Paper-3 Generalised LP-Sasakian Manifolds
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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 [email protected] 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 , on which there are defined a tensor field of type (1, 1), two contravariant vector fields 1 and 2 , two covariant vector fields 1 and 2 and a Lorentzian metric g, satisfying for arbitrary vector fields , , ,β¦ (1.1) = + 1 () 1 + 2 () 2 , 1 = 0, 2 = 0, 1 ( 1 )= β1, 2 ( 2 )= β1, β , 1 ( ) = 0, 2 ( ) = 0, rank = n-2 (1.2) g ( , ) = g (, )+ 1 () 1 ()+ 2 () 2 (), where 1 ()= (, 1 ), 2 ()= (, 2 ) `(, ) β οΏ½ , οΏ½ =`(, ), Then is called a generalised Lorentzian Para-Contact manifold (a generalised LP-Contact manifold). Let D be a Riemannian connection on , then we have (1.3) (a) ( `)οΏ½ , οΏ½ +( `)οΏ½, οΏ½β 1 ()( 1 )() β 2 ()( 2 )() β 1 ()( 1 )() β 2 ()( 2 )()=0 (b) ( `)οΏ½ , οΏ½ +( `)(, ) β 1 ()( 1 )οΏ½ οΏ½β 2 ()( 2 )οΏ½ οΏ½β 1 ()( 1 )οΏ½ οΏ½β 2 ()( 2 )οΏ½ οΏ½ =0 (1.4) (a) ( `) οΏ½ , οΏ½ + ( `) οΏ½ , οΏ½ =0 (b) ( `) οΏ½ , οΏ½ +( `)οΏ½ , οΏ½ =0 A generalised LP-Contact manifold is called a generalised Lorentzian Para-Sasakian manifold (a generalised LP- Sasakian manifold) if 20
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Generalised nearly and almost LP-Sasakian manifolds, generalised LP-Co-symplectic manifolds, generalised D-conformal transformation.
Transcript of Paper-3 Generalised LP-Sasakian Manifolds
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
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 ππ, on which there are defined a tensor field πΉof type (1, 1), two contravariant vector fields π1and π2 , two covariant vector fields π΄1and π΄2 and a Lorentzian metric g, satisfying for arbitrary vector fields π,π,π, β¦
(1.1) π = π + π΄1(π)π1 + π΄2(π)π2, π1 = 0, π2 = 0, π΄1(π1) = β1, π΄2(π2) = β1, π β πΉπ, π΄1(π) = 0, π΄2(π) = 0, rank πΉ= n-2
(1.2) g (π,π) = g (π,π) + π΄1(π)π΄1(π) + π΄2(π)π΄2(π), where π΄1(π) = π(π,π1), π΄2(π) = π(π,π2)
`πΉ(π,π) β ποΏ½ π, ποΏ½ = `πΉ(π,π),
Then ππ is called a generalised Lorentzian Para-Contact manifold (a generalised LP-Contact manifold).
Let D be a Riemannian connection on ππ, then we have
(1.3) (a) (π·π`πΉ)οΏ½ π, ποΏ½ + (π·π`πΉ)οΏ½π,ποΏ½ β π΄1(π)(π·ππ΄1)(π) β π΄2(π)(π·ππ΄2)(π) β π΄1(π)(π·ππ΄1)(π) β π΄2(π)(π·ππ΄2)(π) = 0
(b) (π·π`πΉ)οΏ½ π, ποΏ½ + (π·π`πΉ)(π,π) β π΄1(π)(π·ππ΄1)οΏ½ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ποΏ½ β π΄1(π)(π·ππ΄1)οΏ½ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ποΏ½ = 0
(1.4) (a) (π·π`πΉ) οΏ½ π, ποΏ½ + (π·π`πΉ) οΏ½π ,ποΏ½ = 0
(b) (π·π`πΉ) οΏ½ π, π οΏ½ + (π·π`πΉ)οΏ½ π, π οΏ½ = 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(π·ππΉ)(π) β {π΄1(π) + π΄2(π)} π β ποΏ½ π ,ποΏ½(π1 + π2) = 0 β
(b) 2(π·π`πΉ)(π,π) β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ β {π΄1(π) + π΄2(π)}ποΏ½ π ,ποΏ½ = 0 β
(c) π·ππ1 = π β π2, π·ππ2 = π β π1
On this manifold, we have
(1.6) (a) (π·ππ΄1)οΏ½ ποΏ½ = (π·ππ΄2)οΏ½ ποΏ½ = ποΏ½ π, ποΏ½ β
(b) (π·ππ΄1)( π) + π΄2(π) = (π·ππ΄2)( π) + π΄1(π) = `πΉ(π,π) β (c) π·ππ1 = π β π2, π·ππ2 = π β π1
From (1.5) (b), the equation of a generalised LP-Sasakian manifold can be written as
(1.7) (a) 2(π·π`πΉ)οΏ½π,ποΏ½ = {π΄1(π) + π΄2(π)}`πΉ(π,π)
(b) 2(π·π`πΉ) οΏ½π,ποΏ½ = {π΄1(π) + π΄2(π)}ποΏ½ π ,ποΏ½
(c) 2(π·π`πΉ)(π,π) = π΄1(π)(π·ππ΄1)( π ) + π΄2(π)(π·ππ΄2)( π ) + {π΄1(π) + π΄2(π)}ποΏ½ π ,ποΏ½
Nijenhuis tensor in a generalised LP-Contact manifold is given by
(1.8) `π(π,π,π) = οΏ½π·π`πΉοΏ½(π,π) β οΏ½π·π`πΉοΏ½(π,π) β (π·π`πΉ)οΏ½π,ποΏ½ + (π·π`πΉ)(π,π)
Where `π(π,π,π) β π(π( π,π),π)
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(π·π`πΉ)(π,π) β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½
= 2(π·π`πΉ)(π,π) β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½
= 2(π·π`πΉ)(π,π) β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½
The equation of a generalised nearly LP-Sasakian manifold can be modified as
(2.2) (a) 2(π·ππΉ)π β 2(π·ππΉ)π β {π΄1(π) + π΄2(π)}π + {π΄1(π) + π΄2(π)}π = 0 β
(b) 2(π·π`πΉ)(π,π) β 2(π·π`πΉ)(π,π) β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ + {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ = 0
These equations can be written as
(2.3) (a) 2(π·ππΉ)π β 2(π·ππΉ)π + {π΄1(π) + π΄2(π)}π = 0 β
(b) 2(π·π`πΉ)οΏ½π,ποΏ½ β 2οΏ½π·π`πΉοΏ½(π,π) + {π΄1(π) + π΄2(π)} `πΉ(π,π) = 0
(2.4) (a) 2(π·ππΉ)π β 2(π·ππΉ)π + {π΄1(π) + π΄2(π)}π = 0 β
(b) 2(π·π`πΉ) οΏ½ π,ποΏ½ β 2 οΏ½π·π
`πΉοΏ½ (π,π) + {π΄1(π) + π΄2(π)}ποΏ½ π ,ποΏ½ = 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(π·ππΉ)π β 2(π·ππΉ)π β π΄1(π){π·ππ1 + (π·π1πΉ)π} β π΄2(π){π·ππ2 + (π·π2πΉ)π} + π΄1(π){π·ππ1 + (π·π1πΉ)π} + π΄2(π){π·ππ2 + (π·π2πΉ)π} = 0 β
(b) 2(π·π`πΉ)(π,π) β 2(π·π`πΉ)(π,π) β π΄1(π)οΏ½(π·ππ΄1)οΏ½ ποΏ½ + (π·π1`πΉ)( π,π)οΏ½ β π΄2(π)οΏ½(π·ππ΄2)οΏ½ ποΏ½ + (π·π2`πΉ)( π,π)οΏ½ + π΄1(π)οΏ½(π·ππ΄1)οΏ½ ποΏ½ + (π·π1`πΉ)( π,π)οΏ½ + π΄2(π)οΏ½(π·ππ΄2)οΏ½ ποΏ½ + (π·π2`πΉ)( π,π)οΏ½ = 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) (π·π`πΉ)(π,π) + (π·π`πΉ)(π,π) + (π·π`πΉ)(π,π)
β{π΄1(π) + π΄2(π)}ποΏ½ π ,ποΏ½ β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ β {π΄1(π) + π΄2(π)} ποΏ½ π ,ποΏ½ = 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(π·ππΉ)π β π΄1(π)π·ππ1 β π΄2(π)π·ππ2 β (π·ππ΄1)οΏ½ ποΏ½π1 β (π·ππ΄2)οΏ½ ποΏ½π2 = 0 β
(b) 2(π·π`πΉ)(π,π) β π΄1(π)(π·ππ΄1)οΏ½ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ποΏ½ β π΄1(π)(π·ππ΄1)οΏ½ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ποΏ½ = 0
Therefore a generalised LP-Co-symplectic manifold is a generalised LP-Sasakian manifold if
(3.2) (a) (π·ππ΄1)οΏ½ ποΏ½ = (π·ππ΄2)οΏ½ ποΏ½ = ποΏ½ π, ποΏ½ β
(b) (π·ππ΄1)( π) + π΄2(π) = (π·ππ΄2)( π) + π΄1(π) = `πΉ(π,π) β (c) π·ππ1 = π β π2, π·ππ2 = π β π1
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(π·π`πΉ)(π,π) β π΄1(π)(π·ππ΄1)οΏ½ ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ ποΏ½ β π΄1(π)(π·ππ΄1)οΏ½ ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ ποΏ½
= 2(π·π`πΉ)(π,π) β π΄1(π)(π·ππ΄1)οΏ½ ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ ποΏ½ β π΄1(π)(π·ππ΄1)οΏ½ ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ ποΏ½
= 2(π·π`πΉ)(π,π) β π΄1(π)(π·ππ΄1)οΏ½ ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ ποΏ½ β π΄1(π)(π·ππ΄1)οΏ½ ποΏ½ β π΄2(π)(π·ππ΄2)οΏ½ ποΏ½
It is clear that a generalised nearly LP-Sasakian manifold is a generalised nearly LP-Co-symplectic manifold, in which
(3.4) (a) (π·ππ΄1)οΏ½ ποΏ½ = (π·ππ΄2)οΏ½ ποΏ½ = ποΏ½ π, ποΏ½ β
(b) (π·ππ΄1)( π) + π΄2(π) = (π·ππ΄2)( π) + π΄1(π) = `πΉ(π,π) β (c) π·ππ1 = π β π2, π·ππ2 = π β π1
A generalised LP-Contact manifold will be called a generalised almost LP-Co-symplectic manifold if
(3.5) 2(π·π`πΉ)(π,π) + 2(π·π`πΉ)(π,π) + 2(π·π`πΉ)(π,π) β π΄1(π)οΏ½(π·ππ΄1)οΏ½ ποΏ½ + (π·ππ΄1)οΏ½ ποΏ½οΏ½ β
π΄2(π)οΏ½(π·ππ΄2)οΏ½ ποΏ½ + (π·ππ΄2)οΏ½ ποΏ½οΏ½ β π΄1(π)οΏ½(π·ππ΄1)οΏ½ ποΏ½ + (π·ππ΄1)οΏ½ ποΏ½οΏ½ β π΄2(π){(π·ππ΄2)οΏ½ ποΏ½ +
(π·ππ΄2)οΏ½ ποΏ½} β π΄1(π)οΏ½(π·ππ΄1)οΏ½ ποΏ½ + (π·ππ΄1)οΏ½ ποΏ½οΏ½ β π΄2(π)οΏ½(π·ππ΄2)οΏ½ ποΏ½ + (π·ππ΄2)οΏ½ ποΏ½οΏ½ = 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) (π·ππ΄1)οΏ½ ποΏ½ = (π·ππ΄2)οΏ½ ποΏ½ = ποΏ½ π, ποΏ½ β
(b) (π·ππ΄1)( π) + π΄2(π) = (π·ππ΄2)( π) + π΄1(π) = `πΉ(π,π) β (c) π·ππ1 = π βπ2, π·ππ2 = π βπ1
4. Completely Integrable manifolds
Barring π,π,π in (1.8) and using equations (2.1), (1.4) (a), we get `π( π, π, π ) = 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) (π·π`πΉ)(π ,π) + (π·π`πΉ)(π ,π) = (π·π`πΉ)(π ,π)
5. Generalised D- Conformal transformation.
Let the corresponding Jacobian map B of the transformation b transforms the structure (πΉ,π1,π2,π΄1, π΄2,π )
to the structure (πΉ,π1,π2, π1, π2, β ) such that
(5.1) (a) π΅π = π΅π (b) β( π΅π,π΅π)ππ = ππ g οΏ½π,ποΏ½ β π2π π΄1(π)π΄1(π) β π2π π΄2(π)π΄2(π)
(c) π1 = πβπ π΅π1, π2 = πβπ π΅π2 (d) π1( π΅π )ππ = ππ π΄1(π), π2( π΅π )ππ = ππ π΄2(π)
Where π is a differentiable function on ππ , then the transformation is said to be generalised D-conformal
transformation. If π is a constant, the transformation is known as D-homothetic.
Theorem 5.1 The structure (πΉ,π1,π2, π1, π2, β ) is generalised Lorentzian Para-Contact.
Proof. Inconsequence of (1.1), (1.2), (5.1) (b) and (5.1) (d), we have
β(π΅π,π΅π )ππ =ππ g οΏ½π,ποΏ½ = β( π΅π,π΅π)ππ + π2π π΄1(π)π΄1(π) + π2π π΄2(π)π΄2(π)
= β( π΅π,π΅π)ππ + {π1( π΅π )ππ}{π1( π΅π )ππ} + {π2( π΅π )ππ}{π2( π΅π )ππ}
This implies
(5.2) β(π΅π,π΅π ) = β( π΅π,π΅π) + π1( π΅π ) π1( π΅π ) + π2( π΅π ) π2( π΅π )
Making the use of (1.1), (5.1) (a), (5.1) (c) and (5.1) (d), we get
(5.3) π΅π = π΅π = π΅π + π΄1(π)π΅π1+π΄2(π)π΅π2 = π΅π + {π1( π΅π )ππ}π1 + {π2( π΅π )ππ}π2
Also
(5.4) π1 = πβπ π΅π1 = 0, π2 = πβπ π΅π2 = 0
Equations (5.2), (5.3) and (5.4) prove the statement.
Theorem 5.2 Let πΈ and π· be the Riemannian connections with respect to h and g such that
(5.5) (a) πΈπ΅ππ΅π = π΅π·ππ + π΅π»(π,π) (b) `π»(π,π,π) β π(π»( π,π),π)
Then
(5.6) 2πΈπ΅ππ΅π =
2π΅π·ππ β π΅[2ππ {(XΟ) π΄1 (Y) π1+ (ππ) π΄2 (Y) π2 + (YΟ) π΄1 (X) π1 + (YΟ) π΄2 (X) π2 β (β1GβΟ) π΄1 (X) π΄1(π)
β (β1GβΟ) π΄2 (X) π΄2 (Y)}+(ππ β 1)οΏ½(π·ππ΄1)(π) + (π·ππ΄1)(π) β 2π΄1οΏ½π»(π,π)οΏ½οΏ½π1 + (ππ β 1)οΏ½(π·ππ΄2)(π) +
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(π·ππ΄2)(π) β 2π΄2οΏ½π»(π,π)οΏ½οΏ½π2 + (ππ β 1){π΄1(π)(π·ππ1) + π΄2(π)(π·ππ2) + π΄1(π)(π·ππ1)+π΄2(π)(π·ππ2) β
π΄1(π)(β1πΊβπ΄1)(π) β π΄2(π)(β1πΊβπ΄2)(π) β π΄1(π)(β1πΊβπ΄1)(π) β π΄2(π)(β1πΊβπ΄2)(π)}]
Proof. Inconsequence of (5.1) (b), we have
π΅ποΏ½β( π΅π,π΅π)οΏ½ππ = ποΏ½ππ g οΏ½π,ποΏ½ β π2π π΄1(π)π΄1(π) β π2π π΄2(π)π΄2(π)οΏ½
Consequently
(5.7) β(πΈπ΅ππ΅π,π΅π)ππ + β(π΅π,πΈπ΅ππ΅π)ππ =
(ππ)ππ g οΏ½π,ποΏ½ + ππ g οΏ½π·ππ,ποΏ½ + ππ g οΏ½π,π·πποΏ½ β 2(ππ)π2π π΄1(π)π΄1(π) β π2π (π·ππ΄1)(π)π΄1(π)
β π2π (π·ππ΄1)(π)π΄1(π) β π2π π΄1(π·ππ)π΄1(π) β π2π π΄1(π·ππ)π΄1(π)
β 2(ππ)π2π π΄2(π)π΄1 (π) β π2π (π·ππ΄2)(π)π΄2(π) β π2π (π·ππ΄2)(π)π΄2(π)
β π2π π΄2(π·ππ)π΄2(π) β π2π π΄2(π·ππ)π΄2(π)
Also
(5.8) β(πΈπ΅ππ΅π,π΅π)ππ + β(π΅π,πΈπ΅ππ΅π)ππ = ππ g οΏ½π·ππ,ποΏ½ β π2π π΄1(π·ππ)π΄1(π) β π2π π΄2(π·ππ)π΄2(π) +
ππ g οΏ½π»(π,π),ποΏ½ β π2π π΄1οΏ½π»(π,π)οΏ½π΄1(π) β π2π π΄2οΏ½π»(π,π)οΏ½π΄2(π) + ππ g οΏ½π,π»(π,ποΏ½
β π2π π΄1(π)π΄1οΏ½π»(π,π)οΏ½ β π2π π΄2(π)π΄2οΏ½π»(π,π)οΏ½ + ππ g οΏ½π,π·πποΏ½ β π2π π΄1(π·ππ)π΄1(π) β
π2π π΄2(π·ππ)π΄2(π)
Equations (1.3) (a), (5.7) and (5.8) imply
(5.9) (ππ)g οΏ½π,ποΏ½ β 2(ππ)ππ π΄1(π)π΄1(π) β 2(ππ)ππ π΄2(π)π΄2(π) β (ππ β 1){(π·ππ΄1)(π)π΄1(π) +
(π·ππ΄2)(π)π΄2(π) + (π·ππ΄1)(π)π΄1(π) + (π·ππ΄2)(π)π΄2(π)} = `π»(π,π,π) + `π»(π,π,π)
β(ππ β 1) {π΄1οΏ½π»(π,π)οΏ½π΄1(π) + π΄2οΏ½π»(π,π)οΏ½π΄2(π) + π΄1οΏ½π»(π,π)οΏ½π΄1(π) + π΄2οΏ½π»(π,π)οΏ½π΄2(π)}
Writing two other equations by cyclic permutation of π,π,π and subtracting the third equation from the sum of the
first two equations and using symmetry of `π» in the first two slots, we get
(5.10)
2`π»(π,π,π) = β2ππ οΏ½(ππ)π΄1(π)π΄1(π) + (ππ)π΄2(π)π΄2(π) + (ππ)π΄1(π)π΄1(π) + (ππ)π΄2(π)π΄124242(π) β
(ππ)π΄1(π)π΄1(π) β (ππ)π΄2(π)π΄2(π)οΏ½ β (ππ β 1)οΏ½π΄1(π)οΏ½(π·ππ΄1)(π) + (π·ππ΄1)(π) β 2π΄1οΏ½π»(π,π)οΏ½οΏ½ +
π΄2(π)οΏ½(π·ππ΄2)(π) + (π·ππ΄2)(π) β 2π΄2οΏ½π»(π,π)οΏ½οΏ½ + π΄1(π){(π·ππ΄1)(π) β (π·ππ΄1)(π)} + π΄2(π){(π·ππ΄2)(π) β
(π·ππ΄2)(π)} + π΄1(π){(π·ππ΄1)(π) β (π·ππ΄1)(π)} + π΄2(π){(π·ππ΄2)(π) β (π·ππ΄2)(π)}οΏ½
This gives
(5.11)
2π»(π,π) = β2ππ [(ππ)π΄1(π)π1 + (ππ)π΄2(π) π2 + (ππ)π΄1(π)π1 + (ππ)π΄2(π) π2 β (β1πΊβΟ)π΄1(X)π΄1(Y) β
(β1πΊβΟ)π΄2(X)π΄2(Y)] β (ππ β 1)οΏ½οΏ½(π·ππ΄1)(π) + (π·ππ΄1)(π) β 2π΄1οΏ½π»(π,π)οΏ½οΏ½π1 + οΏ½(π·ππ΄2)(π) + (π·ππ΄2)(π) β
2π΄2οΏ½π»(π,π)οΏ½οΏ½ π2 + π΄1(π)(π·ππ1) + π΄2(π)(π·π π2) + π΄1(π)(π·ππ1) + π΄2(π)(π·π π2) β π΄1(π)(β1πΊβπ΄1οΏ½(Y) β
π΄2(π)(β1πΊβπ΄2)(Y) β π΄1(Y)(β1πΊβπ΄1)(X) β π΄2(Y)(β1πΊβπ΄2)(X)]
Substitution of (5.11) into (5.5) (a) gives (5.6).
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International Journal of Computational Intelligence and Information Security, October 2014 Vol. 5, No. 7 ISSN: 1837-7823
References
[1] Matsumoto, K. (1989) βOn Lorentzian Paracontact Manifoldsβ, Bull. Of Yamagata Univ.,Nat Sci.,Vol. 12, . No.2, pp. 151-156. [2] Matsumoto, K. and Mihai, I. (1988) βOn a certain transformation in a Lorentzian Para-Sasakian Manifoldβ, Tensor N. S., Vol. 47, pp. 189-197. [3] Nivas, R. and Bajpai, A. (2011) βStudy of Generalized Lorentzian Para-Sasakian Manifoldsβ, Journal of International Academy of Physical Sciences, Vol. 15 No.4, pp. 405-412. [4] Sato, I. (1976) βOn a structure similar to almost contact structure Iβ, Tensor N.S.,30, pp. 219-224. [5] Sato, I. (1977) βOn a structure similar to almost contact structure IIβ, Tensor N.S.,31, pp. 199-205. [6] Suguri, T. and Nakayama, S. (1974) βD-conformal deformation on almost contact metric structuresβ, Tensor N. S., 28, pp. 125-129.
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