ON φ-CONHARMONICALLY FLAT LORENTZIAN PARA-KENMOTSU MANIFOLDS Текст научной статьи по специальности «Химические науки»

ON f-CONHARMONICALLY FLAT LORENTZIAN PARA-KENMOTSU MANIFOLDS

I. V. Venkateswara Rao1, S. Sunitha Devi2 and K. L. Sai Prasad3^

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Department of Mathematics 1 P. B. Siddhart ha College of Arts and Science,V ijayawada, Andhra Pradesh, India 2 KL University, Vijayawada, Andhra Pradesh, 520002, INDIA 3,t Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, 530 048, INDIA

venkat_inturi@r ediffmail.com 1 sunithamallakula@y ahoo.com 2 klsprasad@y ahoo.com 3,t

Abstract

The present paper deals with a class of Lorentzian almost paracontact metric manifolds namely Lorentzian para-Kenmotsu (briefly LP-Kenmotsu) manifolds. We study and have shown that a quasi-conformally flat Lorentzian para-Kenmotsu manifold is locally isomorphic with a unit sphere Sn (1). Further it is shown that an LP-Kenmotsu manifold which is f-conharmonically flat is an y-Einstein manifold with the zero scalar curvature. At the end, we have shown that a f-projectively flat LP-Kenmotsu manifold is an Einstein manifold with the scalar curvature r = n(n — 1).

Keywords: Lorentzian para-Kenmotsu manifold, Weyl-pr ojective curvature tensor, conformal cur vatur e tensor , Einstein manifold.

2010 Mathematics Subject Classification: 53C07, 53C05, 53C15

I. Introduction

In 1989, K. Matsumoto [3] introduced the notion of Lorentzian paracontact and in particular , Lorentzian para- Sasakian (briefly 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 Bagewadi [16], Venkatesha, Pradeep Kumar and Bage-wadi [17, 18] and obtained several results of these manifolds.

In 1995, Sinha and Sai Prasad [14] 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, Abdul Haseeb and Rajendra Prasad defined a class of Lor entzian almost paracontact metric manifolds namely Lor entzian para-Kenmotsu (briefly LP- Kenmotsu) manifolds [1]. As an extension, Rajendra Prasad et al, [10] have studied f-semisymmetric LP-Kenmotsu manifolds with a quarter -symmetric non-metric conne ction admitting Ricci solitons.

On the other hand, In 1970, Pokhariy al and Mishra [9] introduced new tensor fields, called the Weyl-pr ojectiv e curvature tensor P(X, Y)Z of type (1, 3) and the tensor field E on a Rie-mannian manifold. Further many geometers have studied the properties of these tensor fields [2, 4, 8, 11, 12, 13, 15] as they play an important role in the theory of projective transfor mations of connectio ns.

The projective curvature tensor P (X, Y) Z, with respect to the Riemannian connection on a Rie-mannian manifol d (Mn, g), is given by:

P(X, Y)Z = R(X, Y)Z + -—y \g(X, Z)QY — g(Y, Z)QX],

where QX = (n — 1)X, and the Riemannian Christoffel curvature tensor R of type (1, 3) is giv en by:

R(X, Y)Z=VxVyZ—VyVxZ—V[XtY] Z. (1)

Here V is said to be the Levi-Civita connection.

In the present work, we study a class of LP-Kenmotsu manifolds and it is organized as follows. Section 2 is equipped with some prerequisites about Lorentzian para-Kenmotsu manifolds. In section 3, we study the quasi-confor mally flat Lorentzian para-Kenmotsu manifolds. Sections 4 and 5 respectiv ely deals with ^-conhar monically flat and ^-projectiv ely flat LP-Kenmotsu manifolds.

II. Preliminaries

An n-dimensional dif fer entiable manifold Mn admitting a (1, 1) tensor field <, contra variant vector field a 1-form n and the Lorentzian metric g (X, Y) satisfying

n (£ ) = — 1, (2)

<2X = X + n (X) £, (3)

g (<X, <Y) = g (X, Y)+ n (X) n (Y), (4)

g (X, I) = n (X), (5)

= 0, n (<X) = 0, rank < = n — 1; (6) is called Lorentzian almost paracontact manifold [3].

In a Lor entzian almost paracontact manifold, we have

$(X, Y) = $(Y, X) where $(X, Y) = g(<X, Y). (7)

A Lor entzian almost paracontact manifold Mn is called Lor entzian para-Kenmotsu manifold if [1]

(Vx<) Y = —g (<X, Y) I — n (Y) <X, (8)

for any vector fields X and Y on Mn, and V is the operator of covariant differentiation with respect to the Lor entzian metric g.

It can be easily seen that in a LP-Kenmotsu manifold Mn, the following relations hold [1]:

Vx I = —< X = —X — n (X) £, (9)

(Vxn) Y = —g (X, Y) I — n (X) n (Y), (10)

for any vector fields X and Y on Mn.

Also, in an LP-Kenmotsu manifold, the follo wing relations hold [1]:

g(R(X, Y)Z, I) = n(R(X, Y)Z) = g(Y, Z)n(X) — g(X, Z)n(Y) (11)

R(£, X)Y = g(X, Y)£ - n(Y)X, (12)

R(X, Y)£ = n(Y)X - n(X)Y, (13)

S(X, £) = (n - l)n(X), (14)

S(QX, QY) = S(X, Y) + (n - 1)n(X)n(Y), (15)

S (X, Y) = ag (X, Y)+ b n (X) n (Y); (16)

for any vector fields X,Y and Z, wher e R is the Riemannian curvature tensor and S is the Ricci tensor of Mn.

III. LP-Kenmotsu manifolds with C (X, Y) Z = 0

The quasi-confor mal curvature tensor C is defined as

C(X, Y)Z=aR(X, Y)Z+b{S(Y, Z)X-S(X, Z)Y+g(Y, Z)QX

-g(X, Z)QY}-1 (n-T +2b) {g(Y, Z)X-g(X, Z)Y}

wher e a,b are constants such that ab=0 and

S (Y, Z)=g (QY, Z).

From (17), we get

R(X, Y)Z= -ba{S(Y, Z)X-S(X, Z)Y+g(Y, Z)QX

-g(X, Z)QY} + n (n-r +2^ {g(Y, Z)X-g(X, Z)Y}.

Taking Z=£ in (18) and on using (5), (13), (14), we get

(17)

(18)

n(Y)X-n(X)Y= -1 {n(Y)QX-n(X)QY} j+2^ -b(n-1^ {n(Y)X-n(X)Y}.

(19)

Taking Y=£ and applying (2) we have

QX= {bk(-+2b)-(n-1)-¡}x

+ { fn(nrr +2b) -1-2(n-1)} n (X)5.

(20)

Contracting (20), we get after a few steps

r=n(n-1). (21)

Using (21) in (20), we get

QX= (n-1)X. (22)

Finally, using (22), we find from (18)

R(X, Y)Z=g(Y, Z)X-g(X, Z)Y.

Thus, we state

Theorem 3.1:A quasi-confor mally flat LP-Kenmotsu manifold is locally isometric with a unit spher e Sn(1).

IV. LP-Kenmotsu manifolds with y-conharmonically flat curvature

tensor

The conhar monic curvature tensor K is defined as

K (X, Y) Z=R (X, Y) Z-[S (Y, Z) X-S (X, Z) Y+g (Y, Z) SX-g (X, Z) SY].

A differentiable manifold (Mn,g), n> 3, satisfying the condition

y2K(yX, yY)yZ= 0 (23)

is called y-conhar monically flat.

In this secti on, we study LP-Kenmotsu manifolds with the condition (23).

Theorem 4.1:Let Mn be an n-dimensional, (n> 3),y-conhar monically flat LP-Kenmotsu manifold. Then Mn is an ^-Einstein manifold with the zero-scalar curvature.

Proof: Assume that (Mn,g), n> 3, is a ç-conformally flat LP-Kenmotsu manifold. It can be easily seen that ç2K(çX, çY) çZ= 0 holds if and only if

g(K(çX, çY) çZ, çW) = 0,

for any X, Y, Z, Wex (Mn ).

g(R(çX, çY)çZ, çW) = n-2 çY, çZ)S(çX, çW)-g(çX, çZ)S(çY, çW) +g(çX, çW)S(çY, çZ)-g(çY, çW)S(çX, çZ)].

We suppose that {e^... ,en-1, Ç} is a local orthonor mal basis of vector fields in Mn. By using the fact that {çe1,..., çe2n, Ç} is also a local orthonormal basis, if we put X=W=ei in (23) and sum up with respect to i, then

Y?- g (R (çei, çY) çZ, çet)= — ^-i1 çY, çZ)S (çe>, çe>) (25)

-g (çei, çZ) S (çY, çei) +g (çei, çei) S(çY, çZ)-g (çY, çei) S (çei, çZ)},

wher

e

n—1

£ g (R (ye,, yY) yZ, ye,)=S (yY, yZ)+g (yY, yZ), (26)

i=1

n—1

£ S (yei, yei) =r+n 1, (27)

i=1

n—1

£ g (yei, yZ) S (yY, yei)=S(yY, yZ), (28)

i=1

n—1

£ g (yei, yei)=n+1. (29)

i=1

So, by the use of (26)-(29) the equation (25) turns into

-S(yY, yZ) = (r+1 )g(yY, yZ). (30)

Then by using (4) and (15), from equation (30) we get

S(Y, Z) = -(r+1)g(Y, Z) - (n+r)n(Y)n(Z), (31)

which gives us, from (16), Mn is an ^-Einstein manifold. Hence on contracting (31) we obtain nr= 0, which implies the scalar curvature r= 0, which proves the theorem.

Rao, Sunitha and Sai Prasad RT&A, No 1 (77)

ON q-CONHARMONICALL Y FLAT LPK- MANIFOLDS Volume 19, March 2024

V. LP-Kenmotsu manifolds with ^-projectively flat curvature tensor

A differentiable manifold (Mn,g), n> 3, satisfying the condition

q2P(qX, qY)qZ= 0 (32)

is called q-projectiv ely flat, wher e P (X, Y) Z is the Weyl-pr ojectiv e curvatur e tensor of (Mn, g).

Theorem 5.1: Let Mn be an n-dimensional, (n> 3), q-projectiv ely flat LP-Kenmotsu manifold. Then Mn is an Einstein manifold with the scalar curvature r=n(n-1).

Proof:It can be easily seen that q2P(qX, qY)qZ= 0 holds if and

g(P(qX, qY)qZ, qW) = 0,

for any X, Y, Z, Wex (Mn).

g(R(qX, qY)qZ, qW) = - qY, qZ)S(qX, qW)-g(qX, qZ)S(qY, qW). (33)

By choosing {e1,... ,en-1, £} as a local orthonor mal basis of vector fields in Mn and using the fact that {qe1,...,qe2n, £} as a local orthonor mal basis, on putting X=W=ei in (33) and summing up with respect to i, we have

n-1 1 n-1

E g(R (q^ qY) qZ, qei) = E [g(qY, qZ)S (qei, qei) -g(qe¿, qZ) S (qY, qei)]. (34)

i=1 i=1

Ther efor e, by using (26)-(29) into (34) we get

nS (qY, qZ) =rg (qY, qZ). Hence by virtue of (4) and (15) we obtain

S(Y, Z) = rg(Y, Z)+(n-(n-1)) n(Y)n(Z). (35)

Ther efor e from (35), by contraction, we obtain

r=n(n-1). (36)

Then by substituting (36) into (35) we get

S (Y, Z) = (n-1) g (Y, Z),

which implies Mn is an Einstein manifold with the scalar curvature r= n(n-1). This completes the proof of the theor em.

Acknowledgements: The authors acknowledge Dr. A. Kamesw ara 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 declar e that ther e is no conflict of inter ests regar ding the publication of this paper .

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