On LP-Sasakian Manifolds - Mathematical Sciences

BULLETIN of theMALAYSIANMATHEMATICALSCIENCESSOCIETYBull. Malaysian Math. Sc. Soc. (Second Series) 27 (2004) 17−26On LP-Sasakian ManifoldsA.A. SHAIKH AND SUDIPTA BISWASDepartment of Mathematics, University of North Bengal, P.O. NBU – 734430, Darjeeling, West Bengal, Indiae-mail: aask@epatra.com or aask2003@yahoo.co.inAbstract. The object of the present paper is to study LP-Sasakian manifolds satisfying certainconditions.2000 Mathematics Subject Classification: 53C15, 53C50.1. IntroductionIn 1989 K. Matsumoto [1] introduced the notion of LP-Sasakian manifold. ThenI. Mihai and R. Rosca [2] introduced the same notion independently and they obtainedseveral results in this manifold. LP-Sasakian manifolds have also been studied byK. Matsumoto and I. Mihai [3]; U.C. De, K. Matsumoto and A.A. Shaikh [4].In [5] Yano and Sawaki defined and studied a tensor field W on a Riemannianmanifold of dimension n which includes both the conformal curvature tensor C and theconcircular curvature tensor C ~ as special cases. This tensor field W is known as quasiconformalcurvature tensor. The quasi-conformally flat Riemannian manifold has beenstudied in [6]. The present paper deals with a study of LP-Sasakian manifolds satisfyingcertain conditions. After preliminaries, in section 3 we study an LP-Sasakian manifoldsatisfying the condition R ( X , Y ) ⋅W = 0,where R ( X , Y ) is considered as a derivationof the tensor algebra at each point of the manifold for tangent vectors X, Y and it is shownthat such a manifold is an Einstein manifold. Also in such a manifold we obtain anecessary and sufficient condition for the characteristic vector field ξ to be a harmonicvector field. Section 4 is devoted to the study of quasi-conformally recurrentLP-Sasakian manifolds. The last section deals with an LP-Sasakian manifold satisfyingthe condition S( X , ξ ) ⋅ R = 0 and it is proved that such a manifold is η-Einstein.2. PreliminariesAn n-dimensional differentiable manifold M is called an LP-Sasakian manifold [1], [3] ifit admits a ( 1, 1)tensor field φ , a contravariant vector field ξ , a 1-form η and aLorentzian metric g which satisfy

18A.A. Shaikh and Sudipta Biswasη ( ξ ) = −1,(2.1)2φ X = X + η(X ) ξ ,(2.2)g( φ X , φY) = g(X , Y ) + η(X ) η(Y )(2.3)g( X , ξ ) = η(X ), ∇ Xξ= φ X ,(2.4)( ∇ φ)(Y ) = g(X , Y ) ξ + η(Y ) X 2η( X ) η(Y ) ξ ,(2.5)X +where ∇ denotes the operator of covariant differentiation with respect to the Lorentzianmetric g.It can be easily seen that in an LP-Sasakian manifold, the following relations hold:Again if we putφξ = 0 , η(φX) = 0,(2.6)rank φ = n − 1.(2.7)Ω ( X , Y ) = g(X , φY)(2.8)for any vector fields X and Y, then the tensor field Ω ( X , Y ) is a symmetric ( 0, 2)tensorfield [1]. Also since the vector field η is closed in an LP-Sasakian manifold, we have[1] [4](i) ( ∇ η )( Y ) = Ω ( X , Y ),(ii) Ω ( X , ξ ) = 0(2.9)Xfor any vector fields X and Y.An LP-Sasakian manifold M is said to be η-Einstein if its Ricci tensor S is of theformS( X , Y ) = α g(X , Y ) + β η(X ) η(Y )(2.10)for any vector fields X, Y where α, β are functions on M. Let M be an n-dimensionalLP-Sasakian manifold with structure ( φ , ξ,η,g). Then we have [3] [4]:g( R(X , Y ) Z , ξ ) = η(R(X , Y ) Z)= g(Y,Z)η(X ) − g(X , Z)η(Y ), (2.11)R( ξ,X ) Y = g(X , Y ) ξ − η(Y ) X ,(2.12)R( X , Y ) ξ = η(Y ) X − η(X ) Y ,(2.13)R ( ξ,X ) ξ = X + η(X ) ξ ,(2.14)S( X , ξ ) = ( n − 1) η(X ),(2.15)S( φX, φY) = S(X , Y ) + ( n − 1) η(X ) η(Y )(2.16)for any vector fields X, Y, Z whereR ( X , Y ) Z is the Riemannian curvature tensor.

On LP-Sasakian Manifolds 19The quasi-conformal curvature tensor W on a manifold M of dimension n is definedby [5]~W ( X , Y ) Z = −(n − 2) b C(X , Y ) Z + [ a + ( n − 2) b]C(X , Y ) Z , (2.17)where a, b are arbitrary constants such that a and b are not zero simultaneously, C and C ~are conformal curvature tensor and concircular curvature tensor respectively, given by1C ( X , Y ) Z = R(X , Y ) Z − [ S(Y,Z)X − S(X , Z)Y + g(Y,Z)QX (2.18)n − 2r− g( X , Z)QY ] +[ g(Y,Z)X − g(X , Z)Y ],( n − 1) ( n − 2)C~ ( X , Y)Zr= R(X , Y)Z − [ g(Y,Z)X − g(X , Z)Y ], (2.19)n(n − 1)Q is the Ricci-operator i.e., g ( QX , Y ) = S(X , Y ) and r is the scalar curvature of themanifold. Using (2.18) and (2.19) in (2.17) we getW ( X , Y)Z = a R(X , Y)Z + b[ S(Y,Z)X − S(X , Z)Y + g(Y,Z)QXr ⎡ a ⎤− g( X , Z)QY ] − 2b{ g(Y,Z)X − g(X , Z)Y}.n ⎢+⎣n− 1 ⎥(2.20)⎦The above results will be used in the next sections.3. LP-Sasakian manifolds satisfying R ( X,Y ) ⋅W = 0Let us consider an LP-Sasakian manifold ( , g)M nsatisfying the conditionR ( X , Y ) ⋅ W = 0 . (3.1)Now,( R ( X , Y ) ⋅ W ) ( U,V ) Z = R(X , Y ) W ( U,V ) Z − W ( R(X , Y ) U,V ) Z− W ( U,R(X , Y ) V ) Z − W ( U,V ) R(X , Y ) Z . (3.2)From (3.1) and (3.2) we haveg ( R( ξ,Y ) W ( U,V ) Z , ξ ) − g(W ( R(ξ,Y ) U , V ) Z,ξ ) −(3.3)g ( W ( U,R(ξ,Y ) V ) Z,ξ ) − g(W ( U,V ) R(ξ,Y ) Z,ξ ) = 0 .

22A.A. Shaikh and Sudipta BiswasR ( X , Y ) ⋅ C = 0. Again, LP-Sasakian manifolds satisfying the conditionR ( X , Y ) ⋅ C = 0 has been studied in [4].This leads to the following:Corollary 3.2. An LP-Sasakian manifold ( , g)( n > 3)satisfying the conditionR ( X , Y ) ⋅ W = 0 is a space of constant curvature if a + ( n − 2) b = 0.Let us now consider an LP-Sasakian manifold ( , g)( n > 3)satisfying thecondition R ( X , Y ) ⋅ W = 0 which is not an Einstein one. Then (3.11) holds good.Differentiating (3.11) covariantly along X and then using (2.9) (i) we get( a − b)( ∇ X S)(V , Z)= −bdr(X ) [ g(V , Z)+ η ( V ) η ( Z)]+b[ n(n − 1) − r][ Ω(X , V ) η(Z)+ Ω(X , Z)η(V )].(3.13)Putting X = Z = eiin (3.13) and then taking summation for 1 ≤ i ≤ n we obtain byvirtue of (2.9) (ii)where ψ =n∑ ii = 1M n1( a + b)dr(V ) = b{[n(n − 1) − r]ψ − dr(ξ )} η(V ),(3.14)2ε Ω ( e , e ) = tr ⋅ φ .Replacing V by ξ in (3.14) we getBy virtue of (3.14) and (3.15) we obtainiiM n2bdr ( ξ ) = [ r − n(n − 1) ] ψ , if a − b ≠ 0.(3.15)a − b2bdr( V ) = [ n(n − 1) − r] ψ η ( V ),if a + b ≠ 0.(3.16)a − bIf r is constant then for b ≠ 0,(3.16) yields either r = n( n − 1) , or ψ = 0.If r = n( n − 1) , then from (3.12), it follows that the manifold is Einstein. Hence if themanifold is not Einstein then we must have ψ = 0,which means that the vector field ξis harmonic. Again, if ψ = 0,then from (3.16), it follows that r is constant.Thus we can state the following:

On LP-Sasakian Manifolds 23Theorem 3.3. Let ( , g)( n > 3)be an LP-Sasakian manifold satisfying thecondition R ( X , Y ) ⋅ W = 0 which is not an Einstein one. Then the scalar curvature ofthe manifold is constant if and only if the timelike vector field ξ is harmonic providedthat a ± b ≠ 0 and b ≠ 0.In particular, ifM na = b then we have the following:Corollary 3.3. Let ( , g)( n > 3)be an LP-Sasakian manifold satisfying thecondition R ( X , Y ) ⋅ W = 0 which is not an Einstein one. If a = b then the manifold isof constant scalar curvature.M n4. Quasi-conformally recurrent LP-Sasakian manifoldsA non-flat Riemannian manifold M is said to be quasi-conformally recurrent [7] if thequasi-conformal curvature tensor W satisfies the condition ∇ W = A ⊗ W , where A is aneverywhere non-zero 1-form. We now define a function f on M by f = g(W , W ),where the metric g is extended to the inner product between the tensor fields in thestandard fashion.2Then we know that f ( Yf ) = f A(Y ).So from this we haveYf = f A(Y ) (because f ≠ 0 ). (4.1)From (4.1) we have1X ( Yf ) = ( Xf ) ( Yf ) + ( XA(Y )) f .f2HenceX ( Yf ) − Y ( Xf ) = { XA(Y ) − YA(X )} f .Therefore we get( ∇ ∇ − ∇ ∇ − ∇ ) f = { XA(Y ) − YA(X ) − A([X , Y ])} .XYYX[ X , Y ]fSince the left hand side of the above equation is identically zero and f ≠ 0 on M by ourassumption we obtainthat is, the 1-form A is closed.dA ( X , Y ) = 0,(4.2)

24A.A. Shaikh and Sudipta BiswasNow, from ( ∇ W ) ( U,V ) Z A(X ) W ( U,V ) Z , we getX =Hence from (4.2) we get( ∇ U ∇VW) ( X , Y ) Z = { UA(V ) + A(U ) A(V )} W ( X , Y ) Z .( R ( X , Y ) ⋅ W ) ( U,V ) Z = [ 2dA(X , Y )] W ( U,V ) Z = 0.Therefore, for a quasi-conformally recurrent manifold, we haveThus we can state the following:R ( X , Y ) ⋅ W = 0 for all X ,Y .Theorem 4.1. A quasi-conformally recurrent LP-Sasakian manifold ( , g)( n > 3)is an Einstein manifold provided that a − b ≠ 0 and a + ( n − 2) b ≠ 0.Corollary 4.1. A quasi-conformally recurrent LP-Sasakian manifold ( , g)( n > 3)is of constant scalar curvature for a = b and is a space of constant curvature ifa + ( n − 2) b = 0.Since for a quasi-conformally symmetric LP-Sasakian manifold ( , g)( n > 3) ,we have ( ∇U W ) ( X , Y ) Z = 0 which implies R ( X , Y ) ⋅ W = 0,we can state thefollowing:Corollary 4.2. A quasi-conformally symmetric LP-Sasakian manifold ( , g)( n > 3)is an Einstein manifold provided that a − b ≠ 0 and a + ( n − 2) b ≠ 0.M nM nM nM n5. LP-Sasakian manifolds satisfying the condition S ( X,ξ)⋅ R = 0We now consider an LP-Sasakian manifold ( , g)( n > 3)M nsatisfying the condition( S(X , ξ ) ⋅ R)( U,V ) Z = 0.(5.1)By definition we have( S(X , ξ ) ⋅ R) ( U,V ) Z = (( X ∧ S ξ ) ⋅ R) ( U,V ) Z= ( X ∧ S ξ ) R(U,V ) Z + R((X ∧ S ξ ) U,V ) Z+ R( U,( X ∧ ξ ) V ) Z + R(U,V ) ( X ∧ ξ ) Z , (5.2)SS

where the endomorphismXOn LP-Sasakian Manifolds 25∧ Y is defined byS( X ∧ S Y ) Z = S(Y,Z)X − S(X , Z)Y .(5.3)Using the definition of (5.3) in (5.2) we get by virtue of (2.15)( S ( X , ξ ) ⋅ R) ( U,V ) Z = ( n − 1)[ η(R(U,V ) Z)X + η(U ) R(X , V ) Z+ η ( V ) R(U,X ) Z + η(Z)R(U,V ) X ]− S ( X , R(U , V ) Z)ξ − S(X , U ) R(ξ,V ) Z− S( X , V ) R(U,ξ ) Z − S(X , Z)R(U,V ) ξ . (5.4)In view of (5.1) and (5.4) we have[ η ( R(U,V ) Z)X + η(U ) R(X , V ) Z + η(V ) R(U,X ) Z + ( Z)R(U,V X ]( n − 1)η )− S ( X , R(U , V ) Z)ξ − S(X , U ) R(ξ,V ) Z− S ( X , V ) R(U,ξ ) Z − S(X , Z)R(U,V ) ξ = 0. (5.5)Taking the inner product on both sides of (5.5) by ξ we obtain( n − 1) [ η ( R(U,V ) Z)η(X ) + η(U ) η(R(X , V ) Z)+ η(V ) η(R(U,X ) Z)+ η ( Z)η(R(U,V ) X ) ] + S(X , R(U,V ) Z)− S(X , U ) η(R(ξ,V ) Z)− S ( X , V ) η(R(U,ξ ) Z)− S(X , Z)η(R(U,V ) ξ ) = 0.(5.6)Putting U = Z = ξ in (5.6) and using ( 2.11) − (2.15)we getS( X , V ) = (1 − n)g(X , V ) + 2(1 − n)η(X ) η(V ),(5.7)which means that the manifold is η-Einstein. This leads to the following:Theorem 5.1. An LP-Sasakian manifold ( , g)( n > 3)S( X , ξ ) ⋅ R = 0 is an η-Einstein manifold.M nsatisfying the conditionAgain, differentiating (5.7) covariantly along Y and using (2.9) we get( ∇ S) ( X , V ) = 2(1 − n)[ Ω(X , Y ) η(V ) + Ω(Y,V ) η(X )].(5.8)YTaking an orthonormal frame field and contracting (5.8) over Y and V we obtaindr( X ) = 4(1 − n)ψ η ( X ),(5.9)

26A.A. Shaikh and Sudipta Biswaswhere ψ = tr.φ . From (5.9), it follows thatdr ( X ) = 0 if and only if ψ = 0.Thus we have the following:Theorem 5.2. Let ( , g)( n > 3)be an LP-Sasakian manifold satisfying thecondition S( X , ξ ) ⋅ R = 0.Then the scalar curvature of the manifold is constant if andonly if the vector field ξ is harmonic.M nAcknowledgement. The authors are grateful to the referee for his valuable suggestionsin the improvement of the paper.References1. K. Matsumoto, On Lorentzian paracontact manifolds, Bull. of Yamagata Univ., Nat. Sci.12 (1989), 151−156.2. I. Mihai and R. Rosca, On Lorentzian P-Sasakian manifolds, Classical Analysis, WorldScientific Publi. (1992), 155−169.3. K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para Sasakianmanifold, Tensor, N.S. 47 (1988), 189−197.4. U.C. De, K. Matsumoto and A.A. Shaikh, On Lorentzian para-Sasakian manifolds, Rendicontidel Seminario Matematico di Messina, Serie II, Supplemento al n. 3 (1999), 149−158.5. K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group,J. Diff. Geom. 2 (1968), 161−184.6. K. Amur and Y. B. Maralabhavi, On quasi-conformally flat spaces, Tensor, N.S. 31 (1977),194−198.7. T. Adati and T. Miyazawa, On a Riemannian space with recurrent conformal curvature,Tensor, N.S. 18 (1967), 348−354.