EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 823-833 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Some Results on Projective Curvature Tensor of Nearly Cosymplectic Manifold Nawaf Jaber Mohammed1, Habeeb Mtashar Abood2,∗ 1,2 Department of Mathematics, Faculty of Education for Pure Sciences, University of Basra, Basra, Iraq Abstract. In the nearly cosymplectic manifold, defined a tensor of type (4,0), it’s called a pro- jective curvature tensor. In this article we discuss an interesting question; what the geometric meaning of this tensor when it’s act on nearly cosymplectic manifold? The answer of this question leads to get an application on Einstein space. In particular, the necessary and sufficient conditions that a projective tensor is vanishes are found. 2010 Mathematics Subject Classifications: 53C55, 53B35 Key Words and Phrases: Projective curvature tensor, almost contact manifold, nearly cosym- plectic manifold 1. Introduction Almost contact manifold contains many varieties, one of the most important of them is called a nearly cosymplectic manifold (NC-manifold). There have many studies about this manifold. In 1974, Blair and Showers [4] have got some of the characteristics of NC-manifold. Later, appeared many studies on NC -manifold, for more details we refer to [2], [7] and [8]. In 2011, Kirichenko and Kusova [14] studied NC-manifold in G- adjoined structure space. This method allowed the researchers to study different geometric properties. Apart from conformal curvature tensor, the projective curvature tensor is another important tensor from the differential geometric point of view. In 1953, Yano and Bochner [19], proved that a manifold is projectively flat if and only if, it is of constant curvature. Thus the projective tensor measures a Riemannian manifold to be of constant curvature. In 2009, Abood and Mohammed [1], studied the projective tensor on almost Hermitian manifold and they are found some properties of this tensor. In 2010, Ghosh [9] found some properties of the projective curvature tensor on (k, µ)-contact manifolds. In 2012, De and De A. [6] studied the projective curvature tensor on K - contact manifolds. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3279 Email addresses: nawafjaber80@yahoo.com (N. J. Mohammed), iraqsafwan2006@gmail.com (H. M. Abood) http://www.ejpam.com 823 c© 2018 EJPAM All rights reserved. N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 824 2. Preliminaries This section shows a simple overview of the basic concepts that pertain to the subject of our study. Definition 2.1. [3] Suppose that M is 2n + 1-dimensional smooth manifold. The set of smooth manifold and tensors (M,η, ξ, Φ, g) is called an almost contact metric manifold (AC-manifold) if such that: η(ξ) = 1, Φ(ξ) = 0, η ◦ Φ = 0 and Φ2 = −id + η ⊗ ξ, where η is differential 1-form called a contact form, ξ be a vector field called a characteristic, Φ endomorphism of X(M) called a structure endomorphisim and there is a Riemannian structure g = 〈., .〉 on M such that: 〈ΦX,ΦY 〉 = 〈X,Y 〉 − η(X)η(Y ), X, Y ∈ X(M). Definition 2.2. [5] Almost contact manifold is called a nearly cosymplectic manifold (NC-manifold) if the equality ∇X(Φ)Y +∇Y (Φ)X = 0, X, Y ∈ X(M), holds. Definition 2.3. [12] Let (M, η, Φ, g) be an almost contact metric manifold (AC-manifold). In the module Xc(M) (complexification of the module X(M) ) define two endomorphisms σ and σ̄ as follows: σ = 1 2(id− √ −1Φ) and σ̄ = −1 2(id+ √ −1Φ). Further, depending on σ, we can define two projections as follows: Π = σ ◦ ` = −1 2 (Φ2 − √ −1Φ) and Π̄ = σ̄ ◦ ` = 1 2 (Φ2 + √ −1Φ), where σ ◦Φ = Φ ◦ σ = iσ and σ̄ ◦ Φ = Φ ◦ σ̄ = −iσ̄. Therefore, If we denote ImΠ = D √ −1 Φ and ImΠ̄ = D− √ −1 Φ , then Xc(M) = D √ −1 Φ ⊕D− √ −1 Φ ⊕D0 Φ, where D √ −1 Φ , D− √ −1 Φ and D0 Φ are proper submodules of endomorphism Φ with proper values√ −1,− √ −1 and 0 respectively. Definition 2.4. [15] At each point p ∈ M, we can construct a frame in T cp (M) by the form (p, ε0, ε1, ..., εn, ε1̂, ..., εn̂), where εa = √ 2σp(ep), εâ = √ 2σ̄(ep) and ε0 = ξp. The frame (p, ε0, ε1, ..., εn, ε1̂, ..., εn̂) is called an A-frame. The principle fiber bundle of all A-frames with structure group {1} × U(n) is called an G-adjoined structure space. Lemma 2.1. [13] Given an AC-manifold. Then the matrices of the tensors Φ and Rie- mannian metric g in A-frame are given by the following forms: (Φij) =  0 0 0 0 √ −1In o 0 0 − √ −1In  , (gij) =  1 0 0 0 0 −In 0 In 0  , where In is the identity matrix of order n. N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 825 The following theorem describes the structure equations of NC-manifold in the G- adjoined structure space. Theorem 2.1. [14] In G-adjoined structure space, the structure equations of NC-manifold are given by the following forms: (i) dωa = ωab ∧ ωb +Babcωb ∧ ωc + 3 2C abωb ∧ ω; (ii) dωa = −ωba ∧ ωb +Babcω b ∧ ωc + 3 2Cabω b ∧ ω; (iii) dω = Cbcωb ∧ ωc + Cbcω b ∧ ωc; (iv) dωab = ωac ∧ ωcb + [Aadbc − 2BadhBhbc + 3 2C adCbc]ω c ∧ ωd , where Babc = √ −1 2 Φa b̂,ĉ , Cab = √ −1Φa 0,b̂ , Cab = − √ −1Φâb,0 and Babc = − √ −1 2 Φâb,c. The tensors B, C and A are called the first, second and third structure tensors respec- tively. Definition 2.5. [16] A Riemann-Christoffel tensor of a smooth manifold M is a tensor of type (4, 0) which is defined by R(X,Y, Z,W ) = g(R(Z,W )Y,X), where R(X,Y )Z = ([∇X ,∇Y ]−∇[X,Y ])Z, and satisfies the following properties: (i) R(X,Y, Z,W ) = −R(Y,X,Z,W ) (ii) R(X,Y, Z,W ) = −R(X,Y,W,Z) (iii) R(X,Y, Z,W ) = R(Z,W,X, Y ) (iv) R(X,Y, Z,W ) +R(X,Z,W, Y ) +R(X,W, Y, Z) = 0. The components of Riemann-Christoffel tensor of NC-manifold are given in theorem below. Theorem 2.2. [14] In the G-adjoined structure space, the components of Riemann- Christoffel tensor of NC-manifold have the following forms: (i) Râbcd = 0; (ii) Rabcd = −2Bab[cd]; (iii) Râb̂cd = −2BabhBhcd; (iv) Râ0b0 = CacCbc; (v) Râbcd̂ = Aadbc −BadhBhbc − 5 3C adCbc. N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 826 The other components of Riemann-Christoffel tensor R can be obtained by the property of symmetry for R or equal to zero. Definition 2.6. [18] A tensor of type (2, 0) which is a contracting of Riemann-Christoffel tensor and defined as rij = Rkijk = gklRkijl is called a Ricci tensor. Lemma 2.2. [14] In the G-adjoined structure space, The components of Ricci tensor of NC-manifold are given by the following forms: (i) rab = 0; (ii) rab̂ = −Acbac + 3BcbhBhac + 2 3C bcCac; (iii) ra0 = 0; (iv) roo = −2CcdCcd. and the others are conjugate to the above components or equal to zero. The previous definitions of Riemann-Christoffel and Ricci tensors completed the re- quirements of the projective tensor which is embodied in the next definition. Definition 2.7. [10] Let M be an AC-manifold. A tensor of type (4, 0) which is defined as Pijkl = Rijkl − 1 2n [rikgjl − rjkgil] is called a projective curvature tensor, where Pijkl = −Pjikl = −Pijlk = Pklij . We will demonstrate the projective tensor on one of the AC -manifolds which is NC - manifold. Definition 2.8. [12] An AC-manifold M is called vanishing projective tensor, if the projective tensor is vanishes. Definition 2.9. An NC-manifold has Φ-invariant Ricci tensor, if Φ ◦ r = r ◦ Φ. Lemma 2.3. An NC-manifold has Φ-invariant Ricci tensor if and only if, in the G- adjoined structure space the following condition râb = rab = 0 holds. Definition 2.10. [11] Let M be a Riemannian manifold, t be a non-zero tensor field of the type (r,s) on M. A tensor t is said to be a recurrent if there is 1-form ρ on M such that ∇t = ρ ⊗ t, where ∇ is the Riemannian connection on M. The 1-form ρ is called a recurrence covector. An NC-manifold which allows a field of the recurrent tensor t is called t-recurrent. N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 827 Lemma 2.4. [11] If ρ = 0, then the manifold is called t-symmetrical, and if ρ 6= 0 then it is called nontrivially t-symmetrical. Now, we are in position to introduce the next definition. Definition 2.11. Let M be NC-manifold, M is called Pr-recurrent if M is P-recurrent and r-recurrent with the same recurrence convector. Definition 2.12. [17] A Riemannian manifold is called an Einstein manifold, if the Ricci tensor satisfies the equation rij = egij, where, e is an Einstein constant. Lemma 2.5. [12] In the G-adjoined structure space, an NC-manifold is a manifold of class (i) CR1 if and only if, Rabcd = Râbcd = Râb̂cd = 0; (ii) CR2 if and only if, Rabcd = Râbcd = 0; (iii) CR3 if and only if, Râbcd = 0. It easy to see that CR1 ⊂ CR2 ⊂ CR3 Concerning the projective tensor, we defined three special classes of NC-manifold which are given in the definition below. Definition 2.13. In the G-adjoined structure space, an NC-manifold is a manifold of class (i) PR1 if and only if, Pabcd = Pâbcd = Pâb̂cd = 0; (ii) PR2 if and only if, Pabcd = Pâbcd = 0; (iii) PR3 if and only if, Pâbcd = 0. 3. The main results In the present section, we concentrate our attention on projective tensor of NC- manifold, and study the notion of projective-recurrent NC-manifold. Lemma 3.1. In the G-adjoined structure space, the components of projective curvature tensor of NC-manifold are given by the following forms: (i) Pabcd = −2Bab[cd] ; (ii) Pâb̂cd = −2BabhBhcd − 1 2n [rac δ b d − rbcδad ]; (iii) Pâbcd̂ = Aadbc −BadhBhac − 5 3C ad ac − 1 2nr a c δ d a; (iv) Pâ0b0 = CacCbc − 1 2nr a b . N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 828 and the others are conjugate to the above components or equal to zero. Proof: By using the Theorem 2.2 , Lemma 2.2 and Definition 2.7 , directly we obtain the above components. Theorem 3.1. Let M be NC-manifold with vanishing projective tensor. If M is a manifold of vanishing Ricci tensor, then the fist structure tensor is vanishing in the first canonical connection. Proof: Suppose that M is projectively vanishing NC-manifold. Making use of Defi- nition 2.8 and Lemma 3.1, we have −2BabhBhcd − 1 2n [rac δ b d − rbcδad ] = 0 (3.1) Since M has vanishing Ricci tensor, so (3.1) becomes −2BadhBhcd = 0 (3.2) Contracting the equation (3.2) by the induces (a, c) and (b, d), it follows that −2BabhBhab = 0 Since B abh and Bhab are antisymmetric tensors, then we get∑ a,b,h |Babh|2 = 0 Consequently, we deduce that Babh = 0. Theorem 3.2. If M is a projectively vanishing NC-manifold and Φ-invariant Ricci tensor, then the necessary and sufficient condition that M is an Einstein manifold is Aadac = 5 3C ad ac + C0δ d c , where C0 = e 2n . Proof: Let M be projectively vanishing NC -manifold. According to the Definition 2.8 and Theorem 3.1, we have Aadbc −BadhBhbc − 5 3 Cadbc − 1 2n rac δ d b = 0 (3.3) Symmetrizing and antisymmetrizing the equation (3.3) by the induces (h, d), we deduce Aadbc − 5 3 Cadbc − 1 2n rac δ d b = 0 (3.4) Suppose that M is Einstein manifold. Using the Definition 3.2, so the equation (3.4) becomes Aadbc − 5 3 Cadbc − e 2n δac δ d b = 0 (3.5) N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 829 Contracting (3.5) by induces (a, b) , it follows that Aadac = 5 3 Cadac + C0δ d c (3.6) Conversely, let the equation (3.6) holds. Contracting the equation (3.4) by indices (a,b), we deduce Aadac − 5 3 Cadac − 1 2n rdc = 0 (3.7) Making use of the equations (3.6) and (3.7), it follows that rac = eδac According to the Φ-invariant Ricci tensor, we get that M is Einstein manifold. Theorem 3.3. Suppose that M is a projectively vanishing NC-manifold and Φ-invariant Ricci tensor. If M is an Einstein manifold then the first structure tensor is vanishing in the first canonical connection. Proof: Let M be NC-manifold with vanishing projective tensor. Making use of the Definition 2.8 and Lemma 3.1, then we have Aadbc −BadhBhbc − 5 3 Cadbc − 1 2n rac δ d b = 0 (3.8) Contracting (3.8) with respect to the induces (a, b), it follows that Aadac −BadhBhac − 5 3 Cadac − 1 2n rac δ d a = 0 (3.9) Since M is an Einstein manifold. So according to the Theorem 3.2, the equation (3.9) reduced to −BadhBhac = 0 (3.10) Contracting the equation (3.10) by the induces (d, c), it follows that −BadhBhad = 0 Since B abh and Bhab are antisymmetric tensors, then we get∑ a,d,h |Bhad|2 = 0 Consequently, we deduce that Bhad = 0. N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 830 Theorem 3.4. Let M be NC-manifold, then the classes CR3 and PR3 are coincide if and only if, M is Φ-invariant Ricci tensor. Proof: Suppose that CR3 and PR3 are coincide, then we have 1 2n rbcδ a d (3.11) Contracting the equation (3.11) by the induces (a, b), we get rdc = 0 Suppose that M is Φ-invariant Ricci tensor. Making use of Lemmas 3.1 and 2.2, it follows that Pȧbcd = Rȧbcd Therefore, CR3 and PR3 are coincide. Theorem 3.5. Suppose that M is Pr-recurrent NC-manifold. Then M is either projective symmetrical manifold or vanishing first structure tensor. Proof: Let M be Pr-recurrent NC -manifold. According to the Definition 2.11, M is P -recurrent and r-recurrent NC -manifold. From Definition 2.10, we have ∇P = ρ⊗ P Which has the following coordinate form Pijk`,h = ρhPijk` (3.12) Consider the equation (3.12) in the G-adjoined structure space, so we have Pâb̂cd,k = ρkPâb̂cd,k (3.13) According to the Lemma 3.1, the equation (3.13) becomes −2BabhBhcd,k − 1 2n [rac,kδ b d − rbc,kδad ] = ρk[−2BabhBhcd − 1 2n [rac δ b d − rbcδad ]] Making use of the Definition 2.10, it follows that BabhBhcd,k = −ρkBabhBhcd Symmetrization and antisymmetrization by induces (a, b) , we obtain ρkB abhBhcd = 0 Contracting the last equation by induces (a, c) and (b, d),we get ρkB abhBhab = 0 N. J. Mohammed, H. M. Abood / Eur. J. Pure Appl. Math, 11 (3) (2018), 823-833 831 Consequently, either ρk = 0, i.e ∇P = 0 which means that M is projective symmetrical manifold. Or, BabhBhab = 0 so by using the same technique in proof of the Theorem 3.1, we have Babh = 0 Therefore, the first structure tensor is vanishing. Theorem 3.6. Suppose that M is Pr-recurrent NC-manifold. Then the sectional curvature tensor is recurrent if and only if, the second structure tensor is recurrent. Proof: Let M be Pr-recurrent NC -manifold. According to the Definition 2.11, M is P -recurrent and r-recurrent NC -manifold. Now the Definition 2.10 implies, ∇P = ρ⊗ P The previous tensor has the following coordinate form Pijk`,h = ρhPijk` (3.14) Consider the equation (3.14) in the G-adjoined structure space, it follows that Pâbcd̂,k = ρkPâbcd̂,k (3.15) By using Lemma 3.1, then the equation (3.15) becomes Aadbc,k −BadhBhbc,k − 5 3 Cadbc,k − 1 2n rac,kδ d b = ρk[A ad bc −BadhBhbc − 5 3 Cadbc − 1 2n rac δ d b ] According to the Definition 2.10, we have Aadbc,k −BadhBhbc,k − 5 3 Cadbc,k = ρk[A ad bc −BadhBhbc − 5 3 Cadbc ] (3.16) Symmetrization and antisymmetrization the equation (3.16) by the induces (h, b), we get Aadbc,k − 5 3 Cadbc,k = ρk[A ad bc − 5 3 Cadbc ] (3.17) Now, If the structure tensor is recurrent so (3.17) becomes Aadbc,k = ρkA ad bc Conversely, If the sectional curvature tensor is recurrent, the the equation (3.17) gives the following desired Cadbc,k = ρkC ad bc . REFERENCES 832 References [1] Abood H. M., Mohammed N. 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