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Sunitha and Sai Prasad RT&A, № 4 (76)
, , -r Volume 18, December 2023
On a class or LP-Kenmotsu manifolds
ON A CLASS OF LORENTZIAN PARA-KENMOTSU MANIFOLDS ADMITTING QUARTER-SYMMETRIC METRIC CONNECTION
S. Sunitha Devi1, K. L. Sai Prasad2
Department of Mathematics
1 Vardhaman College of Engineering, Hyderabad, Telangana, 501218, INDIA
2 Gayatri Vidya Parishad College of Engineering for Women, Madhurawada,
Visakhapatnam, 530 048, INDIA [email protected],* [email protected]
Abstract
In this present paper, a class of Lorentzian almost paracontact metric manifolds known as the LP-Kenmotsu (Lorentzian para-Kenmotsu) is considered that accepts a connection of quarter-symmetric. In this work, it was found that an LP-Kenmotsu manifold is either <p-symmetric or concircular <p-symmetric with respect to quarter-symmetric metric connection if and only if it is symmetric with respect to the Riemannian connection, provided the scalar curvature ofRiemannian connection is constant.
Keywords: Lorentzian para-Kenmotsu manifold, quarter- symmetric metric connection, concircu-iar curvature tensor.
2010 Mathematics Subject Classification: 53C05, 53C20, 53C50
I. Introduction
In 1989, Matsumoto [4] introduced the notion of Lorentzian paracontact and in particular, Lorentzian para-Sasakian (LP-Sasakian) manifolds. Later, these manifolds have been widely studied by many geometers such as Matsumoto and Mihai [5], Mihai and Rosca [6], Mihai, Shaikh and De [7], Venkatesha and C. S. Baggewadi [13], Venkatesha, Pradeep Kumar and Bagewadi [14,15] and obtained several results on these manifolds.
In 1995, Sinha and Sai Prasad [11] defined a class of almost paracontact metric manifolds namely para-Kenmotsu (briefly P-Kenmotsu) and special para-Kenmotsu (briefly SP- Kenmotsu) manifolds in similar to P-Sasakian and SP- Sasakian manifolds. In 2018, Abdui Haseeb and Rajendra Prasad [1] defined a class of Lorentzian almost paracontact metric manifolds namely Lorentzian para-Kenmotsu (briefly LP- Kenmotsu) manifolds. As an extension, many geometers have studied these Lorentzian para-Kenmotsu manifolds [8, 10,12]. Sai Prasad, Sunitha Devi and Deekshituiu have considered LP-Kenmotsu manifolds admitting the Weyi-projective curvature tensor W2 and shown that these manifolds admitting a Weyi-flat projective curvature tensor, an irrotationai Weyi-projective curvature tensor and a conservative Weyi-projective curvature tensor are an Einstein manifolds of constant scalar curvature [9].
A linear connection v in an n-dimensionai differentiabie manifold is said to be a quarter-
symmetric connection [3] if its torsion tensor T is of the form
T(X, Y) = vxY -vYX - [X, Y] = n(Y)$X - n(X)QY,
where n is a 1-form and ty is a tensor field of type (1,1). In particular, if we replace tyX by X and tyY by Y, then the quarter-symmetric connection reduces to the semi-symmetric connection [2]. Thus, the notion of quarter-symmetric connection generalizes the idea of semi-symmetric connection, and if quarter-symmetric linear connection v satisfies the condition ^vxg^J (Y, Z) = 0 for all
X, Y, Z g x (Mn), where x (Mn) is the Lie algebra of vector fields on the manifold Mn, then v is said to be a quarter-symmetric metric connection.
Motivated by these studies, in the present paper, we study the geometry of Lorentzian para-Kenmotsu (LP-Kenmotsu) manifolds with respect to quarter-symmetric metric connection. The present paper is organized as follows. Section 2 is equipped with some prerequisites about Lorentzian para-Kenmotsu manifolds.
Further on, in relation to the quarter-symmetric metric connection in an Lorentzian para-Kenmotsu manifold, we derive the relations for the Ricci tensor S and the Riemannian curvature tensor R in section 3.
Further in sections 4 and 5, we study the ty-symmetry and concircular ty-symmetry of Lorentzian para-Kenmotsu manifolds with respect to quarter-symmetric metric connection respectively.
II. Preliminaries
An n-dimensional differentiable manifold Mn admitting a (1,1) tensor field contravariant vector field e, a 1-form n and the Lorentzian metric g(X, Y) satisfying
= -1, (2)
<£2 X = X + n (X)e, (3)
g(QX, QY) = g(X, Y)+ n(X)n(Y), (4)
g(X, e ) = n (X), (5)
№ = 0, n(QX) = 0, ranty = n - 1. (6) is called Lorentzian almost paracontact manifold [4].
In a Lorentzian almost paracontact manifold, we have
$(X,Y) = ®(Y,X) where ®(X,Y) = g(<pX,Y). (7)
A Lorentzian almost paracontact manifold Mn is called Lorentzian para-Kenmotsu manifold if [1]
(Vx$) Y = -g (<pX, Y) e - n (Y) QX, (8)
for any vector fields X and Y on Mn, and v is the operator of covariant differentiation with
respect to the Lorentzian metric g.
It can be easily seen that in a LP-Kenmotsu manifold Mn, the following relations hold [1]:
Vx e = -f x = -X - n (X) e, (9)
(Vxn) Y = -g (X, Y) e - n (X) n (Y), (10)
for any vector fields X and Y on Mn.
Also, in an LP-Kenmotsu manifold, the following relations hold [1]:
g(R(X, Y)Z, e) = n(R(X, Y)Z) = g(Y, Z)n(X) - g(X, Z)n(Y) (11)
<2 (VwR) (X, Y) Z = 0, (17)
R(£, X)Y = g(X, Y)£ - n(Y)X, (12)
R(X, Y)£ = n(Y)X - n(X)Y, (13)
S(X,£) = (n - 1)n(X), (14)
S(<X,<Y) = S(X, Y) + (n - 1)n(X)n(Y) (15)
for any vector fields X, Y and Z, where R is the Riemannian curvature tensor and S is the Ricci
tensor of Mn.
Definition 1. An LP- Kenmotsu manifold Mn is said to be symmetric if
(VwR)(X, Y) Z = 0, (16)
for all vector fields X, Y, Z and W.
Definition 2. An LP-Kenmotsu manifold Mn is said to be ^-symmetric if
W
for all vector fields X, Y, Z and W.
Definition 3. An LP-Kenmotsu manifold Mn is said to be concircular symmetric if
(VwC) (X, Y) Z = 0, (18)
for all vector fields X, Y and Z. Here C is the concircular curvature tensor and is given by [16]
r
C (X, Y) Z = R (X, Y) Z - -n^Y) [g (Y, Z) X - g (X, Z) Y], (19)
for all vector fields X, Y and Z, where R and r are the Riemannian curvature tensor and scalar curvature respectively.
Definition 4. An LP-Kenmotsu manifold Mn is said to be concircular ^-symmetric if
W
for all vector fields X, Y, Z and W
<2 (VwC) (X, Y) Z = 0, (20)
III. Expression of R (X, y) z in terms of r (x, y) z
In this section we express RR (X, Y) Z, the curvature tensor with respect to quarter-symmetric metric connection, in terms of R (X, Y) Z which is the curvature tensor with respect to Riemannian connection.
Let v be a linear connection and v be a Riemannian connection of an almost contact metric manifoid Mn such that
vxY = VxY + U (X, Y), (21)
where U is a tensor of type (1,1). For v to be a quarter-symmetric metric connection in Mn, we have
i
U (X, Y) = - [T (X, Y) + T' (X, Y) + T' (Y, X^ (22)
and
g(T'(X, Y), Z)= g(T(Z, X), Y). (23)
From (1) and (23), we get
T(X, Y) = n(Y)QX - g(QX, Y)e. (24)
Using (1) and (24) in (22), we obtain
U (X, Y)= n (Y) QX - g (QX, Y) e.
Thus the quarter-symmetric metric connection v in an LP-Kenmotsu manifold is given by
VxY = VxY + n (Y) QX - g (QX, Y) e, (25)
which is the relation between Riemannian connection and the quarter-symmetric metric connection on Lorentzian para-Kenmotsu manifolds.
Similarly, on simplication, we obtain the relation between the curvature tensor R (X, Y) Z of Mn with respect to the quarter-symmetric metric connection v and the curvature tensor R (X, Y) Z of Riemannian connection v as follows:
R(X, Y)Z = R(X, Y)Z + [g(QY, Z) + g(Y, Z)e]QX]
- [g(QX, Z)+ g(X, Z)e]QY (26)
+ Xg(QY, Z) - Yg(QX, Z).
Then from (26), it follows that
S (Y, Z)= S (Y, Z), (27)
where S and S are the Ricci tensors of the connections v and v respectively.
Further contracting (27), we get
r = r, (28)
where r and r are the scalar curvatures of the connections v and v respectively.
IV. Symmetry of LP-Kenmotsu manifold with respect to quarter-symmetric metric connection
By the definition of symmetric LP-Kenmotsu manifold with respect to Riemannian connection, we define a symmetric LP-Kenmotsu manifold with respect to quarter-symmetric metric connection by
fVwR) (X, Y) Z = 0, (29)
(30)
where
(vwR) (X, Y) Z = vW (R (X, Y) Z) - R ( (vwX, y) Z
- R ( (X, vwY) z) - R ((X, Y) VWZ^ ,
for all vector fields X, Y, Z and w.
vW (R (X, Y) Z) = Vw (R (X, Y) Z) + n (R (X, Y) Z) QW
- g (QW, (R (X, Y) Z)) e,
R ( (vwX, Y^Z = R (VwX, Y) Z + n (X) R (QW, Y) Z
- g (QW, X) R (e, Y) Z,
R ( (x, vw^ z) = R (X, VwY) Z + n (Y) R (X, QW) Z
- g (QW, Y) R (X, e) Z,
(31)
(32)
R ((X, Y) vwz) =R (X, Y) VWZ + n (Z) R (X, Y) <W
(34)
g (<pW, Z) R (X, Y) l,
R(l, Y)Z = g(Y, Z)l - n(Z)Y - n(Z)<Y£ + g(<Y, Z)l, (35)
R(X, l)Z = n(Z)X - g(X, Z)l - n(z)<xi + g(<X, Z)l, (36)
R(X, Y)l = n(Y)X - n(X)Y + n(Y)<Xl - n(X)<Yl. (37) By using (25), (31) to (37) in (30), we get
(vwR) (X, Y) Z = (VwR) (X, Y) Z + n (R (X, Y) Z) <W
- g (<W, R (X, Y) Z) l - n (X) R (<W, Y) Z
- n (Y) R (X, <W) Z + g (W, <X) R (l, Y) Z (38) + g (W, <Y) R (X, l) Z + g (W, <Z) R (X, Y) l
- n (Z) R (X, Y) <W.
Then by differentiating (26) with respect to W and on using (6), (7) and (10), we get
(VwR)(X,Y)Z = (VwR)(X,Y)Z - [g(<W,Y)n(Z) + g(<W,Z)n(Y) + Wg(Y, Z) + n(W)g(Y, Z)l]<X + [g(<W, X)n(Z) + g(<W, Z)n(X) + Wg(X, Z)+ n(W)g(X, Z)l]<Y
+ g(<X, Z)[g(<W, Y)l + n(Y)<W] (39)
- g(<Y, Z)[g(<W, X)l + n(X)<W ] + [n(Y)g(X, Z)l - n(X)g(Y, Z)l]<W + [Yg(<W, X) - Xg(<W, Y)]n(Z) + g(<W, Z)[n(X)Y - n(Y)X]. Therefore, by using (2), (8) and (39) in (38), we obtain
(VwR) (X, Y) Z = (VwR) (X, Y) Z. (40)
Thus we can state the following:
Theorem 1. An LP-Kenmotsu manifold is symmetric with quarter-symmetric metric connection v if and only if it is symmetric with respect to Riemannian connection v.
Corollary 1. An LP-Kenmotsu manifold is ^-symmetric with respect to quarter-symmetric metric connection v if and only if it is symmetric with respect to Riemannian connection v.
V. Concircular symmetry of LP-Kenmotsu manifold with respect to quarter-symmetric metric connection
An LP-Kenmotsu manifold Mn is said to be a concircular symmetric with respect to quarter-symmetric metric connection if
(VwC)(X, Y)Z = 0, (41)
for all vector fields X, Y, Z and W. Here the concircular curvature tensor C with respect to quarter-symmetric metric connection is given by
?(X, Y)Z = R(X, Y)Z - [g(Y, Z)X - g(X, Z)Y], (42)
where R is the Riemannian curvature tensor and r is the scalar curvature of the quarter-symmetric metric connection v.
Using (25), we can write
(v wC)(X, Y)Z = (VW£)(X, Y)Z + n(£(X, Y)Z)$W
- g($(Z(X, Y)Z, W))£ - n(X)Z(#W, Y)Z
- n(Y)C(X, )Z - n(Z)C(X, Y)$W (43) + g($W, X)C(£, Y)Z + g($W, Y)C(X, £ )Z
+ g($W, Z)C(X, Y)£. Now on differentiating (42) with respect to W, we obtain
(VwC)(X, Y)Z = (VwR)(X, Y)Z - -V-Y)[g(Y, Z)X - g(X, Z)Y]. (44)
Therefore, by using of (28) and (39), we get from (44) that
(VwC)(X, Y)Z = (VwR)(X, Y)Z - [g(<pW, Y)n(Z) + g($W, Z)n(Y) + Wg(Y, Z) + n(W)g(Y, Z)£]$X + [g($W, X)n(Z) + g(#W, Z)n(X) + Wg(X, Z) + n(W )g(X, Z)£]$Y + g($X, Z)[g($W, Y)£ + n(Y)$W]
- g($Y, Z)[g(<pW, X)£ + n(X)<pW] (45)
+ [n(Y)g(X, Z)£ - n(X)g(Y, Z)£]$W + [Yg(4>W, X) - Xg($W, Y)]n(Z) + g($W,Z)[n(X)Y - n(Y)X] Vwr
-[g(Y, Z)X - g(X, Z)Y].
(46)
n(n -1)
Then by making use of (19), we rewrite the above equation (45) as
(VwC)(X, Y)Z = (VwC)(X, Y)Z - [g(0W, Y)n(Z) + g($W, Z)n(Y) + Wg(Y, Z) + n(W)g(Y, Z)^X + [g($W, X)n(Z) + g(#W, Z)n(X) + Wg(X, Z)+ n(W)g(X, Z)£]$Y + g(#X, Z)[g(#W, Y)Z + n(Y)$W] - g($Y, Z)[g(#W, X)£ + n(X)$W ] + [n(Y)g(X, Z)£ - n(X)g(Y, Z)£]$W + [Yg($W, X) - Xg($W, Y)]n(Z) + g($W, Z)[n(X)Y - n(Y)X].
Using (2), (6) and (46) in (43), we get
(VwC)(X, Y)Z = (VwC)(X, Y)Z. (47)
Hence we can state the following:
Theorem 2. An LP-Kenmotsu manifold is concircular symmetric with respect to v if and
only if it is so with respect to Riemannian connection v.
Corollary 2. An LP-Kenmotsu manifold is concircular ^-symmetric with respect to v if and only if it is so with respect to Riemannian connection v.
Now taking (2), (6) and (45) in (43), we get
(VW£)(X, Y)Z = (VWR)(X, Y)Z - -V-^W, Z)X - g(X, Z)Y]. (48)
If scalar curvature r is constant, the above equation (48) reduces to
(V wC)(X, Y)Z = (VwR)(X, Y)Z. (49)
Thus we have the following assertion.
Theorem 3. An LP-Kenmotsu manifold is concircular symmetric with respect to quarter-symmetric metric connection v if and only if it is symmetric with respect to Riemannian
connection v, provided the scalar curvature r is constant.
Corollary 3. An LP-Kenmotsu manifold is concircular ^-symmetric with respect to quarter-symmetric metric connection v if and only if it is symmetric with respect to Riemannian
connection v, provided the scalar curvature r is constant.
VI. Conclusion
We explore a class of Lorentzian almost paracontact metric manifolds known as the Lorentzian para-Kenmotsu that accepts a quarter-symmetric connection. In relation to the quarter-symmetric metric connection, the relations for the Ricci tensor and the Riemannian curvature tensor in a Lorentzian para-Kenmotsu manifold were derived. Further, it was found that an LP-Kenmotsu manifold is either ^-symmetric or concircular ^-symmetric with respect to quarter-symmetric metric connection if and only if it is symmetric with respect to the Riemannian connection, provided the scalar curvature of Riemannian connection is constant. The paper ends with a handful of bibliography.
Acknowledgements: The authors acknowledge Dr. A. Kameswara Rao, Assistant Professor of G. V. P. College of Engineering for Women for his valuable suggestions in preparation of the manuscript.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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