known formula
$(\mathcal{L}_{\xi}g)(X,Y)=g(\nabla_{X}\xi,Y)+g(\nabla_{Y}\xi,X)$ of Lie-
derivative and using $(2.8)$, we obtain for a trans-Sasakian $3$-manifold

(3.2) $(\mathcal{L}_{\xi}g)(X,Y)=2\beta[g(X,Y)-2\beta\eta(X)\eta(Y)].$

Now in view of the equations $(3.1)$ and $(3.2)$ we get

(3.3) $S(X,Y)=(\frac{r}{2}-\lambda-\beta)g(X,Y)+(\beta-\mu)\eta(X)\eta(Y).$

This shows that the manifold $(M,g)$ is an $\eta$-Einstein manifold.
Also from equation $(3.3)$ replacing $Y=\xi$ we find that

(3.4) $S(X,\xi)=(\frac{r}{2}-\lambda-\mu)\eta(X).$

Compairing the above equation $(3.4)$ with $(2.15)$ yields

(3.5) $r=4(\alpha^{2}-\beta^{2})+2\lambda+2\mu.$

Again, considering an orthonormal basis $\\{e_{1},e_{2},e_{3}\\}$ of $(M,g)$
and then setting $X=Y=e_{i}$ in equation $(3.3)$ and summing over $i=1,2,3$ we
get

(3.6) $r=6\lambda+4\beta+2\mu.$

Finally combining equations $(3.5)$ and $(3.6)$ we arrive at

(3.7) $\lambda=(\alpha^{2}-\beta^{2})-\beta.$

Thus the above discussion leads to the following

######  Theorem 3.1.

If a trans-Sasakian $3$-manifold $(M,g)$ admits an $\eta$-Einstein soliton
$(g,\xi,\lambda,\mu)$, then the manifold $(M,g)$ becomes an $\eta$-Einstein
manifold of constant scalar curvature $r=6\lambda+4\beta+2\mu$. Furthermore,
the soliton is shrinking, steady or expanding according as;
$\alpha^{2}<\beta(\beta+1)$, $\alpha^{2}=\beta(\beta+1)$,
$\alpha^{2}>\beta(\beta+1)$ respectively.

######  Example 3.1.

Let us consider the $3$-dimensional manifold
$M=\\{(u,v,w)\in\mathbb{R}^{3}:w\neq 0\\}$. Define a linearly independent set
of vector fields $\\{e_{i}:1\leq i\leq 3\\}$ on the manifold $M$ given by

$e_{1}=e^{2w}\frac{\partial}{\partial
u},~{}~{}~{}~{}e_{2}=e^{2w}\frac{\partial}{\partial
v},~{}~{}~{}~{}e_{3}=\frac{\partial}{\partial w}.$

Let us define the Riemannian metric $g$ on $M$ by

$g(e_{i},e_{j})=\left\\{\begin{array}[]{rcl}1,&\mbox{for}&i=j\\\
0,&\mbox{for}&i\neq j\end{array}\right.$

for all $i,j=1,2,3$. Now considering $e_{3}=\xi$, let us take the $1$-form
$\eta$, on the manifold $M$, defined by

$\eta(U)=g(U,e_{3}),~{}~{}~{}\forall U\in TM.$

Then it can be observed that $\eta(\xi)=1$. Let us define the $(1,1)$ tensor
field $\phi$ on $M$ as

$\phi(e_{1})=e_{2},~{}~{}~{}~{}\phi(e_{2})=-e_{1},~{}~{}~{}~{}\phi(e_{3})=0.$

Using the linearity of $g$ and $\phi$ it can be easily checked that

$\phi^{2}(U)=-U+\eta(U)\xi,~{}~{}g(\phi U,\phi
V)=g(U,V)-\eta(U)\eta(V),~{}~{}~{}\forall U,V\in TM.$

Hence the structure $(g,\phi,\xi,\eta)$ defines an almost contact metric
structure on the manifold $M$. Now, using the definitions of Lie bracket,
after some direct computations we get

$[e_{1},e_{2}]=0,~{}~{}~{}~{}[e_{1},e_{3}]=-2e_{1},~{}~{}~{}~{}[e_{2},e_{3}]=-2e_{2}.$

Again the Riemannian connection $\nabla$ of the metric $g$ is defined by the
well-known Koszul’s formula which is given by

$\displaystyle 2g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)$ $\displaystyle-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]).$

Using the above formula one can easily calculate that

(3.8)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nabla_{e_{1}}e_{1}=2e_{3},~{}~{}~{}\nabla_{e_{1}}e_{2}=0,~{}~{}~{}\nabla_{e_{1}}e_{3}=-2e_{1},\\\
\nabla_{e_{2}}e_{1}=0,~{}~{}~{}\nabla_{e_{2}}e_{2}=2e_{3},~{}~{}~{}\nabla_{e_{2}}e_{3}=-2e_{2},\\\
\nabla_{e_{3}}e_{1}=0,~{}~{}~{}\nabla_{e_{3}}e_{2}=0,~{}~{}~{}\nabla_{e_{3}}e_{3}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Thus from the above relations it follows that the manifold $(M,g)$ is a trans-
Sasakian $3$-manifold. Now using the well-known formula
$R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$ the non-
vanishing components of the Riemannian curvature tensor $R$ can be easily
obtained as

(3.9)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R(e_{1},e_{2})e_{2}=R(e_{1},e_{3})e_{3}=-4e_{1},\\\
R(e_{2},e_{3})e_{3}=R(e_{3},e_{1})e_{1}=-4e_{2},\\\
R(e_{3},e_{2})e_{2}=4e_{2},~{}~{}~{}~{}R(e_{2},e_{1})e_{1}=4e_{3}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Hence we can calculate the components of the Ricci tensor as follows

$S(e_{1},e_{1})=0,~{}~{}~{}S(e_{2},e_{2})=0,~{}~{}~{}S(e_{3},e_{3})=-8.$

Therefore in view of the above values of the Ricci tensor, from the equation
$(1.2)$ we can calculate $\lambda=-2$ and $\mu=6$. Hence we can say that the
data $(g,\xi,-2,6)$ defines an $\eta$-Einstein soliton on the trans-Sasakian
$3$-manifold $(M,g)$. Also we can see that the manifold $(M,g)$ is a manifold
of constant scalar curvature $r=-8$ and hence the theorem $(3.1)$ is verified.

Next we consider a trans-Sasakian $3$-manifold $(M,g)$ and assume that it
admits an $\eta$-Einstein soliton $(g,V,\lambda,\mu)$ such that $V$ is
pointwise collinear with $\xi$, i.e; $V=b\xi$, for some function $b$; then
from the equation $(1.2)$ it follows that

(3.10)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}bg(\nabla_{X}\xi,Y)+(Xb)\eta(Y)+bg(\nabla_{Y}\xi,X)+(Yb)\eta(X)\\\
+2S(X,Y)+(2\lambda-r)g(X,Y)+2\mu\eta(X)\eta(Y)=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Then using the equation $(2.8)$ in above we arrive at

(3.11)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2b\beta+2\lambda-r)g(X,Y)+(Xb)\eta(Y)+(Yb)\eta(X)\\\
+2S(X,Y)+2(b\beta+\mu)\eta(X)\eta(Y)=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Replacing $Y=\xi$ in the above equation yields

(3.12) $(Xb)+(\xi b)\eta(X)+2S(X,\xi)+(2\lambda+2\mu-r)\eta(X)=0.$

Again taking $X=\xi$ in $(3.9)$ and by virtue of $(2.15)$ we arrive at

(3.13) $2(\xi b)=(r-2\lambda-2\mu)-4(\alpha^{2}-\beta^{2}).$

Using this value from $(3.10)$ in the equation $(3.9)$ and recalling $(2.15)$
we can write

(3.14) $db=[\frac{r}{2}-\lambda-\mu-2(\alpha^{2}-\beta^{2})]\eta.$

Now taking exterior differentiation on both sides of $(3.11)$ and using the
famous Poincare’s lemma i.e; $d^{2}=0$, finally we arrive at

(3.15) $r=2\lambda+2\mu+4(\alpha^{2}-\beta^{2}).$

In view of the above $(3.12)$ the equation $(3.11)$ gives us $db=0$ i.e; the
function $b$ is constant. Then the equation $(3.8)$ reduces to

(3.16) $S(X,Y)=(\frac{r}{2}-\lambda-b\beta)g(X,Y)+(b\beta-\mu)\eta(X)\eta(Y),$

for all $X,Y\in TM$. Hence we can state the following

######  Theorem 3.2.

If a trans-Sasakian $3$-manifold $(M,g)$ admits an $\eta$-Einstein soliton
$(g,V,\lambda,\mu)$ such that $V$ is pointwise collinear with $\xi$, then $V$
is constant multiple of $\xi$ and the manifold $(M,g)$ becomes an
$\eta$-Einstein manifold of constant scalar curvature
$r=2\lambda+2\mu+4(\alpha^{2}-\beta^{2})$.

##  4 $\eta$-Einstein solitons on trans-Sasakian $3$-manifolds with Codazzi
type and cyclic parallel Ricci tensor

The purpose of this section is to study $\eta$-Einstein solitons in trans-
Sasakian $3$-manifolds having certain special types of Ricci tensor namely
codazzi type Ricci tensor and cyclic parallel Ricci tensor.

######  Definition 4.1.

[8] A trans-Sasakian $3$-manifold is said to have Codazzi type Ricci tensor if
its Ricci tensor $S$ is non-zero and satisfies the following relation

(4.1) $(\nabla_{X}S)(Y,Z)=(\nabla_{Y}S)(X,Z),~{}~{}~{}\forall X,Y,Z\in TM.$

Let us consider a trans-Sasakian $3$-manifold having Codazzi type Ricci tensor
and admits an $\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$, then equation
$(3.3)$ holds. Now covariantly differentiating the equation $(3.3)$ with
respect to an arbitrary vector field $X$ and then using $(2.9)$ we get

(4.2) $\displaystyle(\nabla_{X}S)(Y,Z)$ $\displaystyle=$ $\displaystyle
2(\beta-\mu)[\eta(Y)(-\alpha g(\phi X,Z)+\beta g(\phi X,\phi Z))$
$\displaystyle+\eta(Z)(-\alpha g(\phi X,Y)+\beta g(\phi X,\phi Y))].$

Similarly we can compute

(4.3) $\displaystyle(\nabla_{Y}S)(X,Z)$ $\displaystyle=$ $\displaystyle
2(\beta-\mu)[\eta(X)(-\alpha g(\phi Y,Z)+\beta g(\phi Y,\phi Z))$
$\displaystyle+\eta(Z)(-\alpha g(\phi Y,X)+\beta g(\phi Y,\phi X))].$

Since the manifold has Codazzi type Ricci tensor, using $(4.2)$ and $(4.3)$ in
the equation $(4.1)$ and then recalling $(2.4)$ we arrive at

$\displaystyle 2(\beta-\mu)[\eta(Y)(-\alpha g(\phi X,Z)+\beta
g(X,Z))-\eta(X)(-\alpha g(\phi Y,Z)$ (4.4) $\displaystyle+\beta
g(Y,Z))-2\alpha\eta(Z)g(\phi X,Y)]$ $\displaystyle=$ $\displaystyle 0.$

Putting $Z=\xi$ in above and in view of $(2.2)$ we finally obtain

(4.5) $4\alpha(\mu-\beta)g(\phi X,Y)=0,$

forall $X,Y\in TM$. Therefore from $(4.4)$ we can conclude that either
$\alpha=0$ or $\mu=\beta$. Hence we have the following

######  Theorem 4.1.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the Ricci tensor of the manifold is of
Codazzi type then the manifold becomes a $\beta$-Kenmotsu manifold provided
$\mu\neq\beta$.

Now using $\alpha=0$ in equation $(3.7)$ we get $\lambda=-\beta(\beta+1)$.
Thus we can state the following

######  Corollary 4.2.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$ with $\mu\neq\beta$. If the Ricci tensor of the
manifold is of Codazzi type then the soliton is shrinking if $\beta<-1$ or,
$\beta>0$; steady if $\beta=-1$ or $\beta=0$; and expanding if $-1<\beta<0$
respectively.

Again from the equation $(4.4)$ we can write that $\mu=\beta$ if $\alpha\neq
0$. Then from equation $(3.3)$ we obtain

(4.6) $S(X,Y)=(\frac{r}{2}-\lambda-\beta)g(X,Y),$

for all $X,Y\in TM$. Then contracting the equation $(4.5)$ we get
$r=6\lambda+6\beta$. Hence in view of this and equation $(4.5)$ we have the
following

######  Theorem 4.3.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the Ricci tensor of the manifold is of
Codazzi type then the manifold becomes an Einstein manifold of constant scalar
curvature $r=6\lambda+6\beta$ provided $\alpha\neq 0$.

######  Definition 4.2.

[8] A trans-Sasakian $3$-manifold is said to have cyclic parallel Ricci tensor
if its Ricci tensor $S$ is non-zero and satisfies the following relation

(4.7)
$(\nabla_{X}S)(Y,Z)+(\nabla_{Y}S)(Z,X)+(\nabla_{Z}S)(X,Y)=0~{}~{}~{}\forall
X,Y,Z\in TM.$

Let us consider a trans-Sasakian $3$-manifold, having cyclic parallel Ricci
tensor, admits an $\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$, then equation
$(3.3)$ holds. Now taking covariant differentiation of $(3.3)$ and using
equation $(2.9)$ we obtain relations $(4.2)$ and $(4.3)$. In a similar manner
we get the following

(4.8) $\displaystyle(\nabla_{Z}S)(X,Y)$ $\displaystyle=$ $\displaystyle
2(\beta-\mu)[\eta(X)(-\alpha g(\phi Z,Y)+\beta g(\phi Z,\phi Y))$
$\displaystyle+\eta(Y)(-\alpha g(\phi Z,X)+\beta g(\phi Z,\phi X))].$

Now since the manifold has cyclic parallel Ricci tensor, using the values from
$(4.2)$, $(4.3)$ and $(4.8)$ in the equation $(4.7)$ and then making use of
$(2.4)$ we arrive at

(4.9) $4\beta(\beta-\mu)[\eta(X)g(\phi Y,\phi Z)+\eta(Y)g(\phi Z,\phi
X)+\eta(Z)g(\phi X,\phi Y)]=0.$

Replacing $Z=\xi$ in the above equation $(4.9)$ yields

(4.10) $4\beta(\beta-\mu)g(\phi X,\phi Y)=0,$

for all $X,Y\in TM$. Since $g(\phi X,\phi Y)\neq 0$ always, the above equation
$(4.10)$ implies that either $\beta=0$ or, $\mu=\beta$. Thus we can state the
following

######  Theorem 4.4.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold has cyclic parallel Ricci
tensor, then the manifold becomes an $\alpha$-Sasakian manifold provided
$\mu\neq\beta$.

Now using $\beta=0$ in equation $(3.7)$ we get $\lambda=\alpha^{2}>0$.
Therefore we have

######  Corollary 4.5.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$ with $\mu\neq\beta$. If the manifold has cyclic
parallel Ricci tensor then the soliton is expanding.

Again if $\beta\neq 0$ then from $(4.10)$ it follows that $\mu=\beta$.
Therefore after a similar calculation like equation $(4.6)$ we can state

######  Theorem 4.6.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold has cyclic parallel Ricci
tensor, then the manifold becomes an Einstein manifold of constant scalar
curvature $r=6\lambda+6\beta$ provided $\beta\neq 0$.

##  5 $\eta$-Einstein solitons on trans-Sasakian $3$-manifolds satisfying
$R(\xi,X)\cdot S=0$ and $W_{2}(\xi,X)\cdot S=0$

Let us first consider a trans-Sasakian $3$-manifold which admits an
$\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$ and the manifold satisfies the
curvature condition $R(\xi,X)\cdot S=0$. Then $\forall X,Y,Z\in TM$ we can
write

(5.1) $S(R(\xi,X)Y,Z)+S(Y,R(\xi,X)Z)=0.$

Now using $(3.3)$ in $(5.1)$ we get

$\displaystyle(\frac{r}{2}-\lambda-\beta)g(R(\xi,X)Y,Z)+(\beta-\mu)\eta(R(\xi,X)Y)\eta(Z)~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(5.2)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(\frac{r}{2}-\lambda-\beta)g(R(\xi,X)Z,Y)+(\beta-\mu)\eta(R(\xi,X)Z)\eta(Y)=0.$

In view of $(2.12)$ the previous equation becomes

(5.3)
$(\alpha^{2}-\beta^{2})(\beta-\mu)[g(X,Y)\eta(Z)+g(X,Z)\eta(Y)-2\eta(X)\eta(Y)\eta(Z)]=0.$

Putting $Z=\xi$ in the above equation $(5.3)$ and recalling $(2.4)$ obtain

(5.4) $(\alpha^{2}-\beta^{2})(\beta-\mu)g(\phi X,\phi Y)=0,$

for all $X,Y\in TM$. Since $g(\phi X,\phi X)\neq 0$ always and for non-trivial
case $\alpha^{2}\neq\beta^{2}$, we can conclude from the equation $(5.4)$ that
$\mu=\beta$. Then from equation $(3.3)$ we obtain

(5.5) $S(X,Y)=(\frac{r}{2}-\lambda-\beta)g(X,Y),$

for all $X,Y\in TM$. Then contracting the equation $(5.5)$ we get
$r=6\lambda+6\beta$. Hence in view of this and equation $(5.5)$ we have the
following

######  Theorem 5.1.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold satisfies the curvature
condition $R(\xi,X)\cdot S=0$, then the manifold becomes an Einstein manifold
of constant scalar curvature $r=6\lambda+6\beta$.

Our next result of this section is on $W_{2}$-curvature tensor. It is an
important curvature tensor which was introduced in 1970 by Pokhariyal and
Mishra [14]. For this let us recall the definition of $W_{2}$-curvature tensor
as follows

######  Definition 5.1.

The $W_{2}$-curvature tensor in a trans-Sasakian $3$-manifold $(M,g)$ is
defined as

(5.6) $W_{2}(X,Y)Z=R(X,Y)Z+\frac{1}{2}[g(X,Z)QY-g(Y,Z)QX].$

Now assume that $(M,g)$ is a trans-Sasakian $3$-manifold admitting an
$\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$ and also the manifold satisfies
the curvature condition $W_{2}(\xi,X)\cdot S=0$. Then we can write

(5.7) $S(W_{2}(\xi,X)Y,Z)+S(Y,W_{2}(\xi,X)Z)=0,~{}~{}~{}\forall X,Y,Z\in TM.$

In view of $(3.3)$ the above equation $(5.7)$ becomes

$\displaystyle(\frac{r}{2}-\lambda-\beta)[g(W_{2}(\xi,X)Y,Z)+g(W_{2}(\xi,X)Z,Y)]~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(5.8)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(\beta-\mu)[\eta(W_{2}(\xi,X)Y)\eta(Z)+\eta(W_{2}(\xi,X)Z)\eta(Y)]=0.$

Again from $(3.3)$ it follows that

(5.9) $QX=(\frac{r}{2}-\lambda-\beta)X+(\beta-\mu)\eta(X)\xi,$

which implies

(5.10) $Q\xi=(\frac{r}{2}-\lambda-\mu)\xi.$

Replacing $X=\xi$ in $(5.6)$ and then using equations $(2.12)$, $(5.9)$ and
$(5.10)$ we obtain

(5.11) $W_{2}(\xi,Y)Z=Bg(Y,Z)\xi-A\eta(Z)Y+(A-B)\eta(Y)\eta(Z),$

where $A=(\alpha^{2}-\beta^{2})-\frac{1}{2}(\frac{r}{2}-\lambda-\beta)$ and
$B=(\alpha^{2}-\beta^{2})-\frac{1}{2}(\frac{r}{2}-\lambda-\mu)$. Taking inner
product of $(5.11)$ with respect to the vector field $\xi$ yields

(5.12) $\eta(W_{2}(\xi,Y)Z)=B[g(Y,Z)-\eta(Y)\eta(Z)].$

Using $(5.11)$ and $(5.12)$ in the equation $(5.8)$ and then taking $Z=\xi$ we
arrive at

$(A-B)[2B-(\frac{r}{2}-\lambda-\beta)][g(X,Y)-\eta(X)\eta(Y)]=0,$

which in view of $(2.4)$ implies

(5.13) $(A-B)[2B-(\frac{r}{2}-\lambda-\beta)]g(\phi X,\phi Y)=0,$

for all $X,Y\in TM$. Since $g(\phi X,\phi X)\neq 0$ always, we can conclude
from the equation $(5.13)$ that either $A=B$ or,
$2B=\frac{r}{2}-\lambda-\beta$. Thus recalling the values of $A$ and $B$ it
implies that either $\mu=\beta$ or,

(5.14) $2(\alpha^{2}-\beta^{2})=r-2\lambda-\mu-\beta.$

Now for the case $\mu=\beta$, proceeding similarly as the equation $(5.5)$ we
can say that the manifold becomes an Einstein manifold. Again combining
$(5.14)$ with $(3.5)$ we get

(5.15) $r=2\lambda+2\beta.$

Therefore we can state the following

######  Theorem 5.2.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold satisfies the curvature
condition $W_{2}(\xi,X)\cdot S=0$, then either the manifold becomes an
Einstein manifold or it is a manifold of constant scalar curvature
$r=2\lambda+2\beta$.

Again in view of $(3.6)$, the equation $(5.15)$ implies
$\lambda=-\frac{1}{2}(\mu+\beta)$. Hence we have

######  Corollary 5.3.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$ with $\mu\neq\beta$. If the manifold satisfies
the curvature condition $W_{2}(\xi,X)\cdot S=0$, then the soliton is
expanding, steady or shrinking according as $\mu<-\beta$, $\mu=-\beta$ or,
$\mu>-\beta$ respectively.

##  6 Einstein semi-symmetric trans-Sasakian $3$-manifolds admitting
$\eta$-Einstein solitons

######  Definition 6.1.

A trans-Sasakian $3$-manifold $(M,g)$ is called Einstein semi-symmetric [15]
if $R.E=0$, where $E$ is the Einstein tensor given by

(6.1) $E(X,Y)=S(X,Y)-\frac{r}{3}g(X,Y),$

for all vector fields $X,Y\in TM$ and $r$ is the scalar curvature of the
manifold.

Now consider a trans-Sasakian $3$-manifold is Einstein semi-symmetric i.e; the
manifold satisfies the curvature condition $R.E=0$. Then for all vector fields
$X,Y,Z,W\in TM$ we can write

(6.2) $E(R(X,Y)Z,W)+E(Z,R(X,Y)W)=0.$

In view of $(6.1)$ the equation $(6.2)$ becomes

(6.3) $S(R(X,Y)Z,W)+S(Z,R(X,Y)W)=\frac{r}{3}[g(R(X,Y)Z,W)+g(Z,R(X,Y)W)].$

Replacing $X=Z=\xi$ in the above equation $(6.3)$ and then using $(2.12)$,
$(2.13)$ we arrive at

(6.4)
$(\alpha^{2}-\beta^{2})S(Y,W)=(\alpha^{2}-\beta^{2})[\eta(Y)S(\xi,W)+\eta(W)S(\xi,Y)-g(Y,W)S(\xi,\xi)].$

So, now in view of $(2.15)$ the above equation $(6.4)$ finally yields

(6.5)
$S(Y,W)=-2(\alpha^{2}-\beta^{2})g(Y,W)+4(\alpha^{2}-\beta^{2})\eta(Y)\eta(W),$

for all $Y,W\in TM$. This implies that the manifold is an $\eta$-Einstein
manifold. Hence we have the following

######  Lemma 6.1.

An Einstein semi-symmetric trans-Sasakian $3$-manifold is an $\eta$-Einstein
manifold.

Now let us assume that the Einstein semi-symmetric trans-Sasakian $3$-manifold
$(M,g)$ admits an $\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$. Then equation
$(3.3)$ holds and combining $(3.3)$ with the above equation $(6.5)$ we get

(6.6) $r=2\lambda+\mu+\beta.$

Again recalling the equation $(3.6)$ in the above $(6.6)$ we have

(6.7) $\lambda=-\frac{1}{4}(\mu+3\beta).$

Therefore we can state the following

######  Theorem 6.2.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold is Einstein semi-symmetric,
then the manifold becomes an $\eta$-Einstein manifold of constant scalar
curvature $r=2\lambda+\mu+\beta$ and the soliton is expanding, steady or
shrinking according as $\mu<3\beta$, $\mu=3\beta$ or, $\mu>3\beta$
respectively.

##  7 $\eta$-Einstein solitons on trans-Sasakian $3$-manifolds satisfying
$B(\xi,X)\cdot S=0$

In 1949, S. Bochner [4] introduced the concept of the well-known Bochner
curvature tensor merely as a Kähler analogue of the Weyl conformal curvature
tensor but the geometric significance of it in the light of Boothby-Wangs
fibration was presented later by D. E. Blair [2]. The notion of C-Bochner
curvature tensor in a Sasakian manifold was introduced by M. Matsumoto, G.
Chūman [12] in 1969. The C-Bochner curvature tensor in trans-Sasakian
$3$-manifold $(M,g)$ is given by

(7.1) $\displaystyle B(X,Y)Z$ $\displaystyle=$ $\displaystyle
R(X,Y)Z+\frac{1}{6}[g(X,Z)QY-S(Y,Z)-g(Y,Z)QX$ $\displaystyle+S(X,Z)Y+g(\phi
X,Z)Q\phi Y-S(\phi Y,Z)\phi X-g(\phi Y,Z)Q\phi X$ $\displaystyle+S(\phi
X,Z)\phi Y+2S(\phi X,Y)\phi Z+2g(\phi X,Y))Q\phi Z$
$\displaystyle+\eta(Y)\eta(Z)QX-\eta(Y)S(X,Z)\xi+\eta(X)S(Y,Z)\xi-\eta(X)\eta(Z)QY]$
$\displaystyle-\frac{D+2}{6}[g(\phi X,Z)\phi Y-g(\phi Y,Z)\phi X+2g(\phi
X,Y)\phi Z]$
$\displaystyle+\frac{D}{6}[\eta(Y)g(X,Z)\xi-\eta(Y)\eta(Z)X+\eta(X)\eta(Z)Y-\eta(X)g(Y,Z)\xi]$
$\displaystyle-\frac{D-4}{6}[g(X,Z)Y-g(Y,Z)X],$

where $D=\frac{r+2}{4}$.

Let us consider a trans-Sasakian $3$-manifold $(M,g)$ which admits an
$\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$ and also the manifold satisfies
the curvature condition $B(\xi,X)\cdot S=0$. Then $\forall X,Y,Z\in TM$ we can
write

(7.2) $S(B(\xi,X)Y,Z)+S(Y,B(\xi,X)Z)=0.$

Now using $(3.3)$ in $(7.2)$ we get

$\displaystyle(\frac{r}{2}-\lambda-\beta)[g(B(\xi,X)Y,Z)+g(B(\xi,X)Z,Y)]~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(7.3)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(\beta-\mu)[\eta(B(\xi,X)Y)\eta(Z)+\eta(B(\xi,X)Z)\eta(Y)]=0.$

Again from $(3.3)$ it follows that

(7.4) $QX=(\frac{r}{2}-\lambda-\beta)X+(\beta-\mu)\eta(X)\xi,$

which implies

(7.5) $Q\xi=(\frac{r}{2}-\lambda-\mu)\xi.$

Also taking $X=\xi$ in $(7.1)$ we obtain

(7.6) $\displaystyle B(\xi,Y)Z$ $\displaystyle=$ $\displaystyle
R(\xi,Y)Z\frac{1}{6}[S(\xi,Z)Y-g(Y,Z)Q\xi+\eta(Y)\eta(Z)Q\xi$
$\displaystyle-\eta(Y)S(\xi,Z)\xi]+\frac{4}{6}[\eta(Z)Y-g(Y,Z)\xi].$

Using equations $(2.12)$, $(3.4)$ and $(7.5)$ in the above equation $(7.6)$
yields

(7.7)
$B(\xi,Y)Z=[(\alpha^{2}-\beta^{2})-\frac{1}{6}(\frac{r}{2}-\lambda-\mu)-\frac{4}{6}][g(Y,Z)\xi-\eta(Z)Y].$

In view of $(7.7)$ the equation $(7.3)$ becomes

$\displaystyle[(\alpha^{2}-\beta^{2})-\frac{1}{6}(\frac{r}{2}-\lambda-\mu)-\frac{4}{6}](\beta-\mu)[g(X,Y)\eta(Z)$
$\displaystyle+g(X,Z)\eta(Y)-2\eta(X)\eta(Y)\eta(Z)]$ $\displaystyle=$
$\displaystyle 0.$

Replacing $Z=\xi$ in the above equation and recalling $(2.4)$, finally we
arrive at

(7.8)
$[(\alpha^{2}-\beta^{2})-\frac{1}{6}(\frac{r}{2}-\lambda-\mu)-\frac{4}{6}](\beta-\mu)g(\phi
X,\phi Y)=0,$

for all vector fields $X,Y\in TM$. Hence from $(7.8)$ we can conclude that
either

(7.9)
$[(\alpha^{2}-\beta^{2})-\frac{1}{6}(\frac{r}{2}-\lambda-\mu)-\frac{4}{6}]=0,$

or, $\mu=\beta$. Also for $\mu=\beta$ proceeding similarly as equation $(4.6)$
it can be easily shown that the manifold becomes an Einstein manifold. Again
if $\mu\neq\beta$ using $(3.7)$ in the equation $(7.9)$ we have

(7.10) $r=10\lambda+2\mu+12\beta-8,$

which implies that the manifold becomes a manifold of constant scalar
curvature. Therefore we can state the following

######  Theorem 7.1.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold satisfies the curvature
condition $B(\xi,X)\cdot S=0$, then either the manifold is an Einstein
manifold or it is a manifold of constant scalar curvature
$r=10\lambda+2\mu+12\beta-8$.

Now for the case $\mu\neq\beta$, using the equation $(3.6)$ in $(7.10)$ we
obtain $\lambda=2(1-\beta)$. Hence we have

######  Corollary 7.2.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$ with $\mu\neq\beta$. If the manifold satisfies
the curvature condition $B(\xi,X)\cdot S=0$, then the soliton is expanding,
steady or shrinking according as $\beta<1$, $\beta=1$ or, $\beta>1$
respectively.

##  8 $\eta$-Einstein solitons on trans-Sasakian $3$-manifolds satisfying
$S(\xi,X)\cdot R=0$

In this section we study the curvature condition $S(\xi,X)\cdot R=0$, where by
$\cdot$ we denote the derivation of the tensor algebra at each point of the
tangent space as follows:

(8.1) $\displaystyle S((\xi,X)\cdot R)(Y,Z)W$ $\displaystyle:=$
$\displaystyle((\xi\wedge_{S}X)\cdot R)(Y,Z)W$ $\displaystyle:=$
$\displaystyle(\xi\wedge_{S}X)R(Y,Z)W+R((\xi\wedge_{S}X)Y,Z)W$
$\displaystyle+R(Y,(\xi\wedge_{S}X)Z)W+R(Y,Z)(\xi\wedge_{S}X)W,$

where the endomorphism $X\wedge_{S}Y$ is defined by

$(X\wedge_{S}Y)Z:=S(Y,Z)X-S(X,Z)Y.$

Now let us consider a trans-Sasakian $3$-manifold $(M,g)$ which admits an
$\eta$-Einstein soliton $(g,\xi,\lambda,\mu)$ and also the manifold satisfies
the curvature condition $S(\xi,X)\cdot R=0$. Then using this condition and the
equation $(8.1)$ we can write

$\displaystyle S(X,R(Y,Z)W)\xi-S(\xi,R(Y,Z)W)X+S(X,Y)R(\xi,Z)W$
$\displaystyle-S(\xi,Y)R(X,Z)W+S(X,Z)R(Y,\xi)W-S(\xi,Z)R(Y,X)W$ (8.2)
$\displaystyle+S(X,W)R(Y,Z)\xi-S(\xi,W)R(Y,Z)X$ $\displaystyle=$
$\displaystyle 0,$

for all vector fields $X,Y,Z,W\in TM$. Taking inner product of the above
$(8.2)$ with the vector field $\xi$ and then replacing $W=\xi$ we obtain

$\displaystyle S(X,R(Y,Z)\xi)-S(\xi,R(Y,Z)\xi)\eta(X)+S(X,Y)\eta(R(\xi,Z)\xi)$
$\displaystyle-S(\xi,Y)\eta(R(X,Z)\xi)+S(X,Z)\eta(R(Y,\xi)\xi)-S(\xi,Z)\eta(R(Y,X)\xi)$
$\displaystyle+S(X,\xi)\eta(R(Y,Z)\xi)-S(\xi,\xi)\eta(R(Y,Z)X)$
$\displaystyle=$ $\displaystyle 0,$

In view of $(2.11)$ and $(2.13)$ the above equation becomes

$\displaystyle(\alpha^{2}-\beta^{2})[S(X,Y)\eta(Z)-S(X,Z)\eta(Y)-S(\xi,Y)\eta(X)\eta(Z)$
(8.3) $\displaystyle+S(\xi,Z)\eta(X)\eta(Y)]-S(\xi,\xi)\eta(R(Y,Z)X)$
$\displaystyle=$ $\displaystyle 0.$

Putting $Y=\xi$ in $(8.3)$ and then recalling $(2.12)$ we get

$\displaystyle(\alpha^{2}-\beta^{2})[S(X,\xi)\eta(Z)-S(X,Z)-S(\xi,\xi)\eta(X)\eta(Z)$
(8.4) $\displaystyle+S(\xi,Z)\eta(X)-S(\xi,\xi)[g(X,Z)-\eta(X)\eta(Z)]]$
$\displaystyle=$ $\displaystyle 0.$

Using equations $(3.3)$ and $(3.4)$ in the above $(8.4)$ yields

$(\alpha^{2}-\beta^{2})[(r-2\lambda-2\mu+\beta)\eta(X)\eta(Z)-(r-2\lambda-\mu-\beta)g(X,Z)]=0.$

Replacing $X=\xi$ in above we arrive at

(8.5) $(\alpha^{2}-\beta^{2})(2\beta-\mu)\eta(X)=0,~{}~{}~{}~{}~{}\forall X\in
TM.$

Since for non-trivial case $\alpha^{2}\neq\beta^{2}$, from the above equation
$(8.5)$ it follows that $\mu=2\beta$. Therefore in view of this and recalling
$(3.6)$ we finally obtain $r=6\lambda+8\beta$. Therefore we can state the
following

######  Theorem 8.1.

Let $(M,g)$ be a trans-Sasakian $3$-manifold admitting an $\eta$-Einstein
soliton $(g,\xi,\lambda,\mu)$. If the manifold satisfies the curvature
condition $S(\xi,X)\cdot R=0$, then it becomes a manifold of constant scalar
curvature $r=6\lambda+8\beta$.

##  9 $\eta$-Einstein solitons on trans-Sasakian $3$-manifolds with torse-
forming vector field

This section is devoted to study the nature of $\eta$-Einstein solitons on
trans-Sasakian $3$-manifolds with torse-forming vector field.

######  Definition 9.1.

A vector field $V$ on a trans-Sasakian $3$-manifold is said to be torse-
forming vector field [17] if

(9.1) $\nabla_{X}V=fX+\gamma(X)V,$

where $f$ is a smooth function and $\gamma$ is a $1$-form.

Now let $(g,\xi,\lambda,\mu)$ be an $\eta$-Einstein soliton on a trans-
Sasakian $3$-manifold $(M,g)$ and assume that the Reeb vector field $\xi$ of
the manifold is a torse-forming vector field. Then $\xi$ being a torse-forming
vector field, by definiton from equation $(9.1)$ we have

(9.2) $\nabla_{X}\xi=fX+\gamma(X)\xi,$

$\forall X\in TM$, $f$ being a smooth function and $\gamma$ is a $1$-form.
Recalling the equation $(2.8)$ and taking inner product on both sides with
$\xi$ we can write

(9.3) $g(\nabla_{X}\xi,\xi)=(\beta-1)\eta(X).$

Again from the equation $(9.2)$, applying inner product with $\xi$ we obtain

(9.4) $g(\nabla_{X}\xi,\xi)=f\eta(X)+\gamma(X).$

Combining $(9.3)$ and $(9.4)$ we get, $\gamma=(\beta-1-f)\eta$. Thus from
$(9.2)$ it implies that, for torse-forming vector field $\xi$ in a trans-
Sasakian $3$-manifold, we have

(9.5) $\nabla_{X}\xi=f(X-\eta(X)\xi)+(\beta-1)\eta(X)\xi.$

Now from the formula of Lie differentiation and using $(9.5)$ yields

(9.6) $\displaystyle(\mathcal{L}_{\xi}g)(X,Y)$ $\displaystyle=$ $\displaystyle
g(\nabla_{X}\xi,Y)+g(\nabla_{Y}\xi,X)$ $\displaystyle=$ $\displaystyle
2f[g(X,Y)-\eta(X)\eta(Y)]+2(\beta-1)\eta(X)\eta(Y).$

Since $(g,\xi,\lambda,\mu)$ is an $\eta$-Einstein soliton, the equation
$(1.2)$ holds. So in view of $(9.6)$, the equation $(1.2)$ reduces to

(9.7) $S(X,Y)=(\frac{r}{2}-\lambda+f)g(X,Y)+(f-\mu-\beta+1)\eta(X)\eta(Y).$

This implies that the manifold is an $\eta$-Einstein manifold. Again putting
$Y=\xi$ in $(9.7)$ we get

(9.8) $S(X,\xi)=(\frac{r}{2}-\lambda-\mu-\beta+1)\eta(X).$

Combining $(9.8)$ with the equation $(2.15)$ implies

(9.9) $(\frac{r}{2}-\lambda-\mu-\beta+1)=2(\alpha^{2}-\beta^{2}).$

Again tracing out the equation $(9.7)$ we obtain

(9.10) $r=6\lambda+2\mu+4f+2\beta-2.$

Using the above equation $(9.10)$ in $(9.9)$, finally we get
$\lambda=f-(\alpha^{2}-\beta^{2})$. Therefore we have the following

######  Theorem 9.1.

Let $(g,\xi,\lambda,\mu)$ be an $\eta$-Einstein soliton on a trans-Sasakian
$3$-manifold $(M,g)$, with torse-forming vector field $\xi$, then the manifold
becomes an $\eta$-Einstein manifold and the soliton is expanding, steady or
shrinking according as $f>(\alpha^{2}-\beta^{2})$, $f=(\alpha^{2}-\beta^{2})$
or, $f<(\alpha^{2}-\beta^{2})$ respectively.

Acknowledgements. The first author D. Ganguly is thankful to the National
Board for Higher Mathematics (NBHM), India, for their financial support(Ref
No: 0203/11/2017/RD-II/10440) to carry on this research work.

## References

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Authors’ addresses:

Dipen Ganguly
Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,
India.
E-mail<EMAIL_ADDRESS>

Santu Dey
Department of Mathematics, Bidhan Chandra College, Asansol-713304, West
Bengal, India.
E-mail<EMAIL_ADDRESS>

Arindam Bhattacharyya
Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,
India.
E-mail<EMAIL_ADDRESS>


# ADDRESSING SINGLE OBJECT TRACKING IN SATELLITE IMAGERY THROUGH PROMPT-
ENGINEERED SOLUTIONS

###### Abstract

Object tracking in satellite videos remains a complex endeavor in remote
sensing due to the intricate and dynamic nature of satellite imagery. Existing
state-of-the-art trackers in computer vision integrate sophisticated
architectures, attention mechanisms, and multi-modal fusion to enhance
tracking accuracy across diverse environments. However, the challenges posed
by satellite imagery, such as background variations, atmospheric disturbances,
and low-resolution object delineation, significantly impede the precision and
reliability of traditional Single Object Tracking (SOT) techniques. Our study
delves into these challenges and proposes prompt engineering methodologies,
leveraging the Segment Anything Model (SAM) and TAPIR (Tracking Any Point with
per-frame Initialization and temporal Refinement), to create a training-free
point-based tracking method for small-scale objects on satellite videos.
Experiments on the VISO dataset validate our strategy, marking a significant
advancement in robust tracking solutions tailored for satellite imagery in
remote sensing applications.

Index Terms—  Tracking, SOT, SAM, TAPIR

##  1 Introduction

Single Object Tracking (SOT) in satellite videos stands as a critical yet
intricate task in remote sensing due to the complex and dynamic nature of
satellite imagery. Precisely tracing and defining specific objects within
these videos holds immense significance across diverse applications, ranging
from environmental observation to urban development and security monitoring
[1]. Current state-of-the-art trackers [2] [3] in computer vision integrate
complex architectures [4], attention mechanisms [5] and multi-modal fusion
[6], enhancing accuracy and adaptability across diverse environments.

Algorithms for SOT typically fall into two main categories: bounding box-based
and point-based trackers. Bounding box-based trackers enclose the target
object within a rectangular bounding box and focus on predicting and adjusting
the box’s position, scale, and orientation across frames. However, in the
context of satellite videos capturing small and densely moving objects, the
traditional use of bounding box-based trackers is less effective due to the
minute size of the targets. Objects in satellite imagery often require more
nuanced and precise tracking methodologies, rendering point-based trackers
more suitable. These point-based trackers excel in capturing and tracing
small-scale objects by focusing on individual points or features, offering
enhanced adaptability and accuracy in discerning and consistently tracking
amidst diverse and unpredictable backgrounds.

However, the distinctive challenges inherent in satellite imagery, notably the
pervasive presence of background variations, atmospheric disturbances, varying
lighting conditions, and scale discrepancies, significantly impede the
precision and reliability of traditional SOT techniques. Conventional point-
based trackers, while robust in controlled environments, encounter substantial
difficulties in discerning and consistently tracking objects amidst the
diverse and often unpredictable backgrounds captured in satellite footage.
These inherent complexities underscore the pressing need for innovative
approaches that transcend the limitations of conventional tracking algorithms
and harness novel methodologies, such as prompt engineering, to achieve more
resilient and accurate object tracking in satellite videos.

Fig. 1: Methodology overview of our SOT pipeline. A) demonstrates the
continuous tracking of points of interest, while B) and C) highlight the
essential steps of initializing and sampling points during keyframe
transitions, forming a comprehensive strategy for effective SOT in satellite
videos.

Existing methods for object tracking in satellite imagery often grapple with
these challenges by employing adaptations of generic visual tracking
algorithms. Some approaches focus on refining traditional methods, such as
using motion models or region-based tracking, to accommodate the complexities
of satellite imagery. Others have attempted to integrate machine learning
techniques, although these methods commonly face limitations when dealing with
the low-resolution, sparse, and diverse nature of objects within satellite
videos. The lack of robustness in these approaches highlights the need for
innovative strategies capable of handling the intricate and variable
backgrounds while effectively tracking objects.

To this end, our approach is motivated by the inadequacies of current methods
in addressing the object tracking complexities within satellite videos, aiming
to leverage prompt engineering techniques to overcome these limitations and
advance the precision and adaptability of tracking methodologies in this
domain. Integrating the strengths of the Segment Anything Model (SAM) [7] and
TAPIR (Tracking Any Point with per-frame Initialization and temporal
Refinement) [8], our approach aims to navigate background variations and low-
resolution object delineation, enhancing the adaptability and efficacy of
object tracking. SAM, known for its robust segmentation capabilities,
accurately delineates objects of interest, providing precise initial inputs
for TAPIR. Leveraging TAPIR’s strength in per-frame initialization and
temporal refinement, our approach navigates complexities like background
variations and low-resolution object details, enhancing the adaptability and
efficacy of tracking. By synergizing these techniques, our approach optimizes
object delineation and tracking, enhancing adaptability and robustness within
the intricate landscape of satellite imagery.

Our contribution lies in offering a refined methodology within satellite
imagery based on a prompt-engineering approach, bypassing the need for
training by utilizing pre-trained SAM and TAPIR. This strategy enhances
adaptability, allowing seamless integration into diverse tracking scenarios in
satellite videos. We conducted experiments on the VISO dataset [9], showcasing
promising outcomes. These results validate the efficacy of the prompt-
engineering methodology, demonstrating robust tracking performance in
challenging satellite imagery environments.

##  2 METHODOLOGY

Our proposed SOT methodology leverages the TAPIR point tracker and the SAM
segmentation model, as shown in Figure 1. The tracking pipeline operates on
keyframes, each separated by a fixed interval K. TAPIR tracks a set of points
between keyframes, while a SAM-based strategy renews them at each keyframe.
The initial keyframe employs SAM for precise point initialization based on an
initial bounding box for the target. This structured approach ensures accurate
and adaptive tracking, optimizing the interplay between TAPIR and SAM
throughout the sequence. In the subsequent sections, we delve deeper into the
intricacies of our methodology, while also providing a comprehensive analysis
of the underlying algorithms, SAM and TAPIR.

###  2.1 Pre-trained models

The Segment Anything Model (SAM) plays a pivotal role in our methodology for
SOT in satellite videos. SAM [7] operates as a segmentation-based model
designed to identify and segment objects of interest within video frames. SAM
accepts various prompt inputs, including bounding boxes, individual points, or
textual descriptions. For our approach, we specifically utilize bounding boxes
as the input prompt for SAM. This algorithm starts by receiving an initial
bounding box or region of interest (ROI) containing the object in the first
frame. SAM employs advanced segmentation techniques, such as semantic
segmentation or instance segmentation, to delineate and generate precise masks
outlining the object within the defined ROI. SAM’s capability to generate
accurate masks around objects of interest serves as a critical pre-processing
step within our methodology, providing refined and updated points of interest
for subsequent input into tracking algorithms like TAPIR. By harnessing SAM’s
segmentation capabilities, our methodology aims to improve the robustness and
adaptability of SOT in satellite imagery by refining the initial points for
enhanced tracking accuracy across frames.

Also, we leverage the TAPIR tracker [8] within our methodology. TAPIR operates
by initially selecting specific points of interest within the first frame of
the video. These points, often representing distinctive features or objects,
are tracked across subsequent frames using an optical flow estimation
technique. This method analyzes the pixel-level changes between frames,
estimating the motion of the selected points. TAPIR integrates per-frame
initialization, allowing for precise localization of points in each frame, and
employs temporal refinement mechanisms to enhance tracking accuracy. The
algorithm dynamically adapts the tracking process based on the local
properties of the points being tracked, thus enabling robust and precise
tracking even amidst challenging scenarios such as occlusions, scale
variations, and background changes. By leveraging this algorithm’s
adaptability and accuracy in tracking specific points, our methodology aims to
achieve resilient and effective tracking within the complexities of satellite
imagery.

Our methodology for SOT in satellite videos utilized pre-trained models from
the official repositories of SAM and TAPIR, avoiding any additional training
on these algorithms.

###  2.2 Initialization process

In the initial step of our methodology (part B of Figure 1), we employ SAM to
generate a segmentation mask for the target object. This process is initiated
by prompting SAM with the initial bounding box to derive a precise
segmentation, effectively delineating the object within the frame. In case of
very small objects, e.g. car, we additionally add a crop and resampling step
to aid SAM and consequently TAPIR. Subsequently, to establish a robust set of
tracking points, we strategically sample N random points from the generated
segmentation mask, ensuring a diverse and representative selection. This
approach lays the foundation for initializing an adaptive set of points
crucial for the subsequent tracking stages.

###  2.3 Point propagation between keyframes

In the point propagation between keyframes stage, our methodology seamlessly
integrates the standard TAPIR pipeline to facilitate the tracking of points
from one keyframe to the next. TAPIR excels in this process by leveraging its
per-frame Initialization and temporal refinement capabilities. At each
keyframe transition, the set of tracked points by TAPIR is efficiently
renewed, ensuring continuity and precision throughout the sequence. This
entails updating the set of tracked points dynamically, with TAPIR
meticulously refining their positions to adapt to changes in the object’s
appearance, scale, and orientation. The per-frame initialization ensures a
precise re-initialization of points in each new keyframe, while the temporal
refinement enhances the accuracy of point tracking by accounting for motion
dynamics over consecutive frames. The inherent adaptability of TAPIR in
handling dynamic scenes and nuanced tracking scenarios guarantees a consistent
and precise tracking across successive keyframes, contributing significantly
to the accuracy and reliability of the tracking process.

###  2.4 Keyframe point generation

In the keyframe point generation stage (part C of Figure 1), our methodology
strategically harnesses the information propagated by TAPIR to refine and
optimize the set of tracking points at each keyframe. For every point within
the keyframe, previously propagated through the TAPIR pipeline, we prompt SAM
individually. SAM generates segmentation masks for each point, capturing the
evolving appearance and features of the tracked object. Subsequently, these
individual masks are aggregated into an overlap heatmap through point-wise
addition. By selecting the area within the heatmap where the most segmentation
masks overlap, we indicate the region of maximum consensus among the points.
This region becomes the focal point for a refined set of correctly-tracked
points, allowing us to pinpoint the corresponding masks and, consequently, the
points from which these masks originated.

Building upon this refined set of points, we prompt SAM once again, this time
with all the selected points. The objective is to generate a new segmentation
mask that precisely delineates the object within the keyframe, leveraging the
collective information from the strategically chosen points. To maintain
diversity and adaptability, we subsequently employ a random sampling strategy
to acquire a fresh set of N points on the newly generated mask. This strategic
discarding of previous points ensures continual adaptability and
responsiveness to evolving object dynamics, enhancing the overall accuracy and
robustness of our tracking methodology. The meticulous interplay of SAM and
this keyframe-specific point generation process ensures an optimized and
adaptive point set for the ensuing tracking stages, contributing to the
methodology’s efficacy in handling complex tracking scenarios.

##  3 EXPERIMENTAL RESULTS

VISO dataset [9], tailored specifically for evaluating detection and tracking
of a single or multiple objects within satellite imagery, encompasses a
diverse collection of 47 high-resolution satellite videos featuring numerous
small and dense moving objects. These videos depict different environmental
settings, encompassing object categories such as cars, airplanes, ships and
trains, with varying motion patterns. The dataset comprises 3159 tracklets
spanning 1.12 million frames at 10 fps, derived from satellite videos captured
by the Jilin-1 satellite constellation at different positions of its orbit.
For the dataset’s utilization in SOT, tracklets are generated for individual
instances in the sequences and ground-truth bounding boxes of objects are
provided only in the first frame for initialization, following a standard
visual tracking scheme. Compared to existing generic video datasets, the VISO
dataset poses challenges with objects having low spatial resolution and a high
number of similar instances with variations in point of view, motion blur, and
illumination.

We followed the standard evaluation metrics of the VISO dataset, namely
Distance Precision Rate (DPR) and Overlap Success Rate (OSR). The first one
refers to the percentage of frames whose center location errors are smaller
than a given threshold A, while the second one to the percentage of frames
whose overlap ratios with the ground-truth box are larger than a given
threshold B. We set A=5 pixels due to the small size of objects and B=0.5. Our
method was implemented on Pytorch and evaluated on a machine with Ryzen 9
3900X 12 core @3.7GHz CPU, a Nvidia 2080Ti GPU and 64GB RAM. We set the number
of keyframes K=20 and the number of sampled points N=20.

Method | DPR (%) | OSR (%)
---|---|---
SiamRPN++ [10] | 47.9 | 19.9
SiamBAN [11] | 48.0 | 18.8
CFME [12] | 50.4 | 28.2
MCCT [13] | 60.5 | 34.7
ECO [14] | 61.2 | 34.5
SRNK [15] | 75.8 | 33.0
KYS-ME [15] | 78.8 | 34.0
MACF [16] | 79.4 | 35.5
Ours | 63.9 | 36.5
Table 1: Results of SOT in VISO dataset.

We compare our method to several trackers that are either specifically
designed for satellite data [12], either based on correlation and complex
filters [13] [15] [16] or Siamese networks [10] [11]. Our method scores
competitive DPR and OSR scores against them, yielding 63.9% and 36.5%
respectively, as shown in Table 1. These scores affirm the efficacy of our
method in precisely localizing and tracking objects within the challenging
context of satellite imagery. For an intuitive and qualitative evaluation, we
visualize the tracking results achieved by our method on various sequences of
the VISO dataset, as shown in Figure 2. The visualizations affirm the robust
performance of our approach in handling diverse scenes and challenging
conditions within satellite imagery.

Our results on this dataset are noteworthy, given the additional merit that
our methodology leverages pre-trained models of SAM and TAPIR, without
involving any training process. The reliance on these pre-existing, well-
performing models contributes to the adaptability of our approach,
demonstrating the efficacy of prompt engineering in object tracking without
the need for additional training efforts. However, it should be noted that the
selection of the hyperparameters K and N is crucial to our method and results
may greatly vary for slight disturbances in their optimal values. Overall, our
approach is mostly limited by TAPIR’s short-term point tracking performance.

(a)

(b)

(c)

(d)

Fig. 2: Qualitative results on the VISO dataset. Red and green color indicate
the beggining and the end of the trajectory of a target respectively.

##  4 CONCLUSIONS

Our prompt-engineering approach for object tracking in satellite videos,
utilizing pre-trained models SAM and TAPIR, has demonstrated remarkable
adaptability and showcased strong performance on the VISO dataset,
outperforming other satellite trackers. This work emphasizes the potential of
prompt-guided solutions, providing an efficient strategy for SOT in satellite
imagery. However, our approach encourages further exploration into prompt
engineering and future research should be focused on adapting other point
trackers and exploring more complex tracking scenarios where swift or
unpredictable object movements occur.

##  5 Acknowledgements

Part of this research was supported by the research project BiCUBES “Analysis-
Ready Geospatial Big Data Cubes and Cloud-based Analytics for Monitoring
Efficiently our Land & Water” funded by HFRI (grant: 03943).

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