12_587_abood.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 4, 2010, 730-736 ISSN 1307-5543 – www.ejpam.com Almost Hermitian Manifold with Flat Bochner Tensor Habeeb M. Abood Department of Mathematics, University of Basrah, Basrah-Iraq Abstract. Many researchers investigated the flat Bochner tensor on some kinds of almost Hermitian manifold. In the present paper the author studies this tensor on general class almost Hermitian mani- fold by using a new methodology which is called an adjoint G-structure space. Thus this study general- ize the results which are found out by those researchers. It is proved that if M is an almost Hermitian manifold of class R1 with flat Bochner tensor, then either M is 2-dimensional flat Ricci manifold or n-dimensional (n> 2) flat scalar curvature tensor manifold. As well, it is proved that if M is an almost Hermitian manifold with flat Bochner tensor, then M is a manifold of class R3 if and only if M is a linear complex manifold. Later on, equivalently of classes R2 and R3 is investigated. Finally we prove that if M is flat manifold with flat Bochner tensor, then M is an Einstein manifold with a cosmological constant. 2000 Mathematics Subject Classifications: 53C55, 53B35 Key Words and Phrases: Almost Hermitian manifold, flat Bochner tensor, adjoint G-structure space. 1. Introduction The Bochner tensor was introduced by S. Bochner [3] . He defined this tensor on a Kahler manifold as a formal analogy of the Weyle conformal curvature tensor. S. Tachibana [14] gave It the real form and he proved that the Bochner tensor had a meaning on any almost Hermitian manifold. The Kahler manifold with flat Bochner curvature tensor has been studied by many researchers. M. Mastumoto [8] Proved that a Kahler manifold of constant scalar curvature tensor with flat Bochner tensor is local symmetric. S. Tachibana [15] proved that Kahler manifold of a constant scalar curvature tensor with flat Bochner tensor is local-isometric to the product of complex spaces. L. Vanhecke [18] studied the Bochner curvature tensor on almost Hermitian manifold and he obtained some properties which are proved for Kahler manifold. Z. Olsgak [10] gave the classification of 4-dimensional compact flat Bochner of Kahler manifold with non positive scalar curvature tensor. M. Petrovic and L. Vestraclen [11] studied the flat Bochner of Kahler manifold where the Weyles tensor satisfies some conditions. K. Nam [9] proved that if M is a Kahler manifold with flat Bochner curvature tensor whose length of the Ricci tensor is constant, then M is a space of constant holomorphic sectional curvature Email address: iraqsafwan�yahoo. om (H. Abood) http://www.ejpam.com 730 c© 2010 EJPAM All rights reserved. H. Abood / Eur. J. Pure Appl. Math, 3 (2010), 730-736 731 or a locally product space of two spaces of constant holomorphic sectional curvatures. A. Al-Otman [1] studied the Bochner tensor of the class nearly Kahler manifold. He found the classification of the flat Bochner tensor of this class. In fact you may notice that the most of the mentioned researchers above are studied the Bochner tensor on some kinds of the sixteen classes of almost Hermitian manifold. In the present paper we study the flat Bochner tensor on general class almost Hermitian man- ifold. This study uses the method of adjoint G-structure space which was introduced by V.F. Krichenko who found two tensors which are the structure and virtual tensors [5]. This method helped the researchers to study the different geometrical properties of almost Hermitian man- ifold, therefore we use this method to generalize the results which are given by Vanhecke and by the referred researchers. 2. Preliminaries Let M be 2n-dimensional smooth manifold, X (M) be a module of smooth vector fields on M and C∞(M) be an algebra of smooth functions on M . An almost Hermitian structure (AH-structure) on M is a pair � J , g =< ., . > , where J is an endomorphism of a tangent space Tp(M) with � Jp �2 = −id and g is a Reimannian metric on M such that < JX , JY >=< X , Y >, X , Y ∈ X (M). A smooth manifold provided AH-structure is called an almost Hermitian manifold(AH-manifold). We recall that the funda- mental (Kahlerian [7]) form is given by Ω(X , Y ) =< X , JY >. As it is well known from [6] that the given AH-structure on a manifold M is equivalent to the given an G-structure in principle fiber bundle of all complex frames of M with structure group U(n). This group is called an adjoint G-structure. The frame adapted to the AH- structure is called A-frame look as � p,ǫ1, . . . ,ǫn,ǫ1̂, . . . ,ǫn̂ [5], where ǫa are the eigenvectors corresponded to the eigenvalue i = p−1 and ǫâ are the eigenvectors corresponded to the eigenvalue i = −p−1. Here the index a ranges from 1 to n and â = a+ n. The matrices of the J , G and Ω in A-frame are given as: � gi j � = � 0 In In 0 � , � J i j � = � p−1Jn 0 0 −p−1Jn � , (Ωi j) = � 0 p−1In −p−1In 0 � (1) Where In is the identity matrix of order n. A Bochner tensor on AH-manifold M is a tensor of type (4,0) which is defined as the form: B(X , Y, Z ,W ) = R(X , Y, Z ,W ) + L(X ,W )g(Y, Z)− L(X , Z)g(Y,W ) + L(Y, Z)g(X ,W )− L(Y,W )g(X , Z) + L(JX ,W )g(JX , Z) − L(JX , Z)g(JY,W ) + L(JY, Z)g(JX ,W )− L(JY,W )g(JX , Z) − 2L(JX , Y )g(J Z ,W )− 2L(J Z ,W )g(JX , Y ), Where L (X , Y ) = − 1 2n+ 4 g (rX , Y ) + K 2 (2n+ 2) (2n+ 4) g (X , Y ) , H. Abood / Eur. J. Pure Appl. Math, 3 (2010), 730-736 732 r is the Ricci tensor and K is the scalar curvature tensor, X , Y, Z ,W ∈ X (M). Denote C (X , Y ) = L (JX , Y ). We have g (JX , Y ) = −Ω(X , Y ), where Ω is the fundamen- tal(Kahlerian) form. The components of Bochner tensor at any frame will be as the form: Bi jkl = Ri jkl + Lil g jk − Lik g jl + L jk gil − L jl gik − CilΩ jk + CikΩ jl − C jkΩil + C jlΩik + 2Ci jΩkl + 2CklΩi j (2) Li j = − 1 2n+ 4 ri j + K̃ gi j (3) Ci j = − 1 2n+ 4 J k i rk j + K̃J k i gk j (4) Where K̃ = K 2 (2n+ 2) (2n+ 4) Suppose that the indices a,b,c and d in the range 1,2, . . . , n. Denote â = a+ n. In the following proposition we find the components of Bochner tensor on any AH-manifold in the ajoint G-structure space, i.e. in the A-frame: Proposition 1. The components of Bochner tensor of AH-manifold are given as the following forms: 1. Babcd = Rabcd 2. Bâbcd = Râbcd + 1 n+2 (rbdδ a c − rbcδ a d + rcdδ a b ) 3. Bâ b̂cd = Râ b̂cd + 1 2n+4 � ra c δ b d − ra d δ b c + r b d δ a c − r b c δ a d � 4. Bâbcd̂ = Râbcd̂ + 1 n+2 (ra b δ d c − rd c δ a b ) 5. Bab̂cd = Rab̂cd − 1 n+2 rcdδ c d 6. Babĉd = Rabĉd − 1 n+2 rabδ c d 7. Babcd̂ = Rabcd̂ − 1 n+2 rabδ d c 8. Bâbĉd = Râbĉd + 1 n+2 � ra b δ c d + r c d δ a b � − 4Kδa b δ c d And the other components of the Bochner tensor are conjugate of the above components. Proof. 1 - Set i = a, j = b, k = c, l = d , thus equation (2) becomes: Babcd = Rabcd + Lad gbc − Lac gbd + Lbc gad − Lbd gac − CadΩbc+CacΩbd−CbcΩad + CBDΩac+2CabΩcd + 2CcdΩab According to equations (1) we get Babcd = Rabcd . H. Abood / Eur. J. Pure Appl. Math, 3 (2010), 730-736 733 2 - Set i = â, j = b, k = c, l = d , the equation (2) becomes: Bâbcd = Râbcd + Lâd gbc − Lâc gbd + Lbc gâd − Lbd gâc − CâdΩbc+CbacΩbd−CbcΩbad + CbdΩbac+2CabΩcd + 2CcdΩâb Using (1), (3) and (4), we obtained: Bâ bcd = Râbcd + 1 n+ 2 (rbdδ a c − rbcδ a d + rcdδ a b) In the same manner we can get the other components. 3. Main Results A. Gray [4] defined three special classes of AH-manifold, which are given as the following form: 1. Class R1 if < R (X , Y ) Z ,W >=< R (JX , JY ) Z ,W > . 2. Class R2 if < R (X , Y ) Z ,W >=< R (JX , JY ) Z ,W > + < R (JX , Y ) J Z ,W > +R < JX , Y )Z , JW >. 3. Class R3 if < R (X , Y ) Z ,W >=< R (JX , JY ) J Z , JW > . Gray proved that for a random AH-manifold , the relation among them is, R1 ⊂ R2 ⊂ R3. The manifold of class R1 is called a parakahler manifold [12]. The manifold of class R3 has been studied by the name RK-manifold [17]. The following lemma gives the necessary and sufficient conditions that a random AH-manifold is one of the above classes in the adjoint G-space. Lemma 1 ([16]). In the adjoint G-structure space, an AH-manifold is a manifold of: 1. Class R1 if, and only if, Râbcd = 0 , Rabcd = 0 , Râ b̂cd = 0 2. Class R2 if, and only if, Râbcd = 0 , Rabcd = 0 3. Class R3 if, and only if, Râbcd = 0 Recall that an AH-manifold has J-invariant Ricci tensor if, r ◦ J= J ◦ r [16]. Lemma 2 ([16]). An AH-manifold has J-invariant Ricci tensor if, and only if, in the adjoint G-structure space, rab = 0. Theorem 1. Suppose that M is AH-manifold with flat Bochner tensor, then M is a manifold of class R3 if, and only if, M is linear complex manifold. H. Abood / Eur. J. Pure Appl. Math, 3 (2010), 730-736 734 Proof. By proposition 1 we have: Bâ bcd = Râbcd + 1 n+ 2 (rbdδ a c − rbcδ a d + rcdδ a b ) Suppose that M is AH-manifold of class R3 with flat Bochner tensor. That means Bâbcd = 0 and Râbcd = 0. Thus we get: rbdδ a c − rbcδ a d + rcdδ a b = 0 (5) Contracting (5) by the indexes c and a, we obtained: nrbd − rbd + rbd = 0 Which means that rbd = 0. By Lemma 2 we have rbd = 0 if, and only if, r ◦ J = J ◦ r. Hence, from [2] M is linear complex manifold. Corollary 1. Suppose that M is an AH-manifold with flat Bochner tensor, then M is a manifold of class R3 if, and only if, M is a manifold of class R2. Proof. This is directly from the condition of the class R2 . Theorem 2. Suppose that M is an AH-manifold with flat Bochner tensor. If M is a manifold of class R1, then M is either n-dimensional Ricci flat manifold for n> 2 or 2-dimensional flat scalar curvature manifold. Proof. Suppose that M is AH-manifold of class R1 with flat Bochner tensor. According to Lemma 1 we have Râbcd = 0, Rabcd = 0, Râ b̂cd = 0. Thus ra c δ b d − ra d δ b c + r b d δ a c − r b c δ a d = 0 (6) Contracting the equation (6) by the indexes a and c we get: ra aδ b d − r b d + nr b d − r b d = 0 (7) Contracting the equation (7) by the indexes b and d we obtained: nra a − ra a + nra a − ra a = 0 Hence ra a = 0 Thus, the equation (7) will be as the form: (n− 2) r b d = 0 If n 6= 2 we get: r b d = 0 Therefore M is Ricci flat manifold. If n= 2 , we shall discuss the cases of the values a, b, c, and d in the equation (6): H. Abood / Eur. J. Pure Appl. Math, 3 (2010), 730-736 735 1. Put a = 2, b = 2, c = 1, d = 1 we get: r1 1 + r2 2 = 0 2. Put a = 2, b = 1, c = 1, d = 2 we get: −r1 1− r2 2 = 0 Thus in all possible other cases of the values a, b, c, and d we obtained: r1 1 + r2 2 = 0 or −r1 1 − r2 2 = 0 This means r i i = 0. It is well known, that the scalar curvature tensor is given by the form K = r i i . Therefore M is a manifold of flat scalar curvature tensor. Theorem 3. Suppose that M is AH-manifold with flat Bochner tensor, if M is flat manifold, then M is an Einstein manifold with cosmological constant K 2n . Proof. By the proposition 1 we have: Bâbcd̂ = Râbcd̂ + 1 n+ 2 (ra bδ d c − rd c δ a b) Suppose that M is flat manifold with flat Bochner tensor. This means that the Riemannian and Bochner tensors are vanishing. Thus we obtained: ra b δ d c − rd c δ a b = 0 (8) Contracting (8) by the indexes c and d , we get: nra b = r c cδ a b (9) We have K = r i i = ra a + r â â = 2ra a . Thus r c c = K 2 So the equation (9) becomes: ra b = K 2n δ a b K 2n δ â b̂ = K 2n δ a b = K 2n δ a b = ra b = r â b̂ = K 2n δ â b̂ Hence r i j = K 2n δ i j Therefore, from [13] M is Einstein manifold with cosmological constant K 2n . REFERENCES 736 References [1] A. Al-Otman and V. F. Kirichenko, On the geometry of the Bochner tensor of Nearly Kahler, Russian Mathematical Surveys V.48 , p.155-156. 1993. [2] V. I. Arnold, Mathematical methods of classical mechanics, Springer. 1989. [3] S. Bochner, Curvature and Betti numbers, II, Ann of Math., V.50, p.77-93. 1949. [4] A. Gray,Curvature identities for Hermitian and almost Hermitian manifolds, Tohoku Math.J. V. p.601-612. 1976. [5] V. F. Kirichenko, New results of K-spaces theory, Ph.D.thesis, Moscow State University. 1975. [6] V. F. Kirichenko, K-spaces of constant type,Seper. Math. J. V.17 No.2, p.282-289. 1976. [7] S. Kobayashi and K. Nomizu, Foundation of differential geometry, V2, John Wiley and sons. 1969. [8] M. Mastumoto, On Kahlerian spaces with parallel or vanishing curvature tensor,Tensor, N. S. V.20, p. 25-28. 1969. [9] K. Nam, Note on Kahlerian manifold whose Bochner curvature tensor vanishes, Bull. Korean Math. Soc., 25, No.1, p.99-105. 1988. [10] Z. Olsag, Bochner flat manifolds, Diff. geom.. Banash C. P., V.12, p.219-223. 1984. [11] M. Petrovic and L. Vestraclen, On the concircular tensor, the projective curvature tensor of Bochner-Kahler manifolds, Math. Rep. Toyama Univ., V.10, p.37-61. 1987. [12] G. B Rizza, Varieta Parakahleriane, Ann. Math. Pura et Appl., V.98, No.4, p.47-61. 1974. [13] M. C. Sean, The cosmological conatant, Irr, Max-Planek Institut Fur Gravitations Physik, 1. 2001. [14] S. Tachibana, On Bochner curvature tensor, Not. Sci. Ochanomidzu Univ., V.18, p.15-19. 1967. [15] S. Tachibana, Notes on Kahlerian metrics with Vanishing Bochner curvature tensor, Kodi Math. Semin. Repts., V.22, p.313-321. 1970. [16] E. V. Tretiakova, Curvature identities for Almost Kahler manifold, VINITE, Moscow, No.208-B99. 1999 [17] L. Vanhecke, Some almost Hermitian manifolds with constant holomorphic sectional curvature, J. Diff. Geom., V.12, No.4, p.461-467. 1977. [18] L. Vanhecke, The Bochner curvature tensor on almost Hermitian manifold, Hokkaido Math. J., V.7, No.2, p.252-258. 1987.