In the present paper we study some geometrical properties of semi-generalized recurrent P-Sasakian manifolds

Singh, Jay Prakash

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 31, No 1 (2016), 213–225ON A TYPE OF SEMI-GENERALIZED RECURRENT P-SASAKIANMANIFOLDSArchana Singh, J.P. Singh∗and Rajesh KumarAbstract. In the present paper we study some geometrical properties of semi-generalizedrecurrent P-Sasakian manifolds.keywords: Semi-generalized recurrentmanifold, P-Sasakianmanifolds, Concircular cur-vature tensor, Einstein manifold,M-projective curvature tensor.1. IntroductionThe idea of recurrent manifolds was introduced by A.G. Walker in 1950 [16].On the other hand, De and Guha [3] introduced generalized recurrent manifoldwith the non-zero 1-form A and another non-zero associated 1-form B. Such amanifold has been denoted by GKn. If the associated 1-form B becomes zero,then the manifold GKn reduces to a recurrent manifold introduced by Ruse [12]which is denoted by Kn. In 1977, Adati and Matsumoto [1] defined P-Sasakian andSpecial Para Sasakianmanifolds, which are special classes of an almost para-contactmanifold introduced by Sato [13]. Para Sasakian manifolds have been studied byDe and Pathak [4], Matsumoto et.al. [8], Matsumoto [9], Shukla and Shukla [14],Singh [15], De and Sarkar [5] and many others.A Riemannian manifold (Mn, ) is called a semi-generalized recurrent manifoldif its curvature tensor R satisfies the condition(∇XR)(Y,Z)W = A(X)R(Y,Z)W + B(X)(Z,W)Y,(1.1)where A and B are two 1-forms, B is non-zero, P1 and P2 are two vector fields suchthat(X,P1) = A(X), (X,P2) = B(X),(1.2)for any vector field X and ∇ denotes the operator of covariant diﬀerentiation withrespect to the metric .Received August 21, 2015; Accepted November 1, 20152010 Mathematics Subject Classification. Primary 53C25; Secondary 53C15∗Corresponding author213214 Archana Singh, J.P.Singh and R. KumarGeneralizing the notion of recurrency the author Khan [6] introduced the notionof generalized recurrent Sasakianmanifold. In the paper [11], B. Prasad introducedthe notion of semi-generalized recurrent manifold and obtained some interestingresults. Recently Rajesh Kumar et.al. [7] studied semi generalized recurrent LP-Sasakian manifolds. Motivated by the above studies, in this paper we extendthe study of semi-generalized recurrent to Para-Sasakian manifolds. The paperis organized as follows: Section 2, consist the basic definitions of P-Sasakian andEinsteinmanifolds. In Section 3, we studied semi-generalized recurrentP-Sasakianmanifolds. In section 4, we prove that a semi-generalized φ-recurrent P-Sasakianmanifold is an Einstein manifold. Section 5 is devoted to the study of semi-generalized concircular φ-recurrent P-Sasakian manifolds. Section 6 is about thestudy of M-projective φ-recurrent P-Sasakian manifolds respectively. In the lastsection we studied the three-dimensional locally semi-generalized φ-recurrent P-Sasakian manifolds.2. IntroductionAnn-dimensionaldiﬀerentiablemanifoldMn is aPara-Sasakian (brieflyP-Sasakian)manifold if it admits a (1,1) tensor field φ, a contravariant vector field ξ, a covariantvector field η, and a Riemannian metric , which satisfyφ2X = X − η(X)ξ, (X, ξ) = η(X), φξ = 0,(2.1)(φX, φY) = (X,Y) − η(X)η(Y),(2.2)(∇Xφ)Y = −(X,Y)ξ− η(Y)X + 2η(X)η(Y)ξ,(2.3)∇Xξ = φX,(2.4)(∇Xη)(Y) = (φX,Y) = (φY,X),(2.5)for any vectorfieldsX andY, where∇denotes covariant diﬀerentiationwith respectto  ([1],[13]).It can be seen that in a P-Sasakian manifold Mn with the structure (φ, ξ, η, ),the following relations hold:η(ξ) = 1, η(φX) = 0,(2.6)rank(φ) = (n − 1).(2.7)Further in a P-Sasakian manifold the following relations also hold:η(R(X,Y)Z) = (X,Z)η(Y) − (Y,Z)η(X),(2.8)On A Type of Semi-Generalized Recurrent P-Sasakian Manifolds 215R(X,Y)ξ = η(X)Y − η(Y)X,(2.9)R(ξ,X)Y = η(Y)X − (X,Y)ξ,(2.10)R(ξ,X)ξ = X − η(X)ξ,(2.11)Qξ = −(n − 1)ξ,(2.12)S(X, ξ) = −(n − 1)η(X),(2.13)S(φX, φY) = S(X,Y) + (n − 1)η(X)η(Y),(2.14)for any vector fields X, Y, Z, where R and S are the Riemannian curvature tensorand Ricci tensor of the manifold respectively.A P-Sasakian manifold Mn is said to be Einstein if the Ricci tensor S is of the formS(X,Y) = λ(X,Y),(2.15)where λ is a constant.3. Semi-Generalized Recurrent P-SasakianmanifoldsDefinition 3.1. ARiemannianmanifold (Mn, ) is semi-generalizedRicci recurrentmanifold ([3],[2]) if(∇XS)(Y,Z) = A(X)S(Y,Z) + nB(X)(Y,Z).(3.1)Theorem 3.1. The scalar curvature r of a semi-generalized recurrent P-Sasakianmanifoldis related in terms of contact forms η(P1)and η(P2) as given byr = − 1η(P1)[(n2 + 2)η(P2) + 2(n − 1)η(P1)].Proof. Permutting equation (1.1) twice with respect to X, Y, Z; adding the threeequations and using Bianchi’s second identity, we haveA(X)R(Y,Z)W + B(X)(Z,W)Y + A(Y)R(Z,X)W+ B(Y)(X,W)Z+ A(Z)R(X,Y)W + B(Z)(Y,W)X = 0.(3.2)Contracting (3.2) with respect to Y, we getA(X)S(Z,W) + nB(X)(Z,W) − (R(Z,X)P1,W)+ B(Z)(X,W) − A(Z)S(X,W) + B(Z)(X,W) = 0.(3.3)216 Archana Singh, J.P.Singh and R. KumarIn view of S(Y,Z) = (QY,Z), the equation (3.3) reduces toA(X)(QZ,W)+ nB(X)(Z,W) − (R(Z,X)P1,W)+B(Z)(X,W)− A(Z)(QX,W) + B(Z)(X,W) = 0.(3.4)Factoring oﬀW, we get from (3.4)A(X)QZ + nB(X)Z − R(Z,X)P1+ B(Z)X − A(Z)QX + B(Z)X = 0.(3.5)Contracting (3.5) with respect to Z, we getA(X)r + (n2 + 2)B(X) − 2S(X,P1) = 0.(3.6)Putting X = ξ in the equation (3.6) and using the equations (1.2) and (2.13), we getr = − 1η(P1)[(n2 + 2)η(P2) + 2(n − 1)η(P1)].This completes the proof.Theorem 3.2. In a semi-generalized Ricci-recurrent P-Sasakian manifold, the 1-form Aand B are related as−(n − 1)A(X) + nB(X) = 0.Proof. Taking Z = ξ in (3.1), we have(∇XS)(Y, ξ) = A(X)S(Y, ξ) + nB(X)(Y, ξ).(3.7)The left hand side of (3.7), clearly can be written in the form(∇XS)(Y, ξ) = ∇XS(Y, ξ) − S(∇XY, ξ) − S(Y,∇Xξ),which in view of (2.4), (2.5) and (2.13) gives−(n − 1)(Y, φX)− S(Y, φX).While the right hand side of (3.7) equalsA(X)S(Y, ξ) + nB(X)(Y, ξ) = −(n − 1)A(X)η(Y)+ nB(X)η(Y).Hence,−(n − 1)(Y, φX) − S(Y, φX) = −(n − 1)A(X)η(Y)+ nB(X)η(Y).(3.8)Putting Y = ξ in (3.8) and then using (2.1), (2.6) and (2.13), we get−(n − 1)η(φX) + (n − 1)η(φX) = −(n − 1)A(X) + nB(X),Or,−(n − 1)A(X) + nB(X) = 0.(3.9)On A Type of Semi-Generalized Recurrent P-Sasakian Manifolds 217Theorem 3.3. If a semi-generalized Ricci-recurrent P-Sasakian manifold is an Einsteinmanifold then 1-forms A and B are related as λA(Y) + nB(Y) = 0.Proof. For an Einstein manifold, we have S(Y,Z)=λ(Y,Z) and (∇US)=0, where λ isconstant.Hence from (3.6) we have[λA(X) + nB(X)](Y,Z) + [λA(Y) + nB(Y)] (Z,X)+ [λA(Z) + nB(Z)] (X,Y) = 0.(3.10)Replacing Z by ξ in (3.10) and using (1.2) and (2.1), we have[λA(X) + nB(X)] η(Y) + [λA(Y) + nB(Y)]η(X)+[λη(P1) + nη(P2)](X,Y) = 0.(3.11)Again, taking X=Y=ξ in (3.11) and using (1.2), (2.1) and (2.6), we have[λη(P1) + nη(P2)]= 0.(3.12)Using (1.2) and (2.1) in the above relation, it follows thatλA(Y) + nB(Y) = 0.4. Semi-Generalizedφ-Recurrent P-Sasakian manifoldsDefinition 4.1. A P-Sasakian manifold (Mn, ) is called semi-generalized φ recur-rent if its curvature tensor R satisfies the conditionφ2((∇WR)(X,Y)Z) = A(W)R(X,Y)Z + B(W)(Y,Z)X,(4.1)where A and B are two 1-forms, B is non-zero and these are defined byA(W) = (W,P1), B(W) = (W,P2)(4.2)and P1 and P2 are vector fields associated with 1-forms A and B, respectively.Theorem 4.1. A semi generalizedφ-recurrent P-Sasakianmanifold (Mn, ) is an Einsteinmanifold and moreover; the 1-forms A and B are related as (n − 1)A(W) = nB(W).Proof. Let us consider a semi-generalized φ-recurrent P-Sasakian manifold. Thenby virtue of (2.1) and (4.1) we have(∇WR)(X,Y)Z − η((∇WR)(X,Y)Z)ξ= A(W)R(X,Y)Z + B(W)(Y,Z)X.(4.3)218 Archana Singh, J.P.Singh and R. KumarFrom which it follows that((∇WR)(X,Y)Z,U) − η((∇WR)(X,Y)Z)η(U)= A(W)(R(X,Y)Z,U)+ B(W)(Y,Z)(X,U).(4.4)Let {ei}, i = 1, 2, ...n be an orthonormal basis of the tangent space at any point of themanifold. Then putting X = U = ei in (4.4) and taking summation over i, 1 ≤ i ≤ n,we get(∇WS)(Y,Z) −n∑i=1η((∇WR)(ei,Y)Z)η(ei)= A(W)S(Y,Z) + nB(W)(Y,Z).(4.5)The second term of left hand side of (4.5) by putting Z = ξ takes the form((∇WR)(ei,Y)ξ, ξ), which is zero in this case. So, by replacing Z by ξ in (4.5)and using (2.13), we get(∇WS)(Y, ξ) = −(n − 1)A(W)η(Y)+ nB(W)η(Y).(4.6)We know that(∇WS)(Y, ξ) = ∇WS(Y, ξ) − S(∇WY, ξ) − S(Y,∇Wξ),using (2.4), (2.5) and (2.13) in the above relation, it follows(∇WS)(Y, ξ) = −(n − 1)(φW,Y) − S(φW,Y).(4.7)From (4.6) and (4.7) we obtain−(n − 1)(φW,Y) − S(φW,Y) = −(n − 1)A(W)η(Y)+ nB(W)η(Y).(4.8)Replacing Y = ξ in (4.8) then using (2.1) and (2.6), we get(n − 1)A(W) = nB(W).(4.9)Using (4.9) in (4.8), we obtainS(Y, φW) = −(n − 1)(Y, φW).(4.10)Again, replacing Y by φY both sides in the above equation (4.10) and using theequations (2.2) and (2.14), we obtainS(Y,W) = −(n − 1)(Y,W),i.e., the manifold is an Einstein manifold.On A Type of Semi-Generalized Recurrent P-Sasakian Manifolds 2195. Semi-Generalized Concircular φ-Recurrent P-SasakianmanifoldsDefinition 5.1. A P-Sasakian manifold (Mn, ) is called semi-generalized concir-cular ϕ-recurrent if its concircular curvature tensorC(X,Y)Z = R(X,Y)Z − rn(n − 1)[(Y,Z)X − (X,Z)Y](5.1)satisfies the conditionϕ2((∇WC)(X,Y)Z) = A(W)C(X,Y)Z + B(W)(Y,Z)X,(5.2)where A and B are defined as (4.2) and r is the scalar curvature of the manifold(Mn, ).Theorem 5.1. Let (Mn, ) be a semi-generalized concircularϕ-recurrent P-Sasakianman-ifold then [−(n − 1) − rn]A(W) + nB(W) = 0.Proof. Let us consider a semi-generalized ϕ-recurrent P-Sasakian manifold. Thenby virtue of (2.1) and (5.2), we have(∇WC)(X,Y)Z − η((∇WC)(X,Y)Z)ξ= A(W)C(X,Y)Z + B(W)(Y,Z)X,(5.3)from which it follows that((∇WC)(X,Y)Z,U) − η((∇WC)(X,Y)Z)η(U)= A(W)(C(X,Y)Z,U)+ B(W)(Y,Z)(X,U).(5.4)Let {ei}, i = 1, 2, ...n be an orthonormal basis of the tangent space at any point of themanifold. Then putting Y = Z = ei in (5.4) and taking summation over i, 1 ≤ i ≤ n,we get(∇WS)(X,U) = W(r)n (X,U)−W(r)nη(X)η(U)+ (∇WS)(X, ξ)η(U)+ nB(W)(X,U)+[S(X,U) − rn(X,U)]A(W).(5.5)Replacing U by ξ in (5.5) and using (2.1) and (2.13), we have[−(n − 1) − rn]A(W)η(X) + nB(W)η(X) = 0.(5.6)Putting X = ξ in (5.6), we obtain[−(n − 1) − rn]A(W) + nB(W) = 0.This completes the proof.220 Archana Singh, J.P.Singh and R. KumarTheorem 5.2. A semi-generalized concircular ϕ-recurrent P-Sasakian manifold is anEinstein manifold.Proof. Putting X = U = ei in (5.4) and taking summation over i, 1 ≤ i ≤ n, we get(∇WS)(Y,Z) =n∑i=1((∇WR)(ei,Y)Z, ξ)(ei, ξ)+W(r)n(Y,Z) − W(r)n(n − 1)[(Y,Z) − η(Y)η(Z)]+[S(Y,Z) − rn(Y,Z)]A(W) + nB(W)(Y,Z).(5.7)Replacing Z by ξ in (5.7) and using (5.6), we have(∇WS)(Y,Z) = W(r)n η(Y).(5.8)We know that(∇WS)(Y, ξ) = ∇WS(Y, ξ) − S(∇WY, ξ) − S(Y,∇Wξ),using (2.4), (2.5) and (2.13) in the above relation, it follows that(∇WS)(Y, ξ) = −(n − 1)(Y, φW) − S(Y, φW).(5.9)In view of (5.8) and (5.9), we obtainS(Y, φW) = −(n − 1)(Y, φW)− W(r)nη(Y).(5.10)Replacing Y by ϕY in (5.10) then using (2.2), (2.6) and (2.14), we obtainS(Y,W) = −(n − 1)(Y,W).6. Semi-generalizedM-Projective ϕ-recurrent P-Sasakian manifoldsDefinition 6.1. AP-Sasakianmanifold (Mn, ) is called semi-generalizedM-projectivecurvature tensor [10]W∗(X,Y)Z = R(X,Y)Z − 12(n − 1) [S(Y,Z)X − S(X,Z)Y+ (Y,Z)QX − (X,Z)QY].(6.1)satisfies the condition((∇VW∗)(X,Y)Z) − η((∇VW∗)(X,Y)Z)ξ= A(V)W∗(X,Y)Z + B(V)(Y,Z)X,(6.2)whereA and B aredefined as (4.2) and r is a scalar curvature of themanifold (Mn, ).On A Type of Semi-Generalized Recurrent P-Sasakian Manifolds 221Theorem 6.1. Let (Mn, ) be a semi-generalized M-Projective ϕ-recurrent P-Sasakianmanifold then−[n2 − n + r2(n − 1)]A(V) + nB(V) = 0.Proof. Let us consider a semi-generalized ϕ-recurrent P-Sasakian manifold. Thenby virtue of (2.1) and (6.2), we have(∇VW∗)(X,Y)Z− η((∇VW∗)(X,Y)Z)ξ= A(V)W∗(X,Y)Z + B(V)(Y,Z)X,(6.3)from which it follows that((∇VW∗)(X,Y)Z,U) − η((∇VW∗)(X,Y)Z))η(U)= A(V)(W∗(X,Y)Z,U) + B(V)(Y,Z)(X,U).(6.4)Let {ei}, i = 1, 2, ...n be an orthonormal basis of the tangent space at any point of themanifold. Then putting Y = Z = ei in (6.4) and taking summation over i, 1 ≤ i ≤ n,we getn2(n − 1) (∇VS)(X,U) −12(n − 1)V(r)(X,U)− n2(n − 1) (∇VS)(X, ξ)η(U) +12(n − 1) V(r)η(X)η(U)=[n2(n − 1)S(X,U) −r2(n − 1)(X,U)]A(V)+ nB(V)(X,U).(6.5)Replacing U by ξ in (6.5) and using (2.1) and (2.6), we have−A(V)[n2 − n + r2(n − 1)]η(X) + nB(V)η(X) = 0.(6.6)Putting X = ξ in (6.6), we obtain−[n2 − n + r2(n − 1)]A(V) + nB(V) = 0.This completes the proof.Theorem 6.2. A semi-generalized M-Projective ϕ-recurrent P-Sasakian manifold is anEinstein manifold.222 Archana Singh, J.P.Singh and R. KumarProof. Putting X = U = ei in (6.4) and taking summation over i, 1 ≤ i ≤ n, we getn2(n − 1) (∇VS)(Y,Z) =n∑i=1((∇VR)(ei,Y)Z, ξ)(ei, ξ)+12(n − 1)V(r)(Y,Z)− 12(n − 1) [(∇VS)(Y,Z)(ξ, ξ) − (∇VS)(ξ,Z)η(Y)+ (Y,Z)(∇VS)(ξ, ξ) − (∇VS)(Y, ξ)η(Z) ]+[n2(n − 1)S(Y,Z) −r2(n − 1)(Y,Z)]A(V)+ nB(V)(Y,Z).(6.7)Replacing Z by ξ in (6.7) and using (2.1), (2.6) and (2.13), we have(∇VS)(Y, ξ) = −1nV(r)η(Y).(6.8)We know that(∇VS)(Y, ξ) = ∇VS(Y, ξ) − S(∇VY, ξ) − S(Y,∇Vξ),using (2.4), (2.5) and (2.13) in above relation, it follows that(∇VS)(Y, ξ) = −(n − 1)(Y, ϕV)− S(Y, ϕV).(6.9)In view of (6.8) and (6.9)S(Y, ϕV) = −(n − 1)(Y, ϕV)+ 1nV(r)η(Y).(6.10)Replacing Y by ϕY in (6.10) then using (2.2) and (2.14), we getS(Y,V) = −(n − 1)(Y,V).This completes the proof.7. Three Dimensional Locally Semi-Generalizedϕ-recurrent P-SasakianmanifoldsTheorem 7.1. The curvature tensor of three dimensional semi-generalized ϕ-recurrentP-Sasakian manifold is given byR(X,Y,Z) =[dr(ei)2A(ei)− B(ei)A(ei)](Y,Z)X − dr(ei)2A(ei)(X,Z)Y.On A Type of Semi-Generalized Recurrent P-Sasakian Manifolds 223Proof. In a three-dimensional Riemannian manifold (M3, ), we haveR(X,Y)Z = (Y,Z)QX − (X,Z)QY + S(Y,Z)X− S(X,Z)Y + r2[(X,Z)Y − (Y,Z)X],(7.1)where Q is the Ricci operator, i.e., S(X,Y) = (QX,Y) and r is the scalar curvatureof the manifold. Now putting Z = ξ in (7.1) and using (2.13), we getR(X,Y)ξ = η(Y)QX − η(X)QY + (n − 1)[η(X)Y − η(Y)X]+r2[η(X)Y − η(Y)X].(7.2)Using (2.9) in (7.2), we have[(2 − n) − r2][η(X)Y − η(Y)X] = η(Y)QX − η(X)QY.(7.3)Putting Y = ξ in the equation (7.3) and using the equations (2.6) and (2.12), we getQX =[(3 − 2n) − r2]η(X)ξ −[(2 − n) − r2]X.(7.4)Therefore, it follows from (7.4) thatS(X,Y) =[(3 − 2n) − r2]η(X)η(Y) −[(2 − n) − r2](X,Y).(7.5)Thus from (7.1), (7.4) and (7.5), we getR(X,Y)Z =[2(2 − n) − r2][(X,Z)Y − (Y,Z)X]+[(3 − 2n) − r2][(Y,Z)η(X)ξ− (X,Z)η(Y)ξ+ η(Y)η(Z)X − η(X)η(Z)Y].(7.6)Taking the covariant diﬀerentiation to the both sides of the equation (7.6), we get(∇WR)(X,Y)Z = −dr(W)2 [(X,Z)Y − (Y,Z)X + (Y,Z)η(X)ξ− (X,Z)η(Y)ξ + η(Y)η(Z)X − η(X)η(Z)Y]+[(3 − 2n) − r2] [(Y,Z)η)(X)− (X,Z)η)(Y)] (∇Wξ)+[(3 − 2n) − r2] [(Y,Z)ξ − η(Z)Y] (∇Wη)(X)+[(3 − 2n) − r2] [η(Y)X − η(X)Y] (∇Wη)(Z)−[(3 − 2n) − r2] [(X,Z)ξ − η(Z)X] (∇Wη)(Y).(7.7)224 Archana Singh, J.P.Singh and R. KumarNoting, that we may assume that all vector fields X, Y, Z, W are orthogonal to ξand using (2.1), we get(∇WR)(X,Y)Z = −dr(W)2 [(X,Z)Y − (Y,Z)X]+[(3 − 2n) − r2][(Y,Z)(∇Wη)(X)− (X,Z)(∇Wη)(Y)]ξ.(7.8)Applying ϕ2 to the both side of (7.8) and using (2.1) and (2.6), we getϕ2((∇WR)(X,Y)Z) = dr(W)2[(Y,Z)X − (X,Z)Y] .(7.9)By (4.1), the equation (7.9) reduces toA(W)R(X,Y)Z =[dr(W)2− B(W)](Y,Z)X − dr(W)2(X,Z)Y.PuttingW = {ei}, where i=1, 2, 3 is an orthonormal basis of the tangent space at anypoint of the manifold and taking summation over i, 1 ≤ i ≤ 3, we obtainR(X,Y)Z =[dr(ei)2A(ei)− B(ei)A(ei)](Y,Z)X − dr(ei)2A(ei)(X,Z)Y.8. ConclusionThis paper is all about the study of geometrical properties of a semi generalizedrecurrent Para-Sasakian manifold. We prove that a semi generalized φ−recurrentPara-Sasakianmanifold is an Einsteinmanifold. It is also stabilised that a semi gen-eralized M projective φ−recurrent Para-Sasakian manifold and semi generalizedconcircular φ−recurrent Para-Sasakian manifolds are also Einstein manifold.REFERENCES1. T. Adati and K. Matsumoto: On conformally recurrent and conformally symmetricP-Sasakian manifolds. Tru Math. 13 (1977), 25–32.2. D. E. Blair: Contact manifold in Riemannian Geometry. Lecture notes in mathematics,Springer Verlag, (1976), 509.3. U. C. De andN. Guha: On generalized recurrentmanifold. J. Nat. Acad.Math. 9 (1991),85–92.4. U. C. De andG. Pathak: OnP-Sasakianmanifolds satisfying certain conditions. J. IndianAcad. Math. 16 (1994), 27–50.On A Type of Semi-Generalized Recurrent P-Sasakian Manifolds 2255. U. C. De and A. Sarkar: On a type of P-Sasakian manifolds. Math. Reports, 11(61), 2(2009), 139–144.6. Quddus Khan: On generalized recurrent Sasakian manifolds. Kyungpook Math. J. 44(2004),167–172.7. R. Kumar, J. P. Singh and J. Chowdhury: On semi-generalized recurrent LP-Sasakianmanifolds. International Journal of Mathematics and Statistics Invention, 2 (2014),63–66.8. K. Matsumoto, S. Ianus and I. Mihai: On a P-Sasakian manifolds which admit certaintensor fields. Publ. Math. Debrecen, 33 (1986), 61–65.9. K. Matsumoto: Conformal Killing Vector fields in a P-Sasakian manifolds. J. KoreanMath. Soc. 14 1(1977), 135–142.10. G. P. Pokhariyal and R. S. Mishra: Curvature tensor and their relativistic significanceII. Yokohama Mathematical Journal, 19 (1971), 97–103.11. B. Prasad: On semi-generalized recurrent manifold. Mathematica Balkanica, Newseries, 14 (2000), 77–82.12. H. S. Ruse: A Classification of K∗ space. London Mathematical Society. 53 (1951),212–229.13. I. Sato: On a structure similar to the almost contact structure. Tensor, (N.S.). 30 (1976),219–224.14. S. S. Shukla andM. K. Shukla: On ϕ-Symmetric Para-Sasakian manifolds. Int. Journalof Math. Analysis,4, no. 16(2010), 761–769.15. J. P. Singh: On Generalized ϕ-recurrent and Generalized Concircularly ϕ-recurrent P-Sasakian manifolds. Novi Sad J. Math. 44 (2014), 153–163.16. A. G. Walker: On Ruse’s Spaces of Recurrent curvature. Proc. London Math. Soc. 52(1976), 36–64.Archana SinghDepartment of Mathematics and Computer ScienceMizoram UniversityTanhril-796004, Mizoramarchanamath17@gmail.comJ.P. SinghFaculty of MathematicsDepartment of Mathematics and Computer ScienceMizoram UniversityTanhril-796004, Mizoramjpsmaths@gmail.comRajesh KumarFaculty of MathematicsDepartment of MathematicsPachhunga University College, Aizawl,Mizoram-796001, INDIARajesh mzu@yahoo.com

ON A TYPE OF SEMI-GENERALIZED RECURRENT P-SASAKIAN MANIFOLDS

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ON A TYPE OF SEMI-GENERALIZED RECURRENT P-SASAKIAN MANIFOLDS

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