The work towards of the attending paper is to interpret the trans-Sasakian manifold equipped with m-projective curvature tensor and its various geometric properties. First, we observe that m-projectively flat trans-Sasakian manifold is Einstein. In order, we discussed m-projectively conservative and φ-m-projectively flat trans-Sasakian manifold. Following, we found the sufficient condition for quasi m-projectively flat transSasakian manifold to be m-projectively flat. In the end, the m-projectively and φ-mprojectively symmetric trans-Sasakian manifolds are analyzed.The authors pay their sincere gratitute to the reviewer for his/her valuable comments. The authors would also like to express their great appreciation to the National Board for Higher Mathematics (Department of Atomic Energy), Mumbai, India for financial support in the form (No.NBHM/R.P.48/2012/Fresh/364) of research project.Publisher's Versio

Jaiswal, Jai Prakash

Yadav, Arjun Singh

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.7, N.2, 2017, pp. 282-290ON TRANS-SASAKIAN MANIFOLD EQUIPPED WITHm-PROJECTIVE CURVATURE TENSORJ. P. JAISWAL 1, A. S. YADAV2, §Abstract. The work towards of the attending paper is to interpret the trans-Sasakianmanifold equipped with m-projective curvature tensor and its various geometric proper-ties. First, we observe that m-projectively flat trans-Sasakian manifold is Einstein. Inorder, we discussed m-projectively conservative and φ-m-projectively flat trans-Sasakianmanifold. Following, we found the sufficient condition for quasi m-projectively flat trans-Sasakian manifold to be m-projectively flat. In the end, the m-projectively and φ-m-projectively symmetric trans-Sasakian manifolds are analyzed.Keywords: Trans-Sasakian manifold, m-projectively flat, Einstein manifold, m-projectiveconservative.AMS Subject Classification: 53C15, 53B05.1. IntroductionOubina [8] initiated a new class of almost contract manifold, called trans-Sasakianmanifold, which is of type (0, 0), (α, 0) and (0, β) are respectively, familiar as the cosym-plectic, α-Sasakian and β-Kenmotsu manifold, α, β are the scalar smooth functions. Infact if α = 0, β = 1 and α = 1, β = 0, then a trans-Sasakian manifold will enhance aKenmotsu and Sasakian manifold, respectively.In 1971, Pokhariyal and Mishra[9] established a new curvature known as m-projectivelycurvature tensor on Riemannian manifold. Followed that many researcher such as Ojha[6, 7], Singh [12], Choubey and Ojha [3] studied properties of m-projective curvature indifferent manifolds. We say that a Riemannian manifold is flat if its curvature vanishesat each point. Following this sense Ojha [7] and Zengin [15] consider the m-projectiveflat in the Sasakian and LP-Sasakian manifold, respectively. The idea of local symmetryof a Riemannian manifold studied by Cartan [2] and mild version of local symmetry,Takahashi [13] introduced the notion of φ-symmetry on a Sasakian manifold. In this series,we investigate some results about flatness, symmetry and space time with m-projectivecurvature in trans-Sasakian structure.The paper classified as follows: In part 2, we put some basic formulae and definitionof trans-Sasakian manifold. In the next part, we confer about m-projectively flat trans-Sasakian manifold and mentioned a sufficient condition for such a manifold to be Einstein.1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, M. P. India.e-mail: asstprofjpmanit@gmail.com, ORCID1: http://orcid.org/0000-0003-4308-2280;2 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, M. P. India.e-mail: arjunsinghyadav7@gmail.com, ORCID2: http://orcid.org/0000-0003-3594-7846;§ Manuscript received: July 03, 2016; accepted: May 22, 2017.TWMS Journal of Applied and Engineering Mathematics Vol.7, No.2; c© Işık University, Departmentof Mathematics, 2017; all rights reserved.282J. P.JAISWAL, A. S. YADAV: ON TRANS-SASAKIAN MANIFOLD EQUIPPED WITH M -PROJECTIVE...283Then, we found the condition such that the m-projective conservative trans-Sasakian man-ifold is of constant curvature. Successive that, we find the condition for φ-m-projectivelyflat trans-Sasakian manifold to be η-Einstein and quasi m-projectively flat is of constantcurvature. In the last, we examine the m-projective and φ-m-projective symmetric trans-Sasakian manifolds.2. PreliminariesIn this section, we mention some basic formulae and definitions, which will be usedlater.Let Mm be an m = (2n+ 1) dimensional almost contact metric manifold [1], consistingof a (1, 1) tensor field φ, a characteristic vector field ξ, a 1-form η and a Riemannian metricg. Thenφ2X = −X + η(X)ξ, η(ξ) = 1, η(φX) = 0, φξ = 0, (1)g(φX, φY ) = g(X,Y )− η(X)η(Y ), (2)g(ξ, ξ) = 1, φ ◦ ξ = 0, η ◦ φ = 0, (3)for any X, Y in TM . From (1) and (2), it can be easily seen thatg(X,φY ) = −g(φX, Y ), g(X, ξ) = η(X). (4)For an almost contact metric structure (φ, ξ, η, g) on M , we putΦ(X,Y ) = g(X,φY ). (5)Let M2n+1 be almost contact manifold and consider the structure (M × R, J , G)belongs to the class W4 of the Hermitian manifolds, we denote a vector field on M × Rby (X, f ddt), where X is tangent to M, t is the co-ordinates of R and f as C∞ function onM ×R. Define an almost complex structure [4]J(X, fddt)=(φX − fξ, η(X) ddt),for any vector field X on M ×R and G is Hermitian metric on the product M ×R.Thismay be expressed by the condition(∇Xφ)Y = α(g(X,Y )ξ − η(Y )X) + β(g(φX, Y )ξ − η(Y )φX), (6)where ∇ is a Levi-civita connection and α, β are some smooth functions on M2n+1 andwe say that trans-Sasakian structure is type (α, β). From the above, it is follows that(∇Xη)Y = −αg(φX, Y ) + βg(φX, φY ), (7)(∇Xξ) = −αφX + β(X − η(X)ξ). (8)On a trans-Sasakian manifold M2n+1 with structure (φ, ξ, η, g), the following relationshold [11]R(X,Y, ξ) = (α2 − β2)[η(Y )X − η(X)Y ] + 2αβ[η(Y )φX − η(X)φY ]+(Y α)φX − (Xα)φY + (Y β)φ2X − (Xβ)φ2Y, (9)R(ξ,X, ξ) = (α2 − β2 − ξβ)[η(X)ξ −X], (10)2αβ + ξα = 0, (11)η(R(X,Y, ξ)) = η(R(ξ, Y, ξ)) = 0, (12)R(ξ, Y, Z) = (α2 − β2)[g(Z, Y )ξ − η(Z)Y ] + 2αβ[g(φZ, Y )ξ + η(Z)φY ] + (Zα)φY+g(φZ, Y )gradα+ (Zβ)[Y − η(Y )ξ]− g(φZ, φY )gradβ, (13)S(X, ξ) = [2n(α2 − β2)− ξβ]η(X)− (2n− 1)Xβ − (φX)α, (14)284 TWMS J. APP. ENG. MATH. V.7, N.2, 2017S(ξ, ξ) = 2n(α2 − β2 − ξβ), (15)S(φX, φY ) = S(X,Y )− 2n(α2 − β2 − ξβ)η(X)η(Y ), (16)Qξ = (2n(α2 − β2)− ξβ)ξ − (2n− 1)gradβ + φ(gradα), (17)S(X,Y ) = g(QX,Y ), (18)where R is the curvature tensor, S is the Ricci tensor, r is scalar curvature and Q being thesymmetric endomorphism of the tangent space at each point corresponding to Ricci-tensorS. Now, we assume thatφ(gradα) = (2n− 1)gradβ, (19)then [11]S(X, ξ) = 2n(α2 − β2)η(X), (20)S(φX, φY ) = S(X,Y )− 2n(α2 − β2)η(X)η(Y ), (21)Qξ = 2n(α2 − β2)ξ, (22)(∇WS)(Y, ξ) = 2n(α2 − β2)[−αg(Y, φW ) + βg(Y,W )]+αS(Y, φW )− βS(Y,W ). (23)Now we are going to mention the following definition, which will be considered in the laterresults:Definition 2.1. [4] A trans-Sasakian manifold M2n+1 is said to be η-Einstein, if the Riccitensor S satisfies the relationS(X,Y ) = ag(X,Y ) + bη(X)η(Y ), (24)for all X and Z ∈ TM , where a and b are smooth functions on M2n+1.In particular, if b = 0 then it reduce to the Einstein manifold.3. m-projectively flat Trans-Sasakian ManifoldDefinition 3.1. [10] A trans-Sasakian manifold M2n+1 is said to be m-projectively flat,if the m-projective curvature tensor M satisfies the relationM(X,Y, Z) = 0, for all X, Y and Z, (25)where m-projective curvature tensor M is given by [9]M(X,Y, Z) = R(X,Y, Z)− 14n[S(Y,Z)X − S(X,Z)Y+g(Y,Z)QX − g(X,Z)QY]. (26)Theorem 3.1. An m-projectively flat trans-Sasakian manifold M2n+1 is an Einstein man-ifold.Proof. Let M2n+1 be m-projectively flat trans-Sasakian manifold, then the equation (26)becomesR(X,Y, Z) =14n[S(Y, Z)X − S(X,Z)Y+g(Y, Z)QX − g(X,Z)QY]. (27)J. P.JAISWAL, A. S. YADAV: ON TRANS-SASAKIAN MANIFOLD EQUIPPED WITH M -PROJECTIVE...285Proceeds the inner product in above equation both side with respect to U , then we obtaing(R(X,Y, Z), U) =14n[S(Y,Z)g(X,U)− S(X,Z)g(Y, U)+g(Y, Z)S(X,U)− g(X,Z)S(Y,U)]. (28)Taking the contraction over X and U , we getS(Y,Z) =r(2n+ 1)g(Y, Z). (29)Theorem 3.2. An m-projectively flat trans-Sasakian manifold M2n+1 is of constant cur-vature.Proof. Let M2n+1 be m-projectively flat trans-Sasakian manifold. Then by existence ofthe relation (27) and after using the equations (29), we can findR(X,Y, Z) =r2n(2n+ 1)[g(Y, Z)X − g(X,Z)Y ]. (30)By virtue of the Theorem (3.1) and Theorem (3.2), we state the following corollary:Corollary 3.1. An m-projectively flat trans-Sasakian manifold M2n+1, is of constantcurvature iff it is Einstein.4. m-projective conservative Trans- Sasakian ManifoldDefinition 4.1. [5] A trans-Sasakian manifold M2n+1 is said to be m-projective conser-vative, if the m-projective curvature tensor M satisfies the relationdiv(M(X,Y, Z)) = 0, for all X, Y and Z, (31)where div denotes the divergence.Theorem 4.1. An Einstein trans-Sasakian manifold M2n+1 with constant scalar curva-ture is m-projective conservative iff it is conservative.Proof. We assume that M2n+1 be Einstein M -projective trans-Sasakian manifold then byvirtue of relation (26), we obtainM(X,Y, Z) = R(X,Y, Z)− 14n[g(Y,Z)X − g(X,Z)Y ]. (32)By taking covariant derivative both side with respect to W in above equation, we obtain(∇W )M(X,Y, Z) = (∇W )R(X,Y, Z). (33)Contracting the above relation with W , we can finddiv(M(X,Y, Z)) = div(R(X,Y, Z)). (34)If manifold is m-projective conservative, thendiv(R(X,Y, Z)) = 0. (35)Then the converse part is trivial. 286 TWMS J. APP. ENG. MATH. V.7, N.2, 20175. φ-m-projectively flat Trans- Sasakian ManifoldDefinition 5.1. [11] A trans-Sasakian manifold M2n+1 is said to be φ-m-projectively flat,if the m-projective curvature tensor M satisfies the relationφ2(M(φX, φY, φZ)) = 0, for all X, Y and Z. (36)Theorem 5.1. A φ-m-projectively flat trans-Sasakian manifold M2n+1 is an η-Einsteinmanifold.Proof. Let us we assume that M2n+1 be φ-M -projectively flat trans-Sasakian manifold.Then by virtue of the relations (36) and (1), we haveM(φX, φY, φZ) = η(M(φX, φY, φZ))ξ, (37)which impliesg(M(φX, φY, φZ), φU) = η(M(φX, φY, φZ))g(ξ, φU). (38)By the relation (1), the above equation becomesg(M(φX, φY, φZ), φU) = 0. (39)Now, by virtue of the relation (26), we obtaing(R(φX, φY, φZ), φU) =14n[S(φY, φZ)g(φX, φU)− S(φX, φZ)g(φY, φU)+g(φY, φZ)S(φX, φU)− g(φX, φZ)S(φY, φU)]. (40)Let {e1, e2, ........, e2n, ξ} be a local orthonormal basis of vector field in M2n+1 by usingthe fact that {φe1, φe2, ........, φe2n, ξ} is also a orthonormal basis, if we put X=U=ei inabove relation and taking summation with respect to i, then we have2n∑i=1g(R(φei, φY, φZ), φei)=14n[ 2n∑i=1S(φY, φZ)g(φei, φei)−2n∑i=1S(φei, φZ)g(φY, φei)+2n∑i=1g(φY, φZ)S(φei, φei)−2n∑i=1g(φei, φZ)S(φY, φei)]. (41)Now, we find that2n∑i=1g(R(φei, φY, φZ), φei) = S(φY, φZ)− (α2 − β2 − ξβ)g(φY, φZ),2n∑i=1g(φei, φei) = 2n, (42)2n∑i=1S(φei, φZ)g(φY, φei) = S(φY, φZ), (43)2n∑i=1S(φei, φei) = r − 2n(α2 − β2 − ξβ), (44)J. P.JAISWAL, A. S. YADAV: ON TRANS-SASAKIAN MANIFOLD EQUIPPED WITH M -PROJECTIVE...287(2n+ 2)S(φY, φZ) = [r + 2n(α2 − β2 − ξβ)]g(φY, φZ). (45)Using the relations (42)-(45), the equation (41) becomesS(Y, Z) =1(2n+ 2)[r + 2n(α2 − β2 − ξβ)]g(Y, Z)+1(2n+ 2)[2n(2n− 1)(α2 − β2 − ξβ)]η(Y )η(Z). (46)Hence the manifold is η-Einstein. 6. Quasi m-projectively flat Trans-Sasakian ManifoldDefinition 6.1. [10] A trans-Sasakian manifold M2n+1 is said to be quasi m-projectivelyflat, if the m-projective curvature tensor M satisfies the relationg(M(X,Y, Z), φU) = 0, for all X, Y Z and U. (47)Theorem 6.1. A quasi m-projectively flat trans-Sasakian manifold M2n+1 satisfyingφ(gradα) = (2n− 1)gradβ is m-projectively flat if it is of constant curvature .Proof. Let M2n+1 be a quasi m-projectively flat trans-Sasakian manifold. Then by therelations (47) and (26), we obtaing(R(X,Y, Z), φU) =14n[S(Y, Z)g(X,φU)− S(X,Z)g(Y, φU)+g(Y, Z)S(X,φU)− g(X,Z)S(Y, φU)]. (48)Putting X = φX in the above relation, we getg(R(φX, Y, Z), φU) =14n[S(Y, Z)g(φX, φU)− S(φX,Z)g(Y, φU)+g(Y, Z)S(φX, φU)− g(φX,Z)S(Y, φU)]. (49)After putting X=U=ei in above relation and taking summation with respect to i, weattain2n∑i=1g(R(φei, Y, Z), φei)=14n[ 2n∑i=1S(Y, Z)g(φei, φei)−2n∑i=1S(φei, Z)g(Y, φei)+2n∑i=1g(Y, Z)S(φei, φei)−2n∑i=1g(φei, Z)S(Y, φei)]. (50)If M2n+1 satisfies φ(gradα) = (2n− 1)gradβ, we have the following relation2n∑i=1g(R(φei, Y, Z), φei) = S(Y,Z)− (α2 − β2)g(φY, φZ), (51)2n∑i=1S(φei, Z)g(Y, φei) = S(Y, Z)− 2n(α2 − β2)η(Y )η(Z), (52)2n∑i=1S(φei, φei) = r − 2n(α2 − β2). (53)288 TWMS J. APP. ENG. MATH. V.7, N.2, 2017After using the relations (42), (51), (52) and (53) in the equation (50), we obtainS(Y, Z) =[r + 2n(α2 − β2)(2n+ 2)]g(Y, Z). (54)By virtue of the equation (26) and using the above relation, we can getM(X,Y, Z) = R(X,Y, Z)−[r + 2n(α2 − β2)(n+ 1)][g(Y, Z)X − g(X,Z)Y ]. (55)Which shows the statement. 7. m-projectively symmetric Trans-Sasakian ManifoldDefinition 7.1. [4] A trans-Sasakian manifold M2n+1 is said to be m-projectively sym-metric, if the m-projective curvature tensor M satisfies the relation(∇WM)(X,Y, Z) = 0, for all X, Y Z and W. (56)Theorem 7.1. A m-projectively symmetric trans-Sasakian M2n+1 manifold is Ricci-recurrent.Proof. Let M2n+1 is a m-projectively symmetric trans-Sasakian manifold. Then by theequations (56) and (26), we findg((∇WR)(X,Y, Z), U) =14n[(∇WS)(Y, Z)g(X,U)− (∇WS(X,Z)g(Y,U)+(∇WS)(X,U)g(Y, Z)− (∇WS)(Y, U)g(X,Z)]. (57)Taking contraction over X and U , we secure(∇WS)(Y,Z) =14n[(2n+ 1)(∇WS)(Y,Z)− (∇WS(Y,Z)+dr(W )g(Y, Z)− (∇WS)(Y,Z)], (58)which implies(∇WS)(Y,Z) =dr(W )(2n+ 1)g(Y, Z). (59)Hence the manifold is Ricci-recurrent. Suppose the scalar curvature r is constant then we mention the corollary:Corollary 7.1. An m-projective symmetric trans-Sasakian manifold M2n+1 with constantscalar curvature is Einstein.8. φ-m-projectively symmetric Trans-Sasakian ManifoldDefinition 8.1. [4] A trans-Sasakian manifold M2n+1 is said to be φ − m-projectivelysymmetric, if the m-projective curvature tensor M satisfies the relationφ2(∇WM)(X,Y, Z) = 0, for all X, Y Z and W. (60)Theorem 8.1. A φ − m-projectively symmetric trans-Sasakian M2n+1 manifold is anEinstein.J. P.JAISWAL, A. S. YADAV: ON TRANS-SASAKIAN MANIFOLD EQUIPPED WITH M -PROJECTIVE...289Proof. Let us consider M2n+1 is a φ−m-projectively symmetric trans-Sasakian manifold.Then by the equations (60) and (1), we getg((∇WM)(X,Y, Z), U) = η((∇WM)(X,Y, Z))g(ξ, U). (61)The existence of the relation (26), the above equation becomesg((∇WR)(X,Y, Z), U)−14n[(∇WS)(Y,Z)g(X,U)− (∇WS)(X,Z)g(Y,U)+(∇WS)(X,U)g(Y,Z)− (∇WS)(Y,U)g(X,Z)]= g((∇WR)(X,Y, Z), ξ)g(ξ, U)−14n[(∇WS)(Y,Z)g(X, ξ)− (∇WS)(X,Z)g(Y, ξ)+(∇WS)(X, ξ)g(Y,Z)− (∇WS)(Y, ξ)g(X,Z)]g(ξ, U). (62)After contraction over X and Z, we secure(∇WS)(Y,U)− (∇WS)(Y, ξ)η(U) =dr(W )(6n− 1)[−g(Y,U) + g(Y, ξ)η(U)]. (63)Putting Y = ξ, we get(∇WS)(ξ, U) = 0. (64)By virtue of the relation (23) and above equation, we have2n(α2 − β2)[−αg(U, φW ) + βg(U,W )] + αS(U, φW )− βS(U,W ) = 0. (65)We put U=φU andW=φW , respectively in the above relation and then using equations(1), (4), (18), (19) and (22), we find thatS(U,W ) = 2n(α2 − β2)g(U,W )andS(φU,W ) = 2n(α2 − β2)g(φU,W ). (66)Acknowledgment: The authors pay their sincere gratitute to the reviewer for his/hervaluable comments. The authors would also like to express their great appreciation to theNational Board for Higher Mathematics (Department of Atomic Energy), Mumbai, Indiafor financial support in the form (No.NBHM/R.P.48/2012/Fresh/364) of research project.References[1] Blair,D.E., (1976), Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, SpringerVerlag.[2] Cartan,E., (1926), Sur une classes remarquable d’espaces de Riemann, Bull. Soc. Math. France, 54,pp.214-26.[3] Chaubey,S.K. and Ojha,R.H., (2010), On the m-projective curvature tensor of a Kenmotsu manifold,Diff. Geom. Dyn. Sys., 12, pp.52-60.[4] De,U.C. and Shaikh,A.A., (2009), Complex manifolds and contact manifolds, Narosa Publication,New Delhi, India.[5] Hicks,N.J., (1969), Notes on Differential Geometry, Affiliated East West Press Pvt. Ltd.[6] Ojha,R.H., (1975), A note on the mprojective curvature tensor, Indian J. Pure Appl. Math., 8 (12),pp.1531-1534.[7] Ojha,R.H., (1986), m-projectively flat Sasakian manifolds, Indian J. Pure Appl. Math., 17 (4),pp.1531-1534.290 TWMS J. APP. ENG. MATH. V.7, N.2, 2017[8] Oubina,J.A., (1985), New classes of almost contact metric structures, Publ. Math. Debrecen, 32,pp.21-38.[9] Pokhariyal,G.P. and Mishra,R.S., (1971), Curvature tensors and their relativistic significance-ii, Yoko-hama Math. J., 19, pp.97-103.[10] Prakash,A., Ahmad,M., and Srivastava,A., (2013), m-projective curvature tensors on a LP-Sasakianmanifolds, IOSR J. Math., 6, pp.19-23.[11] Prasad,R. and Srivastava,V., (2013), Some results on trans-Sasakian manifolds, Math. Vesnik, 65,pp.346-352.[12] Singh,J.P., (2015), On the M -projective curvature tensor of Sasakian manifold, Sci. Vis., 15 (2),pp.76-79.[13] Takahashi,T., (1977), Sasakian φ-symmetric space, Tohoku Math. J., 29, pp.91-113.[14] Tamassy,L. and Binh,T.Q., (1989), On weakly symmetric and weakly projective symmetric Riemann-ian manifold, Colloquia Math. Soc., 50, pp.663-667.[15] Zengin,F.O., (2013), On m-projectively flat LP-Sasakian manifolds, Ukr. Math. J., 65, pp.1725-1732.Jai Prakash Jaiswal is currently working as an assistant professor in the Depart-ment of Mathematics of Maulana Azad National Institute of Technology, Bhopal,India. He received his Ph. D. degree from Banaras Hindu University, Varanasi,India. His area of interest includes differential geometry of manifolds and numericalfunctional analysis.Arjun Singh Yadav worked as junior research fellow in NBHM-DAE, Mumbai,India sponsored research project in the Department of Mathematics of MaulanaAzad National Institute of Technology, Bhopal, India. He received his B. Sc. andM. Sc. degrees from Vikram University, Ujjain, India. His area of interest includesdifferential geometry of manifolds.

On trans-Sasakian manifold equipped with m-projective curvature tensor

http://acikerisim.isikun.edu.tr/xmlui/bitstream/11729/2630/1/vol.7.no.2-10.pdf

On trans-Sasakian manifold equipped with m-projective curvature tensor

Abstract

Similar works

Full text

Available Versions

Isik University Academic Open Access