Journal of Indian Acad. Math.  
Vol. 47, No. 1 (2025) pp. 89-111  
ISSN: 0970-5120  
Abhijit Mandal1  
SOLITONS ON VARIOUS GEOMETRIC  
STRUCTURES OF PARA-SASAKIAN  
MANIFOLDS ADMITTING  
SCHOUTEN-VAN KEMPEN  
CONNECTION  
Afsar Hossain Sarkar2  
Meghlal Mallik3*  
Ashoke Das4  
and  
Sanjib Kumar Datta5  
Abstract. In this paper we investigate properties of para-Sasakian manifold  
by the help of Schouten-van Kampen connection. We also study para-  
Sasakian manifolds of various equivalent structures admitting conformal  
Ricci soliton and conformal η-Ricci soliton with respect to Schouten-van  
Kampen connection.  
Key words and phrases: Para-Sasakian Manifold, Schouten-van Kampen  
Connection, Conformal Ricci Soliton, Conformal  
η-Ricci Soliton.  
Mathematics Subject Classification (2020) No.: 53C15, 53C25.  
1. Introduction  
In 1979, the notion of para-Sasakian (briefly, P-Sasakian) and special para-  
Sasakian (briefly, SP-Sasakian) manifolds were introduced by Sato and Matsumoto  
[28]. Later, Adati and Matsumoto investigate some interesting results on P-Sasakian  
manifolds and SP-Sasakian manifolds in [1]. The properties of para-Sasakian  
manifold have been studied by many authors. For instance, we see [2, 16, 17, 19, 27,  
30] and their references.  
The notion of Schouten-van Kampen connection (shortly, SVK-connection)  
was introduced in the third decade of last century for a study of non-holomorphic  
manifolds [29, 37]. In 2006, Bejancu [3] studied Schouten-van Kampen connection  
on Foliated manifolds. Recently, Biswas and Baisya [4, 5] investigated some  
90  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
properties of pesudo symmetric Sasakian manifolds with respect to SVK-  
connectiopn. Most recently, this connection has been introduced on para-Sasakian  
manifold by Sundriyal and Upreti [31]. They studied projective curvature tensor,  
concircular curvature tensor and Nijenhuis tensor for the para-sasakian manifold with  
respect to this connection. SVK-connection () for an n-dimensional almost contact  
metric manifold M equipped with an almost contact metric structure (,,,g)  
consisting of a (1, 1) tensor field  
metric , is defined by  
, a vector field  
, a 1-form  
and a Riemannian  
g
XY  XY (X)(Y)(Y)X  
, (1.1)  
for all X,Y (M), where (M) is the set of all vector fields on M and  
being  
the Levi-Civita connection on M .  
The concept of Ricci flow was first introduced by R. S. Hamilton in the early  
1980s. Hamilton [13] observed that the Ricci flow is an excellent tool for simplifying  
the structure of a manifold. It is the process which deforms the metric of a  
Riemannian manifold by smoothing out the irregularities. The Ricci flow equation is  
given by  
g  
2S  
,
(1.2)  
t  
where g is a Riemannian metric, S is Ricci tensor and t is time. The solitons for the  
Ricci flow is the solutions of the above equation, where the metrices at different  
times differ by a diffeomorphism of the manifold. A Ricci soliton is represented by a  
triple (g,V,), where V is a vector field and  
is a scalar, which satisfies the  
equation  
L g 2S 2g 0  
,
(1.3)  
V
where S is Ricci curvature tensor and L g denotes the Lie derivative of g along the  
V
vector field V . A Ricci soliton is said to be shrinking, steady, expanding according  
as0  
,
0  
,
0, respectively. The vector field V is called potential vector  
field and if it is a gradient of a smooth function, then the Ricci soliton (g,V,) is  
called a gradient Ricci soliton and the associated function is called the potential  
function. Ricci soliton was further studied by many researchers. For instance, we see  
[18, 25, 35, 36] and their references.  
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
91  
In 2005, Fischer [12] introduced conformal Ricci flow which is a general-  
isation of the Ricci flow equation that modifies the unit volume constraint to a scalar  
curvature constraint. The conformal Ricci flow equation is given by  
g  
g   
2 S   
 pg  
(1.4)  
t  
n
r(g)  1  
(1.5)  
where r(g) is the scalar curvature of the manifold,  
and  
p
is a non-dynamical scalar field  
n
is the dimension of the manifold. In 2015, corresponding to the conformal  
Ricci flow equation, Basu and Bhattacharyya [7] introduced the notion of conformal  
Ricci soliton as a generalisation of Ricci soliton and it is given by  
2   
L g 2S 2p   
g 0  
,
(1.6)  
V
n
  
where  
is a constant.  
As a generalization of Ricci soliton, the  
-Ricci soliton was introduced by  
Cho and Kimura [9]. This notion has also been studied by Cälin and Crasmarearu  
[10]. Later, remarkable studies on  
-Ricci soliton have been made by Blaga [6] and  
Prakasha [24]. Let  
the equation  
M
be a Riemannian manifold with structure(,,,g). Consider  
L g 2S 2g 2µ0  
,
(1.7)  
V
where  
S
is Ricci curvature tensor, L g denotes the Lie derivative of  
g
along the  
V
vector field  
V
,
and  
µ
are real constants. The data (g,V,,µ) which satisfies the  
equation (1.7) is called an  
-Ricci soliton on  
M
. In particular, when µ 0, the  
notion of η-Ricci soliton simply reduces to the notion of Ricci soliton. And when  
µ 0  
,
(g,V,,µ) is called proper  
-Ricci soliton on  
M
.
In 2018, Siddiqi [34] introduced the notion of conformal  
-Ricci soliton as  
2   
L g 2S 2p   
g 2 0,  
(1.8)  
V
n
  
where L g denotes the Lie derivative of  
g
along the vector field V,and β are real  
V
92  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
is a non-dynamical scalar field.  
constants and  
p
Definition 1.1: Let  
manifold (or, pseudo Riemannian manifold) M is said to be  
type if (X,Y).0 for all smooth vector fields X, Y on M, where  
as derivation of tensor algebra.  
and  
be two tensors of rank 4. A Riemanian  
-semisymmetric  
acts on  
In the above definition if we consider R , then the manifold  
M
is  
called semi-symmetric [32]. Semi-symmetry and other conditions of semi-symmetry  
type are studied in detail in [8, 15, 20, 33]. In 2013, Kundu and Shaikh [26]  
investigated the equivalency of the various geometric structures depending on  
conditions of semi-symmetry. They have established the following conditions  
(i) E.R 0  
,
E.P 0  
,
E.E 0  
,
E.P 0  
,
E.0  
,
E.W 0 and  
i
E.i0 ( for all i 1,2,...9) are equivalent and named such a class by  
C1  
;
(ii) R .R 0  
,
R .P 0  
,
R.E 0  
,
R.P 0  
,
R.M 0  
,
R.W 0 and  
i
R.W i0 (for all i 1,2, ......9 ) are equivalent and named such a class  
by C2  
;
(iii) R.K 0 and R.C 0 are equivalent and named such a class by C3  
;
;
(iv) E .C 0 and E .K 0 are equivalent and named such a class by C4  
where the symbols C,E,P,K,M and Wi stand for conformal curvature tensor [11],  
concircular curvature tensor [38], projective curvature tensor [38], conharmonic  
curvature tensor [14], M-projective curvature tensor [22], Wi -curvature tensor [21,  
22, 23] and W i-curvature tensor [22], respectively.  
C(X,Y) R(X,Y)  
1
r
(X g QY) (QX g Y)   
(X g Y)  
,
(1.9)  
n 2  
n 1  
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
93  
r
E(X,Y) R(X,Y)   
(X g Y)  
,
(1.10)  
n(n 1)  
1
n 1  
1
P(X,Y) R(X,Y)   
K(X,Y) R(X,Y)   
(X,Y) R(X,Y)   
(X g Y)  
,
(1.11)  
(1.12)  
(1.13)  
[(X g QY) (QX g Y)]  
,
n 2  
1
[(X g QY) (QX g Y)]  
,
2(n 1)  
1
0(X,Y) R(X,Y)   
0(X,Y) R(X,Y)   
1(X,Y) R(X,Y)   
(X g QY)  
,
(1.14)  
(1.15)  
(1.16)  
(1.17)  
n 1  
1
(X g QY)  
,
n 1  
1
(X S Y)  
,
,
n 1  
1
1(X,Y) R(X,Y)   
(X S Y)  
n 1  
2(X,Y) R(X,Y)  
1
(QX g Y) (X g QY) (X S Y)  
,
,
(1.18)  
n 2  
2(X,Y) R(X,Y)  
1
(QX g Y) (X g QY) (X S Y)  
(1.19)  
(1.20)  
(1.21)  
(1.22)  
(1.23)  
(1.24)  
n 2  
1
3(X,Y) R(X,Y)   
(Y g QX)  
,
n 1  
1
3(X,Y) R(X,Y)   
(X,Y) R(X,Y)   
(X,Y) R(X,Y)   
7(X,Y) R(X,Y)   
(Y g QX)  
,
n 1  
1
[(X g QY) (X S Y)]  
,
,
n 1  
1
[(X g QY)(X S Y)]  
n 1  
1
[(QX g Y) (X S Y)]  
,
n 1  
94  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
1
7(X,Y) R(X,Y)   
[(QX g Y) (X S Y)]  
,
(1.25)  
(1.26)  
(1.27)  
(1.28)  
(1.29)  
(1.30)  
(1.31)  
(1.32)  
(1.33)  
n 1  
1
4(X,Y)Z R(X,Y)Z   
4(X,Y)Z R(X,Y)Z   
6(X,Y)Z R(X,Y)Z   
6(X,Y)Z R(X,Y)Z   
8(X,Y)Z R(X,Y)Z   
8(X,Y)Z R(X,Y)Z   
9(X,Y)Z R(X,Y)Z   
[g(X,Z)QY g(X,Y)QZ]  
[g(X,Z)QY g(X,Y)QZ]  
[S(X,Z)QY g(X,Y)QZ]  
,
,
n 1  
1
n 1  
1
,
n 1  
1
[S(Y,Z)X g(X,Y)QZ]  
,
n 1  
1
[S(Y,Z)X g(X,Y)Z]  
,
,
n 1  
1
[S(Y,Z)X g(X,Y)Z]  
n 1  
1
[S(X,Y)Z g(Y,Z)QX]  
,
n 1  
1
9(X,Y)Z R(X,Y)Z   
[S(X,Y)Z g(Y,Z)QX],  
n 1  
where  
(X D Y)Z D(Y,Z)X D(X,Z)Y  
.
for allX,Y,Z (M), where  
and is the scalar curvature.  
R
is the Riemannian curvature tensor of type (1, 3)  
r
Definition 1.2: A para-Sasakian manifold M is called an Einstein  
manifold if its Ricci tensor is of the form  
S(Y,Z) kg(Y,Z)  
for all Y,Z (M), where k being a scalar.  
,
Definition 1.3: A para-Sasakian manifold M is called an  
-Einstein  
manifold if its Ricci tensor is of the form  
S(Y,Z) l1g(Y,Z)l2(Y)(Z)  
,
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
95  
for all Y,Z (M), where l1 l2 are scalars.  
,
Definition 1.4: A para-Sasakian manifold M is called a generalized  
-Einstein manifold if its Ricci tensor is of the form  
S(Y,Z) k1g(Y,Z)k2(Y)(Z)k3g(Y,Z)  
,
for all Y,Z (M), where k1  
,
k2 and k3 are scalars.  
This paper is structured as follows:  
First two sections of the paper has been kept for introduction and  
preliminaries. In Section-3, we study properties of para-Sasakian manifold with  
respect to SVK-connection. In Section-4, we introduce conformal Ricci soliton on  
para-Sasakian manifold with respect to SVK-connection. In Section-5, we study  
conformal  
η-Ricci  
soliton  
on  
para-Sasakian  
manifold  
with  
respect  
to  
SVK-connection. Section-6 concerns with conformal η-Ricci soliton with respect to  
SVK-connection on para-Sasakian manifolds of class C1  
,
C2  
,
C3 and C4  
.
2. Preliminaries  
Let  
M
be an  
n
-dimensional differentiable manifold with structure (,,)  
,
where  
is a 1-form,  
is the structure vector field,  
is a (1, 1)-tensor field  
satisfying [28]  
2(X) X (X), () 1  
(2.1)  
(2.2)  
() 0,0  
,
for all vector field  
X
on  
M
is called almost paracontact manifold. If an almost  
paracontact manifold  
M with structure (,,) admits a pseudo-Riemannian metric  
g
such that [39]  
g(X,Y) g(X,Y) (X)(Y)  
,
(2.3)  
then we say that  
M
is an almost paracontact metric manifold with an almost  
paracontact metric structure (,,,g). From (2.3) one can deduce that  
96  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
g(X,Y)  g(X,Y)  
g(X,) ()  
An almost paracontact metric structure of  
,
(2.4)  
(2.5)  
.
M
becomes a paracontact metric  
structure [39] if  
g(X,Y) d(X,Y)  
,
for all vector fields  
X
,Y  
on  
M
, where  
1
d(X,Y) {X(Y) Y(X) ([X,Y])}.  
2
The manifold  
M
is called a para-Sasakian manifold if  
(X)Y  g(X,Y)(Y)X  
for any smooth vector fields on  
In a para-Sasakian manifold the following relations also hold [39]  
(X)Y g(X,Y),XX  
(R(X,Y)Z) g(X,Z)(Y) g(Y,Z)(X)  
R(X,Y)(X)Y (Y)X  
R(,X)Y g(X,Y)(Y)X  
R(X,)Y g(X,Y)(Y)X  
R(,X)X (X)  
S (X,)   (n 1)(X)  
S (,)   (n 1), Q  (n 1)  
S (X, Y) S (X,Y) (n 1)(X)(Y)  
,
(2.6)  
X
,Y  
M
.
,
(2.7)  
(2.8)  
,
,
(2.9)  
,
(2.10)  
,
(2.11)  
,
(2.12)  
(2.13)  
,
,
(2.14)  
(2.15)  
.
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
for any smooth vector fields X,Y,Z on  
3. Schouten-Van Kampen Connection on Para-Sasakian Manifolds  
97  
M
.
In this section we get the relation between SVK-connection and Levi-Civita  
connection on para-Sasakian manifold . Then we obtain Rie-mannian curvature  
tensor, Ricci curvature tensor, Ricci operator and scalar curvature of with respect  
to the SVK-connection. We also establish here the first Bianchi identity with respect  
to SVK-connection on  
M
M
M
.
In view of (1.1), (2.7) and (2.5), we get the expression for SVK-connection  
in a para-Sasakian manifold as  
M
XY  XY g(X,Y)(Y)X  
,
(3.1)  
with torsion tensor  
T (X,Y) 2g(X,Y)(Y)X (X)Y  
.
On para-Sasakian manifold the connection  
has the following properties  
X0,(X)Y g(X,Y)  
,
(3.2)  
(3.3)  
(Xg)(Y,Z) g(X,Y)(Z)g(Y,Z)(X)  
.
for all X,Y (M)  
.
Proposition 3.1: The SVK-connection on a para-Sasakian manifold is  
non metric compatible connection.  
Proposition 3.2: The SVK-connection on a para-Sasakian manifold is  
non symmetric connection.  
Proposition 3.3: The structure vector field of a para-Sasakian manifold  
is parallel with respect to SVK-connection.  
Let  
R
be the Riemannian curvature tensor with respect to SVK-connection  
on a para-Sasakian manifold defined as  
98  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
R(X,Y)Z  XYZ Y XZ  [X,Y ]  
Z
.
(3.4)  
Then using (2.6), (2.7) and (3.1) in (3.4) we get  
R(X,Y)Z R(X,Y)Z g(Y,Z)(X)g(X,Z)(Y)  
g(X,Z)Y g(Y,Z)X  
(Y)(Z)X (X)(Z)Y  
.
(3.5)  
Writing the equation (3.5) by cyclic permutations of X,Y and  
Z
and using  
the fact that R(X,Y)Z R(Y,Z)X R(Z,X)Y 0 , we have  
R(X,Y)Z R(Y,Z)X R(Z,X)Y 0  
,
for all X,Y,Z (M)  
.
Taking inner product of (3.5) with a vector field  
we get  
U
and contracting over  
X
and  
U
S (Y,Z) S(Y,Z) (n 1)(Y)(Z) g(Y,Z)  
,
(3.6)  
where  
S
denotes Ricci curvature tensor with respect to  
and trace()  
.
Proposition 3.4: The SVK-connection on para-Sasakian manifold  
satisfies the first Bianchi identity.  
Lemma 3.5: Let M be an n-dimensional para-Sasakian manifold  
admitting SVK-connection, then  
R(X,Y)0, R(,Y)Z 2 [g(Y,Z)(Z)Y]  
,
(3.7)  
(3.8)  
(3.9)  
R(X,)Z 2[g(X,Z)(Z)X]  
,
S (X,) 0 S (,Y)  
,
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
99  
QX QX (n 1)(X)X, Q0  
,
(3.10)  
r r (n 1)2  
,
(3.11)  
for all X,Y,Z (M), where  
R
,
Q
and r denote Riemannian curvature  
, respectively.  
tensor, Ricci operator and scalar curvature with respect to  
Remark 3.6: Eigen value of Ricci operator with respect to SVK-  
connection corresponding to the eigen vector ξ is zero.  
4. Conformal Ricci Soliton on Para-Sasakian Manifold with Respect to  
SVK-Connection  
In this section we find a para-Sasakian manifold  
Ricci soliton with respect to SVK-connection in which the potential vector field  
being pointwise collinear with the structure vector field of  
M
admitting conformal  
M
.
Let V  , where is some non-zero smooth function. Taking covariant  
derivative of in the direction of and using (2.7) we get  
V
X
XV X()X  
In view of (3.1) and (4.1) we have  
.
(4.1)  
XV X ()X g(X,V)(V)X  
.
(4.2)  
Writing equation (1.6) with respect to SVK-connection we have  
2   
0 (L g)(X,Y)2S (X,Y) 2p   
g(X,Y)  
V
n
  
g(XV,Y)g(X,YV)  
2   
2S(X,Y)2p   
g(X,Y)  
.
(4.3)  
n
  
Using (4.2) in (4.3) we get  
100  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
0 X ()(Y) Y ()(X)  
2   
2S(X,Y)2p   
g(X,Y)  
.
(4.4)  
(4.5)  
n
  
Setting X and using (3.9) in (4.4) we get  
2   
0 ()(Y)Y ()2p   
(Y)  
.
n
  
Replacing  
Y
by  
in (4.5) we obtain  
 p  
1   
()   
  
.
(4.6)  
(4.7)  
2 n  
  
Using (4.6) in (4.5) we get  
 p  
1   
Y()   
  
.
2 n  
  
If we consider Y() 0 , then equation (4.7) yields  
p
1
  
.
2 n  
Therefore we have the following theorem  
Theorem 4.1: Let M(,,,g) be a para-Sasakian manifold admitting  
conformal Ricci soliton (g,V,) with respect to SVK-connection. If is  
V
pointwise collinear with  
, then  
V
is a constant multiple of  
provided  
p
1
n
  
.
2
Now setting V in (4.3) we have  
2   
0 2S(X,Y)2p   
g(X,Y)  
.
(4.8)  
n
  
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
101  
Using (3.6) in (4.8) we get  
 p  
1   
S (X,Y)   
g(X,Y)  
2 n  
  
(n 1)(X)(Y) g(X,Y)  
.
(4.9)  
Corollary 4.2: If a para-Sasakian manifold  
M
admits conformal Ricci  
is generalized  
soliton (g,,) with respect to SVK-connection, then  
M
-Einstein.  
5. Conformal η-Ricci Soliton on Para-Sasakian Manifold with Respect to  
SVK-Connection  
Writing equation (1.6) with respect to SVK-connection we have  
0 (Lg)(X,Y)2S (X,Y)  
2   
2p   
g(X,Y)2 (X)(Y)  
.
.
(5.1)  
n
  
Expanding (5.1) we get  
0 g(X,Y)g(X,Y)2S (X,Y)  
2   
2p   
g(X,Y)2 (X)(Y)  
(5.2)  
(5.3)  
n
  
Using (3.2) in (5.2) we obtain  
2   
0 2S (X,Y)2p   
g(X,Y) 2 (X)(Y)  
.
n
  
Setting X in (5.3) we have  
p  
1   
  
  
.
(5.4)  
2 n  
Hence, we have the following theorem  
102  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
Theorem 5.1: If an n-dimensional para-Sasakian manifold admits a  
conformal  
-Ricci soliton (g,,,) with respect to SVK-connection, then  
the relation between the soliton scalars are given by  
p  
1   
  
  
.
2 n  
Using (3.6) in (5.3) we obtain  
 p  
1   
S (X,Y)   
g(X,Y)  
2 n  
  
(n 1)(X)(Y) g(X,Y)  
,
(5.5)  
which shows that  
M
is generalized  
-Einstein manifold.  
Corollary 5.2: If an n-dimensional para-Sasakian manifold  
M
contains  
a conformal  
generalized  
-Ricci soliton with respect to SVK-connection, then  
M
is  
-Einstein manifold.  
Contracting (5.5) over  
X
andY  
we get  
n
r (p 2)n 2 2  
.
(5.6)  
2
Corollary 5.3: If an n-dimensional para-Sasakian manifold  
M
contains  
a conformal  
curvature of  
-Ricci soliton with respect to SVK-connection, then the scalar  
M
is given by equation (5.6).  
6. Conformal  
-Ricci Soliton with Respect to SVK-Connection on Equivalence  
Classes C1,C2,C3 and C4  
In this section we consider  
-Ricci soliton (g,,,) with respect to SVK-  
connection on the manifolds belong to the equivalence classes C1,C2,C3 and C4 and  
obtain the relation between the soliton constants.  
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
103  
Conformal η-Ricci soliton with respect to  
on class C1: The condition  
that must be satisfied by the Riemannian curvature tensor (R)is  
(E (,X).R)(Y,Z)V 0  
,
(6.1)  
for all X,Y,Z,V (M)  
Equation (6.1) gives  
E (,X).R(Y,Z)V R(E (,X)Y,Z)V  
.
R(Y,E (,X)Z)V R(Y,Z)E (,X)V  
.
(6.2)  
(6.3)  
Setting V and using (1.10), (2.9)-(2.11) in (6.2) we get  
0 [r n(n 1)][g(X,Y)Z g(X,Z)Y] [r n(n 1)]R(Y,Z)X  
.
Taking an inner product of (6.3) with a vector field  
U
we get  
0 [r n(n 1)] [g(X,Y)g(Z,U) g(X,Z)g(Y,U)]  
[r n(n 1)]g(R(Y,Z)X,U)  
.
(6.4)  
(6.5)  
Contracting (6.4) over  
Z
and  
U
we have  
S (X,Y) (n 1)g(X,Y)  
,
if r n(n 1)  
.
In view of (5.5) and (6.5) we obtain  
 p  
1   
0   
n 1 g(X,Y)  
2 n  
  
(n 1)(X)(Y)g(X,Y)  
.
(6.6)  
(6.7)  
Setting Y in (6.6) we have  
p  
1   
  
  
.
2 n  
104  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
Thus, we have the following theorem:  
Theorem 6.1: Let M(,,,g) be an  
manifold of class C1. If  
-dimensional para-Sasakian  
M
admits a conformal  
-Ricci soliton with respect  
to SVK-connection, then the soliton constants are given by  
p  
1   
  
  
,
2 n  
provided r n(n 1)  
.
Corollary 6.2: A para-Sasakian manifold of class C1 is Einstein  
manifold if r n(n 1)  
.
Corollary 6.3: If a para-Sasakian manifold of class C1 contains  
conformal  
-Ricci soliton with respect to SVK-connection, then the manifold  
is generalized  
-Einstein, provided r n(n 1)  
.
Conformal  
-Ricci Soliton with Respect to  
on Class C2 : The  
condition that must be satisfied by the Riemannian curvature tensor (R) is  
(E (,X).R)(Y,Z)V 0  
,
for all X,Y,Z,V (M)  
R(,X).R(Y,Z)V R(R(,X)Y,Z)V  
R(Y,R(,X)Z)V R(Y,Z)R(,X)V  
.
(6.8)  
Setting V and using (2.8)-(2.11) in (6.8) we get  
0 [g(X,Y)Z g(X,Z)Y]R(Y,Z)X  
.
(6.9)  
Taking an inner product of (6.9) with a vector field  
W
we get  
0 [g(X,Y)g(Z,W) g(X,Z)g(Y,W)]g(R(Y,Z)X,W)  
.
(6.10)  
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
Contracting (6.10) over and we have  
S(X,Y) (n 1)g(X,Y)  
105  
Z
W
,
(6.11)  
In view of (5.5) and (6.11) we obtain  
 p  
1   
0   
n 1 g(X,Y)  
2 n  
  
(n 1)(X)(Y)g(X,Y)  
,
(6.12)  
Setting Y in (6.12) we have  
p  
1   
  
  
.
2 n  
This leads to the following theorem:  
Theorem 6.4: Let M(,,,g) be an n-dimensional para-Sasakian  
manifold of class C2 . If  
M
admits a conformal  
-Ricci soliton with respect  
to SVK-connection, then the soliton constants are given by  
p  
1   
  
  
.
2 n  
Corollary 6.5: A para-Sasakian manifold of class C2 is always Einstein  
manifold.  
Conformal η-Ricci Soliton with Respect to  
on Class C3 : The condition  
that must be satisfied by conformal curvature tensor (C) is  
(R(,X).C)(Y,Z)V 0  
,
(6.13)  
for all X,Y,Z,V (M)  
.
Equation (6.13) gives  
106  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
R(,X).C (Y,Z)V C (R(,X)Y,Z)V  
C (Y,R(,X)Z)V C (Y,Z)R(,X)V  
.
(6.14)  
(6.15)  
Setting V in (6.14) we have  
R(,X).C (Y,Z)C (R(,X)Y,Z)  
C (Y,R(,X)Z)C (Y,Z)R(,X)  
.
Using (1.9), (2.9)-(2.11) in (6.14) and taking inner product of (6.15) with a  
vector field  
U
and then contracting over  
Z
,U  
we get  
2
n r 1  
n n r  
S (X,Y)   
g(X,Y)   
(X)(Y)  
.
(6.16)  
(6.17)  
n 1  
n 1  
In consequence of (5.5) and (6.16) we obtain  
 p  
1   
n r 1  
0   
  
g(X,Y)  
2 n  
n 1  
  
2
n n r  
n 1 (X)(Y) g(X,Y)  
,
n 1  
Setting Y in (6.17) we have  
p  
1   
  
  
,
2 n  
which gives the following theorem:  
Theorem 6.6: Let M(,,,g) be an n-dimensional para-Sasakian  
manifold of class C3 . If  
M
admits a conformal  
-Ricci soliton with respect  
to SVK-connection, then the soliton constants are given by  
p  
1   
  
  
.
2 n  
SOLITONS ON VARIOUS GEOMETRIC STRUCTURES  
107  
Corollary 6.7: a para-Sasakian manifold of class C3 is always an  
-Einstein manifold.  
Conformal η-Ricci soliton with respect to  
on class C4 : The condition  
that must be satisfied by conformal curvature tensor (C)is  
(E (,X).C)(Y,Z)V 0  
,
(6.18)  
for all X,Y,Z,V (M)  
.
Equation (6.13) gives  
E (,X).C (Y,Z)V C (E (,X)Y,Z)V  
C (Y,E (,X)Z)V C (Y,Z)E (,X)V . (6.19)  
Setting V in (6.14) we have  
E (,X).C (Y,Z)C (E (,X)Y,Z)  
C (Y,E (,X)Z)C (Y,Z)E (,X)  
.
(6.20)  
Using (1.9), (1.10), (2.9)-(2.11) in (6.14) and taking inner product of (6.15)  
with a vector field U and then contracting over  
Z
,
U
we get  
2
n r 1  
n n r  
S (X,Y)   
g(X,Y)  
(X)(Y)  
,
n 1  
n 1  
if r n(n 1)  
.
In view of (5.5) and (6.16) we obtain  
 p  
1   
n r 1  
0   
  
g(X,Y)  
2 n  
n 1  
  
2
n n r  
n 1 (X)(Y) g(X,Y)  
,
(6.22)  
n 1  
108  
A. MANDAL, A.H. SARKAR, M. MALLIK, A. DAS, & S.K. DATTA  
Setting Y in (6.22) we have  
p  
1   
  
  
,
2 n  
which gives the following theorem:  
Theorem 6.8: Let M(,,,g) be an n-dimensional para-Sasakian  
manifold of class C4 . If  
M
admits a conformal  
-Ricci soliton with respect  
to SVK-connection, then the soliton constants are given by  
p  
1   
  
  
.
2 n  
provided r n(n 1)  
.
Corollary 6.9: A para-Sasakian manifold of class C4 is an η-Einstein  
manifold if r n(n 1)  
.
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1. Department of Mathematics,  
(Received, September 23, 2024)  
Raiganj Surendranath Mahavidyalaya,  
(Revised, October 7, 2024)  
Raiganj, Uttar Dinajpur-733134, West Bengal, India,  
E-mail: abhijit4791@gmail.com  
3*. Coressponding Author  
Department of Mathematics,  
Raiganj Surendranath Mahavidyalaya,  
Raiganj, Uttar Dinajpur-733134, West Bengal, India,  
E-mail: meghlal.mallik@gmail.com  
2, 4. Department of Mathematics,  
Raiganj University, Raiganj,  
Uttar Dinajpur-733134, West Bengal, India,  
2. E-mail: afsarhsarkar1986@gmail.com  
4. E-mail: ashoke.avik@gmail.com  
5. Department of Mathematics,  
University of Kalyani, Nadia-741235, West Bengal, India,  
E-mail: sanjibdatta05@gmail.com