International Journal of Computational Intelligence and Information Security, October 2014 Vol. 5, No. 7 ISSN: 1837-7823

Generalised LP-Sasakian Manifolds

L K Pandey D S Institute of Technology & Management, Ghaziabad, U.P. - 201007

dr.pandeylk@rediffmail.com

Abstract

In 1989, K. Matsumoto [1] introduced the notion of manifolds with Lorentzian paracontact metric structure similar to the almost paracontact metric structure which is defined by I. Sato [4], [5]. Also in 1988, K. Matsumoto and I. Mihai [2] discussed on a certain transformation in a Lorentzian Para-Sasakian manifold. T. Suguri and S. Nakayama [6] considered D-conformal deformations on almost contact metric structure. In this paper generalised Lorentzian Para-Sasakian manifold [3], generalised nearly LP-Sasakian manifolds and generalised almost LP-Sasakian manifolds have been discussed and some of their properties have been established. The purpose of this paper is to introduce a generalised D-conformal transformation in a generalised LP-Contact manifold. Keywords: Generalised nearly and almost LP-Sasakian manifolds, generalised LP-Co-symplectic manifolds, generalised D-conformal transformation.

1. Introduction

An n-dimensional differentiable manifold 𝑀𝑛, on which there are defined a tensor field 𝐹of type (1, 1), two contravariant vector fields 𝑇1and 𝑇2 , two covariant vector fields 𝐴1and 𝐴2 and a Lorentzian metric g, satisfying for arbitrary vector fields 𝑋,𝑌,𝑍, …

(1.1) 𝑋 = 𝑋 + 𝐴1(𝑋)𝑇1 + 𝐴2(𝑋)𝑇2, 𝑇1 = 0, 𝑇2 = 0, 𝐴1(𝑇1) = −1, 𝐴2(𝑇2) = −1, 𝑋 ≝ 𝐹𝑋, 𝐴1(𝑋) = 0, 𝐴2(𝑋) = 0, rank 𝐹= n-2

(1.2) g (𝑋,𝑌) = g (𝑋,𝑌) + 𝐴1(𝑋)𝐴1(𝑌) + 𝐴2(𝑋)𝐴2(𝑌), where 𝐴1(𝑋) = 𝑔(𝑋,𝑇1), 𝐴2(𝑋) = 𝑔(𝑋,𝑇2)

`𝐹(𝑋,𝑌) ≝ 𝑔� 𝑋, 𝑌� = `𝐹(𝑌,𝑋),

Then 𝑀𝑛 is called a generalised Lorentzian Para-Contact manifold (a generalised LP-Contact manifold).

Let D be a Riemannian connection on 𝑀𝑛, then we have

(1.3) (a) (𝐷𝑋`𝐹)� 𝑌, 𝑍� + (𝐷𝑋`𝐹)�𝑌,𝑍� − 𝐴1(𝑌)(𝐷𝑋𝐴1)(𝑍) − 𝐴2(𝑌)(𝐷𝑋𝐴2)(𝑍) − 𝐴1(𝑍)(𝐷𝑋𝐴1)(𝑌) − 𝐴2(𝑍)(𝐷𝑋𝐴2)(𝑌) = 0

(b) (𝐷𝑋`𝐹)� 𝑌, 𝑍� + (𝐷𝑋`𝐹)(𝑌,𝑍) − 𝐴1(𝑌)(𝐷𝑋𝐴1)�𝑍� − 𝐴2(𝑌)(𝐷𝑋𝐴2)�𝑍� − 𝐴1(𝑍)(𝐷𝑋𝐴1)�𝑌� − 𝐴2(𝑍)(𝐷𝑋𝐴2)�𝑌� = 0

(1.4) (a) (𝐷𝑋`𝐹) � 𝑌, 𝑍� + (𝐷𝑋`𝐹) �𝑌 ,𝑍� = 0

(b) (𝐷𝑋`𝐹) � 𝑌, 𝑍 � + (𝐷𝑋`𝐹)� 𝑌, 𝑍 � = 0

A generalised LP-Contact manifold is called a generalised Lorentzian Para-Sasakian manifold (a generalised LP-Sasakian manifold) if

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International Journal of Computational Intelligence and Information Security, October 2014 Vol. 5, No. 7 ISSN: 1837-7823

(1.5) (a) 2(𝐷𝑋𝐹)(𝑌) − {𝐴1(𝑌) + 𝐴2(𝑌)} 𝑋 − 𝑔� 𝑋 ,𝑌�(𝑇1 + 𝑇2) = 0 ⇔

(b) 2(𝐷𝑋`𝐹)(𝑌,𝑍) − {𝐴1(𝑌) + 𝐴2(𝑌)} 𝑔� 𝑋 ,𝑍� − {𝐴1(𝑍) + 𝐴2(𝑍)}𝑔� 𝑋 ,𝑌� = 0 ⇔

(c) 𝐷𝑋𝑇1 = 𝑋 − 𝑇2, 𝐷𝑋𝑇2 = 𝑋 − 𝑇1

On this manifold, we have

(1.6) (a) (𝐷𝑋𝐴1)� 𝑌� = (𝐷𝑋𝐴2)� 𝑌� = 𝑔� 𝑋, 𝑌� ⇔

(b) (𝐷𝑋𝐴1)( 𝑌) + 𝐴2(𝑌) = (𝐷𝑋𝐴2)( 𝑌) + 𝐴1(𝑌) = `𝐹(𝑋,𝑌) ⇔ (c) 𝐷𝑋𝑇1 = 𝑋 − 𝑇2, 𝐷𝑋𝑇2 = 𝑋 − 𝑇1

From (1.5) (b), the equation of a generalised LP-Sasakian manifold can be written as

(1.7) (a) 2(𝐷𝑋`𝐹)�𝑌,𝑍� = {𝐴1(𝑍) + 𝐴2(𝑍)}`𝐹(𝑋,𝑌)

(b) 2(𝐷𝑋`𝐹) �𝑌,𝑍� = {𝐴1(𝑍) + 𝐴2(𝑍)}𝑔� 𝑋 ,𝑌�

(c) 2(𝐷𝑋`𝐹)(𝑌,𝑍) = 𝐴1(𝑌)(𝐷𝑋𝐴1)( 𝑍 ) + 𝐴2(𝑌)(𝐷𝑋𝐴2)( 𝑍 ) + {𝐴1(𝑍) + 𝐴2(𝑍)}𝑔� 𝑋 ,𝑌�

Nijenhuis tensor in a generalised LP-Contact manifold is given by

(1.8) `𝑁(𝑋,𝑌,𝑍) = �𝐷𝑋`𝐹�(𝑌,𝑍) − �𝐷𝑌`𝐹�(𝑋,𝑍) − (𝐷𝑋`𝐹)�𝑌,𝑍� + (𝐷𝑌`𝐹)(𝑋,𝑍)

Where `𝑁(𝑋,𝑌,𝑍) ≝ 𝑔(𝑁( 𝑋,𝑌),𝑍)

2. Generalised nearly and almost Lorentzian Para-Sasakian manifold

A generalised LP-contact manifold will be called a generalised nearly Lorentzian Para-Sasakian manifold (a generalised nearly LP-Sasakian manifold) if

(2.1) 2(𝐷𝑋`𝐹)(𝑌,𝑍) − {𝐴1(𝑌) + 𝐴2(𝑌)} 𝑔� 𝑋 ,𝑍� − {𝐴1(𝑍) + 𝐴2(𝑍)} 𝑔� 𝑋 ,𝑌�

= 2(𝐷𝑌`𝐹)(𝑍,𝑋) − {𝐴1(𝑍) + 𝐴2(𝑍)} 𝑔� 𝑋 ,𝑌� − {𝐴1(𝑋) + 𝐴2(𝑋)} 𝑔� 𝑌 ,𝑍�

= 2(𝐷𝑍`𝐹)(𝑋,𝑌) − {𝐴1(𝑋) + 𝐴2(𝑋)} 𝑔� 𝑌 ,𝑍� − {𝐴1(𝑌) + 𝐴2(𝑌)} 𝑔� 𝑋 ,𝑍�

The equation of a generalised nearly LP-Sasakian manifold can be modified as

(2.2) (a) 2(𝐷𝑋𝐹)𝑌 − 2(𝐷𝑌𝐹)𝑋 − {𝐴1(𝑌) + 𝐴2(𝑌)}𝑋 + {𝐴1(𝑋) + 𝐴2(𝑋)}𝑌 = 0 ⇔

(b) 2(𝐷𝑋`𝐹)(𝑌,𝑍) − 2(𝐷𝑌`𝐹)(𝑋,𝑍) − {𝐴1(𝑌) + 𝐴2(𝑌)} 𝑔� 𝑋 ,𝑍� + {𝐴1(𝑋) + 𝐴2(𝑋)} 𝑔� 𝑌 ,𝑍� = 0

These equations can be written as

(2.3) (a) 2(𝐷𝑋𝐹)𝑌 − 2(𝐷𝑌𝐹)𝑋 + {𝐴1(𝑋) + 𝐴2(𝑋)}𝑌 = 0 ⇔

(b) 2(𝐷𝑋`𝐹)�𝑌,𝑍� − 2�𝐷𝑌`𝐹�(𝑍,𝑋) + {𝐴1(𝑋) + 𝐴2(𝑋)} `𝐹(𝑌,𝑍) = 0

(2.4) (a) 2(𝐷𝑋𝐹)𝑌 − 2(𝐷𝑌𝐹)𝑋 + {𝐴1(𝑋) + 𝐴2(𝑋)}𝑌 = 0 ⇔

(b) 2(𝐷𝑋`𝐹) � 𝑌,𝑍� − 2 �𝐷𝑌

`𝐹� (𝑍,𝑋) + {𝐴1(𝑋) + 𝐴2(𝑋)}𝑔� 𝑌 ,𝑍� = 0

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International Journal of Computational Intelligence and Information Security, October 2014 Vol. 5, No. 7 ISSN: 1837-7823

(2.5) (a) 2(𝐷𝑋𝐹)𝑌 − 2(𝐷𝑌𝐹)𝑋 − 𝐴1(𝑌){𝐷𝑋𝑇1 + (𝐷𝑇1𝐹)𝑋} − 𝐴2(𝑌){𝐷𝑋𝑇2 + (𝐷𝑇2𝐹)𝑋} + 𝐴1(𝑋){𝐷𝑌𝑇1 + (𝐷𝑇1𝐹)𝑌} + 𝐴2(𝑋){𝐷𝑌𝑇2 + (𝐷𝑇2𝐹)𝑌} = 0 ⇔

(b) 2(𝐷𝑋`𝐹)(𝑌,𝑍) − 2(𝐷𝑌`𝐹)(𝑋,𝑍) − 𝐴1(𝑌)�(𝐷𝑋𝐴1)� 𝑍� + (𝐷𝑇1`𝐹)( 𝑍,𝑋)� − 𝐴2(𝑌)�(𝐷𝑋𝐴2)� 𝑍� + (𝐷𝑇2`𝐹)( 𝑍,𝑋)� + 𝐴1(𝑋)�(𝐷𝑌𝐴1)� 𝑍� + (𝐷𝑇1`𝐹)( 𝑍,𝑌)� + 𝐴2(𝑋)�(𝐷𝑌𝐴2)� 𝑍� + (𝐷𝑇2`𝐹)( 𝑍,𝑌)� = 0

A generalised LP-Contact manifold will be called a generalised almost Lorentzian Para-Sasakian manifold (a generalised almost LP-Sasakian manifold) if

(2.6) (𝐷𝑋`𝐹)(𝑌,𝑍) + (𝐷𝑌`𝐹)(𝑍,𝑋) + (𝐷𝑍`𝐹)(𝑋,𝑌)

−{𝐴1(𝑋) + 𝐴2(𝑋)}𝑔� 𝑌 ,𝑍� − {𝐴1(𝑌) + 𝐴2(𝑌)} 𝑔� 𝑋 ,𝑍� − {𝐴1(𝑍) + 𝐴2(𝑍)} 𝑔� 𝑋 ,𝑌� = 0

3. Generalised Lorentzian Para-Co-symplectic manifold

A generalised LP-Contact manifold will be called a generalised Lorentzian Para-Co-symplectic manifold (a generalised LP-Co-symplectic manifold) if

(3.1) (a) 2(𝐷𝑋𝐹)𝑌 − 𝐴1(𝑌)𝐷𝑋𝑇1 − 𝐴2(𝑌)𝐷𝑋𝑇2 − (𝐷𝑋𝐴1)� 𝑌�𝑇1 − (𝐷𝑋𝐴2)� 𝑌�𝑇2 = 0 ⇔

(b) 2(𝐷𝑋`𝐹)(𝑌,𝑍) − 𝐴1(𝑌)(𝐷𝑋𝐴1)�𝑍� − 𝐴2(𝑌)(𝐷𝑋𝐴2)�𝑍� − 𝐴1(𝑍)(𝐷𝑋𝐴1)�𝑌� − 𝐴2(𝑍)(𝐷𝑋𝐴2)�𝑌� = 0

Therefore a generalised LP-Co-symplectic manifold is a generalised LP-Sasakian manifold if

(3.2) (a) (𝐷𝑋𝐴1)� 𝑌� = (𝐷𝑋𝐴2)� 𝑌� = 𝑔� 𝑋, 𝑌� ⇔

(b) (𝐷𝑋𝐴1)( 𝑌) + 𝐴2(𝑌) = (𝐷𝑋𝐴2)( 𝑌) + 𝐴1(𝑌) = `𝐹(𝑋,𝑌) ⇔ (c) 𝐷𝑋𝑇1 = 𝑋 − 𝑇2, 𝐷𝑋𝑇2 = 𝑋 − 𝑇1

A generalised LP-Contact manifold will be called a generalised nearly Lorentzian Para-Co-symplectic manifold (a generalised nearly LP-Co-symplectic manifold) if

(3.3) 2(𝐷𝑋`𝐹)(𝑌,𝑍) − 𝐴1(𝑌)(𝐷𝑋𝐴1)� 𝑍� − 𝐴2(𝑌)(𝐷𝑋𝐴2)� 𝑍� − 𝐴1(𝑍)(𝐷𝑋𝐴1)� 𝑌� − 𝐴2(𝑍)(𝐷𝑋𝐴2)� 𝑌�

= 2(𝐷𝑌`𝐹)(𝑍,𝑋) − 𝐴1(𝑍)(𝐷𝑌𝐴1)� 𝑋� − 𝐴2(𝑍)(𝐷𝑌𝐴2)� 𝑋� − 𝐴1(𝑋)(𝐷𝑌𝐴1)� 𝑍� − 𝐴2(𝑋)(𝐷𝑌𝐴2)� 𝑍�

= 2(𝐷𝑍`𝐹)(𝑋,𝑌) − 𝐴1(𝑋)(𝐷𝑍𝐴1)� 𝑌� − 𝐴2(𝑋)(𝐷𝑍𝐴2)� 𝑌� − 𝐴1(𝑌)(𝐷𝑍𝐴1)� 𝑋� − 𝐴2(𝑌)(𝐷𝑍𝐴2)� 𝑋�

It is clear that a generalised nearly LP-Sasakian manifold is a generalised nearly LP-Co-symplectic manifold, in which

(3.4) (a) (𝐷𝑋𝐴1)� 𝑌� = (𝐷𝑋𝐴2)� 𝑌� = 𝑔� 𝑋, 𝑌� ⇔

(b) (𝐷𝑋𝐴1)( 𝑌) + 𝐴2(𝑌) = (𝐷𝑋𝐴2)( 𝑌) + 𝐴1(𝑌) = `𝐹(𝑋,𝑌) ⇔ (c) 𝐷𝑋𝑇1 = 𝑋 − 𝑇2, 𝐷𝑋𝑇2 = 𝑋 − 𝑇1

A generalised LP-Contact manifold will be called a generalised almost LP-Co-symplectic manifold if

(3.5) 2(𝐷𝑋`𝐹)(𝑌,𝑍) + 2(𝐷𝑌`𝐹)(𝑍,𝑋) + 2(𝐷𝑍`𝐹)(𝑋,𝑌) − 𝐴1(𝑋)�(𝐷𝑌𝐴1)� 𝑍� + (𝐷𝑍𝐴1)� 𝑌�� −

𝐴2(𝑋)�(𝐷𝑌𝐴2)� 𝑍� + (𝐷𝑍𝐴2)� 𝑌�� − 𝐴1(𝑌)�(𝐷𝑋𝐴1)� 𝑍� + (𝐷𝑍𝐴1)� 𝑋�� − 𝐴2(𝑌){(𝐷𝑋𝐴2)� 𝑍� +

(𝐷𝑍𝐴2)� 𝑋�} − 𝐴1(𝑍)�(𝐷𝑋𝐴1)� 𝑌� + (𝐷𝑌𝐴1)� 𝑋�� − 𝐴2(𝑍)�(𝐷𝑋𝐴2)� 𝑌� + (𝐷𝑌𝐴2)� 𝑋�� = 0

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International Journal of Computational Intelligence and Information Security, October 2014 Vol. 5, No. 7 ISSN: 1837-7823

Therefore, A generalised almost LP-Co-symplectic manifold is a generalised almost LP-Sasakian manifold if

(3.6) (a) (𝐷𝑋𝐴1)� 𝑌� = (𝐷𝑋𝐴2)� 𝑌� = 𝑔� 𝑋, 𝑌� ⇔

(b) (𝐷𝑋𝐴1)( 𝑌) + 𝐴2(𝑌) = (𝐷𝑋𝐴2)( 𝑌) + 𝐴1(𝑌) = `𝐹(𝑋,𝑌) ⇔ (c) 𝐷𝑋𝑇1 = 𝑋 –𝑇2, 𝐷𝑋𝑇2 = 𝑋 –𝑇1

4. Completely Integrable manifolds

Barring 𝑋,𝑌,𝑍 in (1.8) and using equations (2.1), (1.4) (a), we get `𝑁( 𝑋, 𝑌, 𝑍 ) = 0,which implies that a

generalised nearly LP-Sasakian manifold is completely integrable.

Barring X, Y, Z in (1.8) and using equations (2.6), (1.4) (a), we see that a generalised almost LP-Sasakian manifold

is completely integrable if

(4.1) (𝐷𝑋`𝐹)(𝑌 ,𝑍) + (𝐷𝑌`𝐹)(𝑍 ,𝑋) = (𝐷𝑍`𝐹)(𝑋 ,𝑌)

5. Generalised D- Conformal transformation.

Let the corresponding Jacobian map B of the transformation b transforms the structure (𝐹,𝑇1,𝑇2,𝐴1, 𝐴2,𝑔 )

to the structure (𝐹,𝑉1,𝑉2, 𝜐1, 𝜐2, ℎ ) such that

(5.1) (a) 𝐵𝑍 = 𝐵𝑍 (b) ℎ( 𝐵𝑋,𝐵𝑌)𝑜𝑏 = 𝑒𝜎 g �𝑋,𝑌� − 𝑒2𝜎 𝐴1(𝑋)𝐴1(𝑌) − 𝑒2𝜎 𝐴2(𝑋)𝐴2(𝑌)

(c) 𝑉1 = 𝑒−𝜎 𝐵𝑇1, 𝑉2 = 𝑒−𝜎 𝐵𝑇2 (d) 𝜐1( 𝐵𝑋 )𝑜𝑏 = 𝑒𝜎 𝐴1(𝑋), 𝜐2( 𝐵𝑋 )𝑜𝑏 = 𝑒𝜎 𝐴2(𝑋)

Where 𝜎 is a differentiable function on 𝑀𝑛 , then the transformation is said to be generalised D-conformal

transformation. If 𝜎 is a constant, the transformation is known as D-homothetic.

Theorem 5.1 The structure (𝐹,𝑉1,𝑉2, 𝜐1, 𝜐2, ℎ ) is generalised Lorentzian Para-Contact.

Proof. Inconsequence of (1.1), (1.2), (5.1) (b) and (5.1) (d), we have

ℎ(𝐵𝑋,𝐵𝑌 )𝑜𝑏 =𝑒𝜎 g �𝑋,𝑌� = ℎ( 𝐵𝑋,𝐵𝑌)𝑜𝑏 + 𝑒2𝜎 𝐴1(𝑋)𝐴1(𝑌) + 𝑒2𝜎 𝐴2(𝑋)𝐴2(𝑌)

= ℎ( 𝐵𝑋,𝐵𝑌)𝑜𝑏 + {𝜐1( 𝐵𝑋 )𝑜𝑏}{𝜐1( 𝐵𝑌 )𝑜𝑏} + {𝜐2( 𝐵𝑋 )𝑜𝑏}{𝜐2( 𝐵𝑌 )𝑜𝑏}

This implies

(5.2) ℎ(𝐵𝑋,𝐵𝑌 ) = ℎ( 𝐵𝑋,𝐵𝑌) + 𝜐1( 𝐵𝑋 ) 𝜐1( 𝐵𝑌 ) + 𝜐2( 𝐵𝑋 ) 𝜐2( 𝐵𝑌 )

Making the use of (1.1), (5.1) (a), (5.1) (c) and (5.1) (d), we get

(5.3) 𝐵𝑋 = 𝐵𝑋 = 𝐵𝑋 + 𝐴1(𝑋)𝐵𝑇1+𝐴2(𝑋)𝐵𝑇2 = 𝐵𝑋 + {𝜐1( 𝐵𝑋 )𝑜𝑏}𝑉1 + {𝜐2( 𝐵𝑋 )𝑜𝑏}𝑉2

Also

(5.4) 𝑉1 = 𝑒−𝜎 𝐵𝑇1 = 0, 𝑉2 = 𝑒−𝜎 𝐵𝑇2 = 0

Equations (5.2), (5.3) and (5.4) prove the statement.

Theorem 5.2 Let 𝐸 and 𝐷 be the Riemannian connections with respect to h and g such that

(5.5) (a) 𝐸𝐵𝑋𝐵𝑌 = 𝐵𝐷𝑋𝑌 + 𝐵𝐻(𝑋,𝑌) (b) `𝐻(𝑋,𝑌,𝑍) ≝ 𝑔(𝐻( 𝑋,𝑌),𝑍)

Then

(5.6) 2𝐸𝐵𝑋𝐵𝑌 =

2𝐵𝐷𝑋𝑌 − 𝐵[2𝑒𝜎 {(Xσ) 𝐴1 (Y) 𝑇1+ (𝑋𝜎) 𝐴2 (Y) 𝑇2 + (Yσ) 𝐴1 (X) 𝑇1 + (Yσ) 𝐴2 (X) 𝑇2 − (−1G∇σ) 𝐴1 (X) 𝐴1(𝑌)

− (−1G∇σ) 𝐴2 (X) 𝐴2 (Y)}+(𝑒𝜎 − 1)�(𝐷𝑋𝐴1)(𝑌) + (𝐷𝑌𝐴1)(𝑋) − 2𝐴1�𝐻(𝑋,𝑌)��𝑇1 + (𝑒𝜎 − 1)�(𝐷𝑋𝐴2)(𝑌) +

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International Journal of Computational Intelligence and Information Security, October 2014 Vol. 5, No. 7 ISSN: 1837-7823

(𝐷𝑌𝐴2)(𝑋) − 2𝐴2�𝐻(𝑋,𝑌)��𝑇2 + (𝑒𝜎 − 1){𝐴1(𝑋)(𝐷𝑌𝑇1) + 𝐴2(𝑋)(𝐷𝑌𝑇2) + 𝐴1(𝑌)(𝐷𝑋𝑇1)+𝐴2(𝑌)(𝐷𝑋𝑇2) −

𝐴1(𝑋)(−1𝐺∇𝐴1)(𝑌) − 𝐴2(𝑋)(−1𝐺∇𝐴2)(𝑌) − 𝐴1(𝑌)(−1𝐺∇𝐴1)(𝑋) − 𝐴2(𝑌)(−1𝐺∇𝐴2)(𝑋)}]

Proof. Inconsequence of (5.1) (b), we have

𝐵𝑋�ℎ( 𝐵𝑌,𝐵𝑍)�𝑜𝑏 = 𝑋�𝑒𝜎 g �𝑌,𝑍� − 𝑒2𝜎 𝐴1(𝑌)𝐴1(𝑍) − 𝑒2𝜎 𝐴2(𝑌)𝐴2(𝑍)�

Consequently

(5.7) ℎ(𝐸𝐵𝑋𝐵𝑌,𝐵𝑍)𝑜𝑏 + ℎ(𝐵𝑌,𝐸𝐵𝑋𝐵𝑍)𝑜𝑏 =

(𝑋𝜎)𝑒𝜎 g �𝑌,𝑍� + 𝑒𝜎 g �𝐷𝑋𝑌,𝑍� + 𝑒𝜎 g �𝑌,𝐷𝑋𝑍� − 2(𝑋𝜎)𝑒2𝜎 𝐴1(𝑌)𝐴1(𝑍) − 𝑒2𝜎 (𝐷𝑋𝐴1)(𝑌)𝐴1(𝑍)

− 𝑒2𝜎 (𝐷𝑋𝐴1)(𝑍)𝐴1(𝑌) − 𝑒2𝜎 𝐴1(𝐷𝑋𝑌)𝐴1(𝑍) − 𝑒2𝜎 𝐴1(𝐷𝑋𝑍)𝐴1(𝑌)

− 2(𝑋𝜎)𝑒2𝜎 𝐴2(𝑌)𝐴1 (𝑍) − 𝑒2𝜎 (𝐷𝑋𝐴2)(𝑌)𝐴2(𝑍) − 𝑒2𝜎 (𝐷𝑋𝐴2)(𝑍)𝐴2(𝑌)

− 𝑒2𝜎 𝐴2(𝐷𝑋𝑌)𝐴2(𝑍) − 𝑒2𝜎 𝐴2(𝐷𝑋𝑍)𝐴2(𝑌)

Also

(5.8) ℎ(𝐸𝐵𝑋𝐵𝑌,𝐵𝑍)𝑜𝑏 + ℎ(𝐵𝑌,𝐸𝐵𝑋𝐵𝑍)𝑜𝑏 = 𝑒𝜎 g �𝐷𝑋𝑌,𝑍� − 𝑒2𝜎 𝐴1(𝐷𝑋𝑌)𝐴1(𝑍) − 𝑒2𝜎 𝐴2(𝐷𝑋𝑌)𝐴2(𝑍) +

𝑒𝜎 g �𝐻(𝑋,𝑌),𝑍� − 𝑒2𝜎 𝐴1�𝐻(𝑋,𝑌)�𝐴1(𝑍) − 𝑒2𝜎 𝐴2�𝐻(𝑋,𝑌)�𝐴2(𝑍) + 𝑒𝜎 g �𝑌,𝐻(𝑋,𝑍�

− 𝑒2𝜎 𝐴1(𝑌)𝐴1�𝐻(𝑋,𝑍)� − 𝑒2𝜎 𝐴2(𝑌)𝐴2�𝐻(𝑋,𝑍)� + 𝑒𝜎 g �𝑌,𝐷𝑋𝑍� − 𝑒2𝜎 𝐴1(𝐷𝑋𝑍)𝐴1(𝑌) −

𝑒2𝜎 𝐴2(𝐷𝑋𝑍)𝐴2(𝑌)

Equations (1.3) (a), (5.7) and (5.8) imply

(5.9) (𝑋𝜎)g �𝑌,𝑍� − 2(𝑋𝜎)𝑒𝜎 𝐴1(𝑌)𝐴1(𝑍) − 2(𝑋𝜎)𝑒𝜎 𝐴2(𝑌)𝐴2(𝑍) − (𝑒𝜎 − 1){(𝐷𝑋𝐴1)(𝑌)𝐴1(𝑍) +

(𝐷𝑋𝐴2)(𝑌)𝐴2(𝑍) + (𝐷𝑋𝐴1)(𝑍)𝐴1(𝑌) + (𝐷𝑋𝐴2)(𝑍)𝐴2(𝑌)} = `𝐻(𝑋,𝑌,𝑍) + `𝐻(𝑋,𝑍,𝑌)

−(𝑒𝜎 − 1) {𝐴1�𝐻(𝑋,𝑌)�𝐴1(𝑍) + 𝐴2�𝐻(𝑋,𝑌)�𝐴2(𝑍) + 𝐴1�𝐻(𝑋,𝑍)�𝐴1(𝑌) + 𝐴2�𝐻(𝑋,𝑍)�𝐴2(𝑌)}

Writing two other equations by cyclic permutation of 𝑋,𝑌,𝑍 and subtracting the third equation from the sum of the

first two equations and using symmetry of `𝐻 in the first two slots, we get

(5.10)

2`𝐻(𝑋,𝑌,𝑍) = −2𝑒𝜎 �(𝑋𝜎)𝐴1(𝑌)𝐴1(𝑍) + (𝑋𝜎)𝐴2(𝑌)𝐴2(𝑍) + (𝑌𝜎)𝐴1(𝑍)𝐴1(𝑋) + (𝑌𝜎)𝐴2(𝑍)𝐴124242(𝑋) −

(𝑍𝜎)𝐴1(𝑋)𝐴1(𝑌) − (𝑍𝜎)𝐴2(𝑋)𝐴2(𝑌)� − (𝑒𝜎 − 1)�𝐴1(𝑍)�(𝐷𝑋𝐴1)(𝑌) + (𝐷𝑌𝐴1)(𝑋) − 2𝐴1�𝐻(𝑋,𝑌)�� +

𝐴2(𝑍)�(𝐷𝑋𝐴2)(𝑌) + (𝐷𝑌𝐴2)(𝑋) − 2𝐴2�𝐻(𝑋,𝑌)�� + 𝐴1(𝑋){(𝐷𝑌𝐴1)(𝑍) − (𝐷𝑍𝐴1)(𝑌)} + 𝐴2(𝑋){(𝐷𝑌𝐴2)(𝑍) −

(𝐷𝑍𝐴2)(𝑌)} + 𝐴1(𝑌){(𝐷𝑋𝐴1)(𝑍) − (𝐷𝑍𝐴1)(𝑋)} + 𝐴2(𝑌){(𝐷𝑋𝐴2)(𝑍) − (𝐷𝑍𝐴2)(𝑋)}�

This gives

(5.11)

2𝐻(𝑋,𝑌) = −2𝑒𝜎 [(𝑋𝜎)𝐴1(𝑌)𝑇1 + (𝑋𝜎)𝐴2(𝑌) 𝑇2 + (𝑌𝜎)𝐴1(𝑋)𝑇1 + (𝑌𝜎)𝐴2(𝑋) 𝑇2 − (−1𝐺∇σ)𝐴1(X)𝐴1(Y) −

(−1𝐺∇σ)𝐴2(X)𝐴2(Y)] − (𝑒𝜎 − 1)��(𝐷𝑋𝐴1)(𝑌) + (𝐷𝑌𝐴1)(𝑋) − 2𝐴1�𝐻(𝑋,𝑌)��𝑇1 + �(𝐷𝑋𝐴2)(𝑌) + (𝐷𝑌𝐴2)(𝑋) −

2𝐴2�𝐻(𝑋,𝑌)�� 𝑇2 + 𝐴1(𝑋)(𝐷𝑌𝑇1) + 𝐴2(𝑋)(𝐷𝑌 𝑇2) + 𝐴1(𝑌)(𝐷𝑋𝑇1) + 𝐴2(𝑌)(𝐷𝑋 𝑇2) − 𝐴1(𝑋)(−1𝐺∇𝐴1�(Y) −

𝐴2(𝑋)(−1𝐺∇𝐴2)(Y) − 𝐴1(Y)(−1𝐺∇𝐴1)(X) − 𝐴2(Y)(−1𝐺∇𝐴2)(X)]

Substitution of (5.11) into (5.5) (a) gives (5.6).

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References

[1] Matsumoto, K. (1989) “On Lorentzian Paracontact Manifolds”, Bull. Of Yamagata Univ.,Nat Sci.,Vol. 12, . No.2, pp. 151-156. [2] Matsumoto, K. and Mihai, I. (1988) “On a certain transformation in a Lorentzian Para-Sasakian Manifold”, Tensor N. S., Vol. 47, pp. 189-197. [3] Nivas, R. and Bajpai, A. (2011) “Study of Generalized Lorentzian Para-Sasakian Manifolds”, Journal of International Academy of Physical Sciences, Vol. 15 No.4, pp. 405-412. [4] Sato, I. (1976) “On a structure similar to almost contact structure I”, Tensor N.S.,30, pp. 219-224. [5] Sato, I. (1977) “On a structure similar to almost contact structure II”, Tensor N.S.,31, pp. 199-205. [6] Suguri, T. and Nakayama, S. (1974) “D-conformal deformation on almost contact metric structures”, Tensor N. S., 28, pp. 125-129.

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