Microsoft Word - Shah _80-88_.docx R C O S T N Publisher: Research Council of Science and Technology, Biratnagar, Nepal: p. 80 BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (print), 2382-5340 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA On contact conformal curvature tensor in trans-Sasakian manifolds Riddhi Jung Shah Department of Mathematics, Janata Campus Nepal Sanskrit University, Dang E-mail: shahrjgeo@gmail.com Accepted for publication: December 18, 2014 Abstract The purpose of this paper is to study some results on contact conformal curvature tensor in trans-Sasakian manifolds. Contact conformally flat trans-Sasakian manifold, ξ -contact conformally flat trans-Sasakian manifold and curvature conditions ( )0 , . 0C X Sξ = and ( )0 0, . 0C X Cξ = are studied with some interesting results. Finally, we study an example of 3- dimensional trans-Sasakian manifold. DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11783 © 2014 RCOST: All rights reserved. Keywords: Contact conformal curvature tensor; Trans-Sasakian manifold; Hermitian manifolds. 1. Introduction In 1978, Gray and Hervella [1] studied on the sixteen classes of almost Hermitian manifolds and their linear invariants. They considered unitary group ( )U n on a certain space W and studied that the representation of ( )U n on W has four irreducible components, 1 2 3 4.W W W W W= ⊕ ⊕ ⊕ From these four components sixteen different invariants subspaces were obtained. Among four components 3 4W W⊕ corresponds to the class of Hermitian manifolds. Oubina [2] studied a new class of almost contact metric structure, called trans-Sasakian which is an analogue of a locally conformal Kaehler structure on an almost Hermitian manifold. An almost contact metric structure ( ), , , gϕ ξ η (where ϕ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form and g is a compatible Rimannian metric) on M is trans-Sasakian [2] if ( ), ,M J G×ℝ belongs to the class 4 ,W where J is the almost complex structure on M ×ℝ defined by (1.1) ( ), , , d d J X f X f X dt dt ϕ ξ η   = −        Shah/BIBECHANA 12 (2015) 80-88 : p. 81 for any vector field X on ,M where G is the product metric on .M ×ℝ Trans-Sasakian manifold is the trans-Sasakian structure of type ( ), ,α β where α and β are smooth functions on .M Trans- Sasakian manifolds of type ( ) ( )0,0 , ,0α and ( )0,β are cosympletic [3],α -Sasakian [4] and β - Kenmotsu manifold [4,5] respectively. Trans-Sasakian manifolds have been studied in [6,7] and by many others. On the other hand, contact conformal curvature tensor field was introduced and defined by Jeong et al. [8] in a ( )2 1n+ -dimensional Sasakian manifold which was constructed from the conformal curvature tensor field defined by Kitahara et al. [9] in a Kaehler manifold by using the Boothby- Wang’s fibration. Contact conformal curvature tensor has also been studied in [10] and [11]. 2. Preliminaries Let M be a ( )2 1n+ -dimensional almost contact metric manifold equipped with an almost contact metric structure ( ), , , ,gϕ ξ η where ϕ is a (1, 1) tensor field, ξ is a vector field, η -is a 1-form and g is a compatible Riemannian metric such that [3] (2.1) ( ) ( ) ( ) ( )2 , 1, =0, 0,X X X Xϕ η ξ η ξ ϕξ η ϕ= − + = = (2.2) ( ) ( ) ( ) ( ), , ,g X Y g X Y X Yϕ ϕ η η= − (2.3) ( ) ( ) ( ) ( ), , , , ,g X Y g X Y g X Xϕ ϕ ξ η= − = for all , .X Y TM∈ The fundamental 2-form Φ of the almost contact metric structure ( ), , , gϕ ξ η is defined as (2.4) ( ) ( ) ( ), , , ,X Y g X Y g X Yϕ ϕΦ = = − since ϕ is a skew-symmetric with respect to .g An almost contact metric manifold M is called trans-Sasakian manifold if [2] (2.5) ( ) ( ) ( ){ } ( ) ( ){ }, , ,X Y g X Y Y X g X Y Y Xϕ α ξ η β ϕ ξ η ϕ∇ = − + − where ∇ is Levi-Civita connection of Riemannian metric g and ,α β are smooth functions on .M From (2.5) it follows that (2.6) ( ){ },X X X Xξ αϕ β η ξ∇ = − + − (2.7) ( ) ( ) ( ), , .X Y g X Y g X Yη α ϕ β ϕ ϕ∇ = − + In a ( )2 1n+ -dimensional trans-Sasakian manifold ,M the following relations hold [6] (2.8) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 , [ ] 2 [ ] , R X Y Y X X Y X Y X Y Y X X Y Y X Y X ξ α β η η α ϕ β ϕ αβ η ϕ η ϕ α ϕ β ϕ = − − − − + − + + (2.9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2, [ , ] 2 [ , ] , [ ] , , R X Y g X Y Y X g Y X Y X Y X g Y X grad Y X X g X Y grad ξ α β ξ η αβ ϕ ξ η ϕ α ϕ ϕ α β η ξ ϕ ϕ β = − − + − + + + − − Shah/BIBECHANA 12 (2015) 80-88 : p. 82 (2.10) ( )2 0,αβ ξα+ = (2.11) ( ) ( ) ( ) ( ) ( )( ) ( )( )2 2 , [2 ] 2 1 ,S X n X X n Xξ α β ξβ η ϕ α β= − − − − − (2.12) ( )( ) ( )( ), , , ,R X Y Z g R X Y Zη ξ= − (2.13) ( )( ) ( )( ) ( )( ), , , 0,R X Y R X R Xη ξ η ξ ξ η ξ ξ= = = (2.14) ( )( ) ( )( ) ( )2 2 , , .R X Y g X Yη ξ α β ξβ ϕ ϕ= − − In a ( )2 1n+ -dimensional trans-Sasakian manifold if we put ( ) ( )2 1 ,grad n gradϕ α β= − then we have (2.15) ( ) 0,ξβ = (2.16) ( ) ( ) ( )2 2 , 2 ,S X n Xξ α β η= − (2.17) ( )( ) ( ) ( )2 2 , , ,R X Y g X Yη ξ α β ϕ ϕ= − (2.18) ( ) ( ) ( ){ }2 2 , ,R X X Xξ ξ α β η ξ= − − (2.19) ( ) ( ), , .R X R Xξ ξ ξ ξ= − Throughout the paper we consider the trans-Sasakian manifold under the condition ( ) ( )2 1 .grad n gradϕ α β= − In a ( )2 1n+ -dimensional trans-Sasakian manifold the contact conformal curvature tensor field 0C of type (1, 3) which is defined by [8] can be written as (2.20) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 , , { , , , 2 , , , , , , , 2 , 2 , C X Y Z R X Y Z S Y Z X S X Z Y g Y Z QX n g X Z QY S X Z Y S Y Z X X Z QY Y Z QX S X Z Y S Y Z X g X Z Q Y g Y Z Q X g X Y Q Z S X η ξ η ξ η η η η ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ = + − + − + − + − + − + − + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) } 21 2 {2 2 }{ , 2 1 2 , 2 , } 3 21 { 2 }{ , , } 2 1 2 Y Z n r n n g Y Z X n n n g X Z Y g X Y Z n r n g Y Z X g X Z Y n n n ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + − − + + − − + + + − − + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 21 2 {4 5 2 }{ 2 1 2 , , }, n r n n Y Z X n n n X Z Y g Y Z X g X Z Y η η η η η ξ η ξ + − + + − + − + − where , ,R S Q and r denote the curvature tensor, the Ricci tensor, the Ricci operator and the scalar curvature respectively. From (2.20), we also have Shah/BIBECHANA 12 (2015) 80-88 : p. 83 (2.21) ( ) ( ) ( ) ( ) ( ){ }2 2 0 , , 2 ,C X Y R X Y Y X X Yξ ξ α β η η= + − − − (2.22) ( ) ( ) ( ) ( ) ( ){ }2 2 0 , , 2 , ,C X Y R X Y g X Y Y Xξ ξ α β ξ η= + − − − (2.23) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ){ } 0 2 2 , , 2 , , , C X Y Z R X Y Z g Y Z X g X Z Y η η α β η η = + − − − (2.24) ( )( )0 , 0,C X Yη ξ = (2.25) ( )( ) ( ) ( ) ( ) ( ){ }2 2 0 , 2 1 , .C X Y g X Y X Yη ξ α β η η= − − − Definition. A ( )2 1n+ -dimensional trans-Sasakian manifold M is said to be an η -Einstein manifold if its Ricci tensor S of type (0, 2) is of the form (2.26) ( ) ( ) ( ) ( ), , ,S X Y ag X Y b X Yη η= + where ,a b are smooth functions on .M If 0,b = then the manifold M becomes an Einstein manifold. 3. Contact Conformally Flat Trans-Sasakian Manifold Definition. A ( )2 1n+ -dimensional trans-Sasakian manifold is said to be contact conformally flat if it satisfies the condition (3.1) ( )0 , 0.C X Y Z = Now, we prove the following result: Theorem 3.1. If a ( )2 1n+ -dimensional trans-Sasakian manifold M is contact conformally flat, then 2 2 1.α β= + Proof. Let M be a ( )2 1n+ -dimensional trans-Sasakian manifold. Suppose M is contact conformally flat then the condition ( )0 , 0C X Y Z = holds. Now, using (3.1) in (2.20) and taking inner product on both sides by ,ξ we get (3.2) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , [ , , , , 2 , , ] 2[ , , ]. R X Y Z g X Z S Y g Y Z S X n X Z S Y Y Z S X g Y Z X g X Z Y η ξ ξ η η ξ η η ξ η η = − − + + − In view of (2.12), (2.16) and (3.2), we get (3.3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 20 2 1 [ , , ] 2 [ , , ] , , , . g Y Z X g X Z Y g Y Z X g X Z Y X g Y Z Y g X Z X g Y Z α β η η αβ ϕ η ϕ η α ϕ α ϕ β ϕ ϕ = − − − + − + − − Putting X ξ= in (3.3) and using (2.1), (2.10) and (2.15), we obtain (3.4) ( ) ( ) ( ) ( )2 2 1 [ , ] 0.g Y Z Y Zα β η η− − − = Since ( ) ( ) ( ), 0,g Y Z Y Zη η− ≠ we have ( )2 2 1 0.α β− − = This implies that Shah/BIBECHANA 12 (2015) 80-88 : p. 84 (3.5) 2 2 1.α β= + This completes the proof of the theorem. 4. ξ -Contact Conformally Flat Trans-Sasakian Manifold Definition. A trans-Sasakian manifold of dimension ( )2 1n+ is said to be ξ -contact conformally flat if the condition (4.1) ( )0 , 0C X Y ξ = holds. Theorem 4.1. Let M be a ( )2 1n+ -dimensional trans-Sasakian manifold satisfying the condition ( )0 , 0,C X Y ξ = then 2 2 1.α β= + Proof. Let us consider a ( )2 1n+ -dimensional trans-Sasakian manifold M which satisfies the condition ( )0 , 0.C X Y ξ = Then by virtue of (2.1), (2.3), (2.8), (2.16) and (4.1) in (2.20), we get (4.2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 20 2 1 { } 2 [ ] . Y X X Y X Y X Y X Y Y X X Y Y X Y X Y X α β η η α ϕ β β η ξ αβ η ϕ η ϕ α ϕ β β η ξ = − − − − + − + − + − + Putting X ξ= in (4.2) and using (2.1), (2.10) and (2.15), we obtain (4.3) ( ) ( ){ }2 2 1 0.Y Yα β η ξ− − − = Since ( ) ( )2 0,Y Y Yη ξ ϕ− = ≠ we have ( )2 2 1 0.α β− − = This yields (4.4) 2 2 1.α β= + Thus the theorem is proved. From theorem 3.1 and theorem 4.1, we can state the following result: Theorem 4.2. Trans-Sasakian manifolds of dimension ( )2 1n+ which satisfy the conditions ( )0 , 0C X Y Z = and ( )0 , 0C X Y ξ = are equivalent. 5. Trans-Sasakian Manifold Satisfying ( )0C ξ,X .S=0 Consider a trans-Sasakian manifold M of dimension ( )2 1 .n+ Let S be the Ricci tensor of type (0, 2). We prove the following result: Theorem 5.1. Let M be a ( )2 1n+ -dimensional trans-Sasakian manifold. If M satisfies the condition ( )0 , . 0,C X Sξ = then it is an Einstein manifold with scalar curvature ( )( )2 22 2 1 .r n n α β= + − Proof. Let M be a ( )2 1n+ -dimensional trans-Sasakian manifold which satisfies the condition Shah/BIBECHANA 12 (2015) 80-88 : p. 85 (5.1) ( ) ( )0 , . , 0.C X S U Vξ = This condition implies that (5.2) ( )( ) ( )( )0 0, , , , 0.S C X U V S U C X Vξ ξ+ = Putting V ξ= in (5.2) and using (2.16) and (2.22), we obtain (5.3) ( ) ( ) ( )2 2, 2 , .S X U n g X Uα β= − Taking an orthonormal frame field at any point of the manifold and contracting over X and U in (5.3), we get (5.4) ( )( )2 22 2 1 .r n n α β= + − From (5.3) and (5.4) it follows that the manifold M is an Einstein manifold with scalar curvature ( )( )2 22 2 1 .r n n α β= + − This completes the proof of the result. 6. Trans-Sasakian Manifold Satisfying ( )0 0C ξ,X .C =0 Let M be a ( )2 1n+ -dimensional trans-Sasakian manifold. Suppose the condition ( )( )( )0 0, . , 0C X C U V Zξ = holds in .M Then we have Theorem 6.1. A ( )2 1n+ -dimensional trans-Sasakian manifold satisfying the condition ( )0 0, . 0C X Cξ = is contact conformally semi-symmetric if ( ) ( ) ( ) ( ){ } ( )( ) ( ) 2 20 2 , , , , , , . g V Z X g X V Z g X Z V g R V Z X R X V Z α β ξ ξ = − − − − − Proof. Let us consider a ( )2 1n+ -dimensional trans-Sasakian manifold which satisfies the condition ( ) ( )0 0, . , 0,C X C U V Zξ = then by definition we have (6.1) ( ) ( ) ( )( ) ( )( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 , , , , , , , , . C X C U V Z C C X U V Z C U C X V Z C U V C X Z ξ ξ ξ ξ = − − − Using (2.22) in (6.1) we get (6.2) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 0 0 0 0 0 0 0 0 0 , . , 2 [ , , , , , , , , , , , , ]. R X C U V Z g X C U V Z C U V Z X g X U C V Z U C X V Z g X V C U Z V C U X Z g X Z C U V Z C U V X ξ α β ξ η ξ η ξ η ξ η = + − − − − + − + − + Taking inner product on both sides of (6.2) by ξ and using (2.24), we obtain Shah/BIBECHANA 12 (2015) 80-88 : p. 86 (6.3) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 2 2 0 0 0 0 0 0 0 0 , . , , 2 [ , , , , , , , , , , ]. og R X C U V Z g X C U V Z X C U V Z g X U C V Z U C X V Z g X V C U Z V C U X Z Z C U V X ξ ξ α β η η η ξ η η η ξ η η η η = + − − − − + − + − Putting U ξ= in (6.3) and using (2.22), (2.23) and (2.25), we get (6.4) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 2 2 2 0 , . , , 2 [ , , , { , 2 , , }]. g R X C V Z g R V Z X R X V Z Z g X V X g V Z V g X Z ξ ξ ξ α β ξ η α β η η η = + − − + + − − + This implies that (6.5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 0 2 2 2 2 , . , 2 [ {2 , , , } , , , ]. R X C V Z g V Z X g X V Z g X Z V g R V Z X R X V Z ξ ξ α β α β ξ ξ = − − − − − − − From this it follows that the manifold is contact conformally semi-symmetric if the right hand side of (6.5) vanishes, i. e., if ( ) ( ) ( ) ( ){ } ( )( ) ( ) 2 20 2 , , , , , , . g V Z X g X V Z g X Z V g R V Z X R X V Z α β ξ ξ = − − − − − This completes the proof of the theorem. 7. An Example of a 3-dimensional Trans-Sasakian Manifold Let us consider a 3-dimensional manifold ( ) ( ){ }3, , : , , , 0,M x y z x y z z= ∈ ≠ℝ where ( ), ,x y z are the standard coordinates in 3.ℝ We choose the vector fields (7.1) 1 2, ,z ze e y e e x z y ∂ ∂ ∂ = + = ∂ ∂ ∂  3 ,e z ∂ = ∂ which are linearly independent at each point of .M Now we define a semi-Riemannian metric g on M as (7.2) ( ) ( ) ( )1 3 2 3 1 2, , , 0,g e e g e e g e e= = = (7.3) ( ) ( ) ( )1 1 2 2 3 3, , , 1.g e e g e e g e e= = = Let η be a 1-form defined by ( ) ( )3,Z g Z eη = for any vector field Z M∈ and ϕ be a (1, 1) tensor field defined by Shah/BIBECHANA 12 (2015) 80-88 : p. 87 (7.4) ( ) ( ) ( )1 2 2 1 3, , 0.e e e e eϕ ϕ ϕ= = − = The linearity property of ϕ and g yields that (7.5) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 3 31, , , ,e Z Z Z e g Z U g Z U Z Uη ϕ η ϕ ϕ η η= = − + = − for any , .Z U M∈ If we take 3e ξ= in (7.5), ( ), , , gϕ ξ η defines an almost contact metric structure on .M By the definition of Lie bracket and (7.1) we have [ ]1 2 1 2 2 1 2 2 3 , . z z z z z z z e e e e e e e y e e e ye x z y y x z ye e e e ∂ ∂ ∂ ∂ ∂ ∂   = − = + − +   ∂ ∂ ∂ ∂ ∂ ∂    = − Proceeding same way we obtain [ ]2 3 2,e e e= − and [ ]1 3 1, .e e e= − Thus we have (7.6) [ ] [ ] [ ]2 1 2 2 3 2 3 2 1 3 1, , , , , . z ze e ye e e e e e e e e e= − = − = − Let ∇ be the Levi-Civita connection with respect to g then we have the Koszul's formula (7.7) ( ) ( ) ( ) ( ) [ ]( ) [ ]( ) [ ]( ) 2 , , , , , , , , , , . Xg Y Z Xg Y Z Yg Z X Zg X Y g X Y Z g Y Z X g Z X Y ∇ = + − + − + By the use of (7.2), (7.3) and (7.6), (7.7) yields (7.8) 1 1 1 2 2 2 3 3 3 2 2 3 1 2 2 3 1 3 2 2 3 2 1 2 3 1 1 2 3 2 2 3 2 1 1 2 1 1 , , , 2 2 1 1 , , , 2 2 1 1 0, , , 2 2 z z z z z z z z e e e e e e e e e e e e e e e e e e e e e e e e ye e e ye e e e e e e e e e e ∇ = − + ∇ = ∇ =   ∇ = − − ∇ = + ∇ = − +   ∇ = ∇ = − ∇ = In view of (2.6), (7.2), (7.3) and (7.4) we have 1 2 1 2 1 2, e ee e e eξ β α ξ α β∇ = − ∇ = + and 3 0e ξ∇ = for 3 .e ξ= Comparing these equations with (7.8) (first column), we get 21 2 zeα = − and 1.β = − Again, by virtue of (2.7) and ( ) ( ) ( )X X XY Y Yη η η∇ = ∇ − ∇ we obtain ( ) ( ) ( ) 1 2 3 2 1 1 1 1 1, , 0. 2 z e e ee e e eη β η α η∇ = = − ∇ = = − ∇ == Thus from above calculation the conditions (2.6) and (2.7) are satisfied and the structure ( ), , , gϕ ξ η is a trans-Sasakian structure of type ( ),α β where 21 2 zeα = − and 1.β = − Consequently ( )3 , , ,M gϕ ξ η is a trans-Sasakian manifold. 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