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ON CERTAIN CLASSES OF CONFORMALLY FLAT LORENTZIAN PARA-KENMOTSU MANIFOLDS
K. L. Sai Prasad1, P. Naveen2 and S. Sunitha Devi3'*
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Department of Mathematics 1 Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, 530 048, INDIA 2 Rajiv Gandhi University of Knowledge Technology, Srikakulam -532402, INDIA 3'* KL University, Vijayawada, Andhra Pradesh, 520002, INDIA
klsprasad@yahoo.com1 potnurul23@gmail.com2 sunithamallakula@yahoo.com3'*
Abstract
In this present paper, we classify and explore the geometrical significance of a class of Lorentzian almost paracontact metric manifolds namely Lorentzian para-Kenmotsu (briefly LP-Kenmotsu) manifolds whenever the manifolds are either conformally flat or conformally symmetric. It was found that a conformally flat LP-Kenmotsu manifold is of constant curvature and a conformally symmetric LP-Kenmotsu manifold is locally isomorphic to a unit sphere. At the end, we obtain the scalar curvature of <p-conformally flat LP-Kenmotsu manifolds.
Keywords: Lorentzian para-Kenmotsu manifold, Weyl-conformal curvature tensor, Riemanni-nan curvature tensor, ^-conformal curvature tensor, ^-Einstein manifold.
2010 Mathematics Subject Classification: 53C05, 53C07, 53C15
I. Introduction
In 1995, Sinha and Sai Prasad [ll] 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 1989, K. Matsumoto [3] 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 Mat-sumoto and Mihai [4], Mihai and Rosca [5], Mihai, Shaikh and De [6], Venkatesha and Bagewadi [13], Venkatesha, Pradeep Kumar and Bagewadi [14,15].
In 2018, Abdul Haseeb and Rajendra Prasad defined a class of Lorentzian almost paracontact metric manifolds namely Lorentzian para-Kenmotsu (briefly LP-Kenmotsu) manifolds [l, 2] and they studied ^-semisymmetric LP-Kenmotsu manifolds with a quarter-symmetric non-metric connection admitting Ricci solitons [7, 8]. As an extension, Sai Prasad et al., [9] have studied LP-Kenmotsu manifolds admitting the Weyl-projective curvature tensor of type (l, 3). Further, they also have studied and shown that the LP-Kenmotsu manifolds admitting both irrotational and conservative pseudo-projective curvature tensors are Einstein manifolds of constant scalar curvature [10].
In 2023, Sunitha and Sai Prasad [12] have defined a class of Lorentzian para-Kenmotsu manifolds admitting a quarter-symmetric metric connection and shown that it 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. Recently, Rao, Sunitha and Sai Prasad [16] have studied <-conharmonically flat and <-projectively flat LP-Kenmotsu manifolds. They have shown that <-conharmonically flat LP-Kenmotsu manifold is an ^-Einstein manifold with zero-scalar curvature and < -projectively flat LP-Kenmotsu manifold is an Einstein manifold with the scalar curvature r = n(n — 1).
In this work we explore a class of conformally flat Lorentzian para-Kenmotsu (LP-Kenmotsu) manifolds. The following is the layout of the current paper: Following the introduction, Section 2 includes some preliminaries on Lorentzian para-Kenmotsu manifolds. In section 3, we study conformally flat Lorentzian para-Kenmotsu manifolds and shown that they are of constant curvature. Further in section 4, we study and have shown that Lorentzian para-Kenmotsu manifold satisfying the condition R(X,Y).C = 0 is locally isomorphic to a unit sphere Sn(1). Finally in section 5, it is shown that <-conformally flat LP-Kenmotsu manifold is an ^-Einstein manifold with the scalar curvature r = n(n — 1).
II. Preliminaries
An n-dimensional differentiable manifold Mn admitting a (1,1) tensor field <, contravariant vector field £, a 1-form q and the Lorentzian metric g(X, Y) satisfying
n(£) = —1, (1)
<2 X = X + q(X)£, (2)
g(<X, <Y) = g(X, Y) + q(X)q(Y), (3)
g(X, £) = n(X), (4)
<£ = 0, n(<X) = 0, rank < = n — 1 (5) is called Lorentzian almost paracontact manifold [3].
In a Lorentzian almost paracontact manifold, we have
$(X, Y) = $(Y, X), where $(X, Y) = g(<X, Y). (6)
A Lorentzian almost paracontact manifold Mn is called Lorentzian para-Kenmotsu manifold if [1]
(Vx <)Y = —g(<X, Y)£ — n (Y)<X, (7)
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 £ = —<2 X = —X — q(X)£, (8)
(VX n)Y = —g(X, Y)£ — n(X)n(Y), (9)
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, £ ) = n(R(X, Y)Z)
= g(Y, Z)n(X) — g(X, Z)n(Y),
R(£, X)Y = g(X, Y)£ - n(Y)X, (11)
R(X, Y)£ = n(Y)X - n(X)Y, (12)
S(X,£) = (n - 1)n(X) (13)
S^X,фY) = S(X, Y) + (n - 1)n(X)n(Y), (14)
S(X, Y) = ag(X, Y) + bn(X)n(Y), (15) for any vector fields X, Y and Z and S is the Ricci tensor of Mn.
The Riemannian Christoffel curvature tensor R of type (1,3) is given by:
R(X, Y)Z = VxVyZ - VYVXZ - V[x,Y\Z, (16)
where V be its Levi-Civita connection.
III. LP-Kenmütsü manifolds with C(X, Y)Z = 0
In this section, we consider conformally flat Lorentzian para-Kenmotsu manifolds.
The Weyl-conformal curvature tensor C(X, Y)Z is given by
C(X, Y)Z = R(X, Y)Z 1
- (^2)[g(Y, Z)QX - g(X, Z)QY + S(Y, Z)X - S(X, Z)Y] ^
r
+ (n - 1)(n - 2) [g(Y,Z)X - g(X,ZY
where
S(X, Y) = g(QX, Y).
Using (16), we get from (17)
1
R(X, Y)Z = ¡n^2) [g(Y, Z)QX - g(X, Z)QY + S(Y, Z)X - S(X, Z)Y] r
+ (n - 1)(n - 2) [g(Y,Z)X - g(X,Z)Y] By taking Z = £ in (18) and on using (4), (12) and (13), we get
1 (n 1)
n(Y)X - n(X)Y = [n(Y)QX - n(X)QY] + [n(Y)X - n(X)Y]
r
[n(Y)X - n(X)Y].
(n - 1)(n - 2) Taking Y = £ and using (1), we get
(18)
(19)
QX = - 0 X +(n-i - 0 (20)
It shows that the manifold is n-Einstein. Further on contracting (20), we have
r = n(n - 1). (21)
Now, by using (21) in (20), we get
QX =(n - 1) X. (22)
Then by putting (22) in (19), we get
R(X, Y)Z = g(Y, Z)X - g(X, Z)Y. (23)
Thus, a conformally flat LP-Kenmotsu manifold is of constant curvature. The value of this constant is +1. Hence, we can state
Theorem 1. A conformally flat LP-Kenmotsu manifold is locally isometric to a unit sphere Sn (1).
IV. LP-Kenmotsu manifold satisfying R(X, Y).C = 0 Using (4), (11) and (13) we find from (17) that
1 r
n(C(X, Y)Z) = — К^ - 1) (g(Y, Z)n(X) - g(X, Z)n(Y)) ^
- (S(Y,Z)n(X) - S(X,Z)n(Y))]. Putting Z = £ in (24) and on using (4), (13) we get
n(C(X, Y)£ ) = 0. (25)
Again, taking X = £ in (24), we get
1
n(C(£, Y)Z) = — [S(Y, Z) + (n - 1)n(Y)n(Z)] 1 ( r
(ПЗГ - l) [g(Y,Z) + n(Y)n(Z)].
(26)
(27)
n-2\n- 1
Now,
(R(X, Y)C)(U, V)W = R(X, Y)C(U, V)W - C(R(X, Y)U, V)W - C(U,R)(X,Y)V)W - C(U, V)R(X, Y)W. Using R (X, Y) .C = 0, we find from above that
g[R(Ç, Y)C(U, V)W,£] - g[C(R(ï, Y)U, V)W, Ç] - g[C(U, R(£, Y)V)W, £] - g[C(U, V)R(£, Y)W, £] = 0. Using (4) and (11) we get
- C(U, V, W, Y) - n(Y)n(C(U, V)W) - g(Y, U)n(C(£, V)W)
+ n(U)n(C(Y, V)W) - g(Y, V)n(C(U,£)W) + n(V)n(C(U, Y)W) (28)
- g(Y, W )n (C(U, V )£ ) + n (W )n (C(U, V )Y) = 0,
where
C(U, V, W, Y)= g(C(U, V)W, Y).
Putting U = Y in (28), we get
- C(U, V, W, U) - n(U)n(C(U, V)W)+ n(U)n(C(U, V)W)
+ n(V)n(C(U, U)W) + n(W)n(C(U, V)U) - g(U, U)n(C(£, V)W) (29)
- g(U, V)(C(U, £)W) - g(U, W)n(C(U, V)£) = 0.
Let [ei: i= 1,...,n} be an orthonormal basis of the tangent space at any point, then the sum for 1<i<n of the relations (29) for U = ei gives
(1 - n)n(C(£, V)W) = 0,
which implies
n(C(g, V)W) = 0 as n > 3. (30)
Using (25) and (30), (28) takes the form
- C(U, V, W, Y) - n(Y)n(C(U, V)W) + n(U)n(C(Y, V)W) + n(V)n(C(U, Y)W) + n()n(C(U, V)Y) = 0. (31)
Using (24) in (31) we get
-C(U, V, W, Y) + n(W) — J (-r— - 1) (n(U)g(V, Y) - n(V)g(U, Y))
n - 2L\n - 1 J (32)
- (n(U)S(V, Y) - n(V)S(U, Y))
In virtue of (30), (26) reduces to
0.
S(Y, Z) = - 1)g(Y,Z) + (^ - n)n(Y)n(Z). (33)
Using (33), (31) reduces to
C(U, V, W, Y) = 0, (34)
which proves that the manifold is conformally flat. Hence, by using the Theorem 1, we state
Theorem 2. If in an LP-Kenmotsu manifold Mn(n> 3) the relation R(X, Y).C= 0 holds, then it is locally isometric with a unit sphere Sn(1).
For a conformally symmetric Riemannian manifold, we have VC= 0. Hence for such a manifold R(X, Y).C= 0 holds. Thus, we have the following corollary of the above theorem.
Corollary 1. A conformally symmetric LP-Kenmotsu manifold Mn (n> 3) is locally isometric with a unit sphere Sn(1).
V. Q-conformally flat LP-Kenmotsu manifold
Let C be the Weyl conformal curvature tensor of Mn. Since at each point peMn the tangent space T (Mn) can be decomposed into the direct sum Tp (Mn) =$ (Tp (Mn)) ®L (£v), where L (£v) is a 1-dimensional linear subspace of Tp (Mn) generated by gp, we have a map:
C :Tp (Mn) xTp (Mn) xTp (Mn) ^q> (Tp (Mn)) ©L (gp)
It may be natural to consider the following particular cases:
1. C:Tp (Mn) x Tp (Mn) x Tp (Mn) -^L (gp), that is, the projection of the image of C in Q (Tp(Mn)) is zero.
2. C:Tp )Mn) xTp (Mn) xTp (Mn) (Tp (Mn)), that is, the projection of the image of C in L (gp) is zero.
3. C:Q (Tp (Mn)) xQ (Tp (Mn)) xQ (Tp (Mn)) —L (gp), that is, when C is restricted to
(Tp (Mn)) xQ {Tp (Mn)) xq {Tp (Mn)), the projection of the image of C in Q (Tp (Mn)) is zero. This condition is equivalent to
Q2C(QX, qY)QZ= 0. (35)
Definition 1. A differentiable manifold (Mn,g), n> 3, satisfying the condition (35) is called <-conformally flat.
Now our aim is to find the characterization of LP-Kenmotsu manifolds satisfying the condition (35).
Theorem 3. Let Mn be an n-dimensional, (n> 3),^-conformally flat LP-Kenmotsu manifold. Then Mn is an ^-Einstein manifold.
Proof. Suppose that (Mn,g), n> 3, is a ^-conformally flat LP- Kenmotsu manifold. It is easy to see that <2C(<X, <Y)<Z= 0 holds if and only if g(C(<X, <Y)<Z, yW) = 0 for any X, Y, Z, Wex (Mn). So, by the use of (17), <-conformally flat means
1
g(R(<X, <Y)<Z, <W) = — [g(<Y, <Z)S(<X,<W) - g(<X, <Z)S(<Y, <W)
+ g(QX, QW)S(QY, QZ) - g(QY, QW)S(QX, QZ)] [g(QY, QZ)g(QX, QW )
r (36)
(n - 1)(n - 2)' - g(QX, QZ)g(QY, QW )].
Let {e1,... ,en-1, £} be a local orthonormal basis of vector fields in Mn. Using that {Qei,... ,Qen-1, £} is also a local orthonormal basis, if we put X=W=ei in (36) and sum up with respect to i, then
n-i i n-i
£ g(R(Qei,QY)QZ,Qei) = — £ [g(QY,QZ)S(Qei,Qei)
i=1 i=1
- g(Qei, QZ)S(QY, Qei ) + g(Qei, Qe( )S(QY, QZ)
- g(QY, Qei)S(Qei, QZ)] (37)
r n-i
(n - 1)(n - 2) £[g(QY,QZ)g(Qei, Qei)
- g(Qeu QZ)g(QY, Qei )].
It can be easily verified that n-1
£ g(R(Qeu QY)QZ, Qe,) = S(QY, QZ) + g(QY,QZ), (38)
i=1
n-1
£ S (Qei, Qei)=r+n-1, (39)
i=1
n-i
£ g (Qei, QZ) S (QY, Qe,)=S (QY, QZ), (40)
i=i
n-i
£ g (Qei, Qei )=n+1, (41)
i=i
n-1
£ g (Qei, QZ) g (QY, çe,)=g(QY, QZ). (42)
i=1
So, by virtue of (38)-(42) the equation (37) can be written as
and
S (QY, QZ)=(--1) g (QY, QZ).
(43)
Then by making use of (3) and (14), the equation (43) takes the form
S
(Y,Z) = (— -1) g (Y'Z) + (—n (Y) V (Z)
(44)
Therefore from (44), by contraction, we obtain
r = n(n — 1).
(45)
Then by substituting (45) in (44), we get
This completes the proof of the theorem.
VI. Conclusion
The present work explores the geometrical significance of a new class of Lorentzian paracontact metric manifolds namely the Lorentzian para-Kenmotsu manifolds whenever these manifolds are either conformally symmetric or conformally flat. The concepts and various geometrical properties of these manifolds can be applied in various aspects of Applied Mathematics such as Computational Fluid Dynamics, in designing the Super Resolution Sensors in Communications Engineering, and also in the field of General Theory of Relativity.
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.
Conflicts of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] Abdul Haseeb and Rajendra Prasad. Certain results on Lorentzian para-Kenmotsu manifolds. Bulletin of Parana's Mathematical Society, (2018), 201-220.
[2] A. Friedmann and J. A. Schouten. Uber die Geometric der halbsymmetrischen Ubertragung. Math. Zeitschr, 21 (1924), 211-223.
[3] K. Matsumoto. On Lorentzian Paracontact manifolds. Bulletin of the Yamagata University Natural Science, 12 (2)(1989), 151-156.
[4] K. Matsumoto and I. Mihai. On a certain transformation in a Lorentzian para-Sasakian manifold. Tensor, N.S., 47(1988), 189-197.
[5] I. Mihai and R. Rosca. On Lorentzian P-Sasakian manifolds. Classical Analysis, World Scientific Publ., Singapore,(1992), 155-169.
[6] I. Mihai, A. A. Shaikh, U.C. De. On Lorentzian para-Sasakian manifolds. Rendicontidel Seminario Mathematico de Messina, Serie II (1999), 75-77.
[7] Rajendra Prasad, Abdul Haseeb and Umesh Kumar Gautam. On Q-semisymmetric LP Ken-motsu manifolds with a QSNM-connection admitting Ricci solitons. Kragujevac Journal of Mathematics, 45(5)(2021), 815-827.
[8] Rajendra Prasad, Shashikant Pandey and Abdul Haseeb. On a Lorentzian Sasakian manifold endowed with a quarter-symmetric metric connection. Sciendo, Seria Matematica, Informatica, LVII, 2 (2019), 61-76.
[9] K. L. Sai Prasad, S. Sunitha Devi and G. V. S. R. Deekshitulu. On a class of Lorentzian para-Kenmotsu manifolds admitting the Weyl-projective curvature tensor of type (1, 3). Italian Journal of Pure and Applied Mathematics, 45(2021), 990-1001.
References
[10] K. L. Sai Prasad, S. Sunitha Devi and G. V. S. R. Deekshitulu. On a class of Lorentzian paracontact metric manifolds. Italian Journal of Pure and Applied Mathematics, 49(2023), 514-527.
[11] B. B. Sinha and K. L. Sai Prasad. A class of almost paracontact metric Manifold. Bulletin of the Calcutta Mathematical Society, 87(1995), 307-312.
[12] S. Sunitha Devi and K. L. Sai Prasad. On a class of Lorentzian para-Kenmotsu manifolds admitting quarter-symmetric metric connection. Reliability Theory and Applications, 18(2023), no.4(76), 525-532.
[13] Venkatesha and C. S. Bagewadi, On concircular ^-recurrent LP-Sasakian manifolds. Differ. Geom. Dyn. Syst., 10(2008), 312-319.
[14] Venkatesha, K. T. Pradeep Kumar and C. S. Bagewadi. On Lorentzian para-Sasakian manifolds satisfying W2 curvature tensor. IOSR J. of Mathematics, 9(2014), no. 6,124-127.
[15] Venkatesha, K. T. Pradeep Kumar and C. S. Bagewadi. On quarter-symmetric metric connection in a Lorentzian para-Sasakian manifold. Ajerbaijan Journal of Mathematics, 5(2015), no. 1, 3-12.
[16] I. V. Venkateswara Rao, S. Sunitha Devi and K. L. Sai Prasad. On <-conharmonically flat Lorentzian para-Kenmotsu manifolds. Reliability Theory and Applications, 19(2024), no.1(77), 764-769.
