Conharmonically Flat Vaisman-Gray Manifold

American Journal of Mathematics and Statistics 207, 7(): 38-43 DOI: 0.5923/j.ajms.207070.06 Conharmonically Flat Vaisman-Gray Manifold Habeeb M. Abood *, Yasir A. Abdulameer Department of Mathematics, College of Education for Pure Sciences, Basra University, Basra, Iraq Abstract This paper is devoted to study some geometrical properties of conharmonic curvature tensor of Vaisman-Gray manifold. In particular, we have found the necessary and sufficient condition that flat conharmonic Vaisman-Gray manifold is an Einstein manifold. Keywords Almost Hermitian Manifold, Vaisman-Gray manifold, Conharmonic tensor. Introduction One of the representative work of differential geometry is an almost Hermitian structure. Gray and Hervalla [] found that the action of the unitary group UU(nn) on the space of all tensors of type (3,0) decomposed this space into sixteen classes. The conditions that determined each one of these classes belongs to the type of almost Hermitian structure have been identified. These conditions were formulated by using the method of Kozel's operator [2]. The Russian researcher Kirichenko found an interesting method to study the different classes of almost Hermitian manifold. This method depending on the space of the principal fiber bundle of all complex frames of manifold MM with structure group is the unitary group UU(nn). This space is called an adjoined GG-structure space, more details about this space can be found in [3-6]. One of the most important classes of almost Hermitian structures is denoted by WW WW 4, where WW and WW 4 respectively denoted to the nearly Ka hler manifold and local conformal Ka hler manifold. A harmonic function is a function whose Laplacian vanishes. Related to this fact, Y. Ishi [7] has studied conharmonic transformation which is a conformal transformation that preserves the harmonicity of a certain function. Agaoka, et al. [8] studied the twisted product manifold with vanishing conharmonic curvature tensor. Agaoka, et al. [9] studied the fibred Riemannian space with flat conharmonic curvature tensor, in particular, they proved that a conharmonically flat manifold is locally the product manifold of two spaces of constant curvature tensor with constant scalar curvatures. Siddiqui and Ahsan [0] gave an interesting application when they studied the conharmonic curvature tensor on the four dimensional space-time that satisfy the Einstein field equations. Abood and Lafta [] * Corresponding author: iraqsafwan2006@gmail.com (Habeeb M. Abood) Published online at http://journal.sapub.org/ajms Copyright 207 Scientific & Academic Publishing. All Rights Reserved studied the conharmonic curvature tensor of nearly Ka hler and almost Ka hler manifolds. The present work devoted to study the flatness of conharmonic curvature tensor of Vaisman-Gray manifold by using the methodology of an adjoined GG-structure space. 2. Preliminaries Suppose that MM is 2nn -dimensional smooth manifold, CC (MM) is a set of all smooth functions on MM, XX(MM) is the module of smooth vector fields on MM. An almost Hermitian manifold (AAAA-manifold) is the set {MM, JJ, gg =<.,. >}, where MM is a smooth manifold, and JJ is an almost complex structure, and gg =<.,. > is a Riemannian metric, such that < JJJJ, JJJJ >=< XX, YY > ; XX, YY XX(MM). Suppose that TT cc pp (MM) is the complexification of tangent space TT pp (MM) at the point pp MM and {ee,, ee nn, JJee,, JJJJ nn } is a real adapted basis of AAAA-manifold. Then in the module TT cc pp (MM) there exists a basis given by {εε,, εε nn, εε,, εε nn} which is called adapted basis, where, εε = σσ(ee ) and εε = σσ (ee ) and σσ, σσ are two endomorphisms in the module XX cc (MM) which are defined by σσ = iiii 2 JJcc and σσ = iiii + 2 JJcc, such that, XX cc (MM) and JJ cc are the complexifications of XX(MM) and JJ respectively. The corresponding frame of this basis is {pp; εε,, εε nn, εε,, εε nn}. Suppose that the indexes ii, jj, kk and ll are in the range,2,,2nn and the indexes, bb, cc, dd and ff are in the range,2,, nn. And = + nn. The GG-structure space is the principal fiber bundle of all complex frames of manifold MM with structure group is the unitary group UU(nn). This space is called an adjoined GG-structure space. In the adjoined GG -structure space, the components matrices of complex structure JJ and Riemannian metric gg are given by the following: JJ ii jj = II nn 0, gg iiii = 0 II nn (2.) 0 II nn II nn 0 where II nn is the identity matrix of order nn.

American Journal of Mathematics and Statistics 207, 7(): 38-43 39 Definition 2. [2] The Riemannian curvature tensor RR of a smooth manifold MM is an 4-covariant tensor RR: TT pp (MM) TT pp (MM) TT pp (MM) TT pp (MM) R which is defined by: RR(XX, YY, ZZ, WW) = gg(rr(zz, WW)YY, XX), where RR (XX, YY)ZZ = [ XX, YY ] [XX,YY] ZZ; XX, YY, ZZ, WW TT pp (MM) and satisfies the following properties: i) RR(XX, YY, ZZ, WW) = RR(YY, XX, ZZ, WW); ii) RR(XX, YY, ZZ, WW) = RR(XX, YY, WW, ZZ); iii) RR(XX, YY, ZZ, WW) + RR(XX, ZZ, WW, YY) + RR(XX, WW, YY, ZZ) = 0; iv) RR(XX, YY, ZZ, WW) = RR(ZZ, WW, XX, YY). Definition 2.2 [3] The Ricci tensor is a tensor of type (2,0) which is defined as follows: kk rr iiii = RR iiiiii = gg kkkk RR kkkkkkkk Definition 2.3 [7] The conharmonic tensor of an AAAA-manifold is a tensor TTof type (4,0) which is defined as the form: TT iiiiiiii = RR iiiiiiii 2(nn ) [rr iiiigg jjjj rr jjjj gg iiii + rr jjjj gg iiii rr iiii gg jjjj ] where rr, RR and gg are respectively Ricci tensor, Riemannian curvature tensor and Riemannian metric. Similar to the properties of Riemannian curvature tensor, the conharmonic tensor has the following properties: TT iiiiiiii = TT jjjjjjjj = TT iiiiiiii = TT kkkkkkkk Definition 2.4. An AAAA -manifold is called a conharmonically flat if the conharmonic tensor vanishes. Definition 2.5 [4] In the adjoined GG-structure space, an AAAA-manifold {MM, JJ, gg =<.,. >} is called a Vaisman-Gray manifold ( VVVV -manifold) if BB = BB bbbbbb, BB cc = αα [ δδ bb] cc ; is called a locally conformal K a hler manifold (LLLLLL-manifold) if BB = 0 and BB cc = αα [ δδ bb] cc ; and is called a nearly Ka hler manifold (NNNN-manifold) if BB = BB bbbbbb and BB cc = 0, where BB =, BB = cc JJ 2 bb,cc 2 JJ [bb,cc ] and αα = δδ FF JJ is a Lie form; FF is a nn Ka hler form which is defined by F(XX, YY) =< JJJJ, YY >, δδ is a coderivative and XX, YY XX(MM) and the bracket [ ] denote to the antisymmetric operation. Theorem 2.6 [5] In the adjoined GG-structure space, the components of Riemannian curvature tensor of VVVV-manifold are given by the following forms: i) RR = 2(ΒΒ [cccc ] + αα [ ΒΒ bb]cccc ); ii) RR bbbbbb = 2ΑΑ bbbbbb ; iii) RR bb cccc = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ); iv) RR bbbb dd = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb, where, {ΑΑ bbbbbb } are some functions on adjoined GG- structure space and AA bbbb are system of fuctions in the adjoined GG -structure space which are symmetric by the lower and upper indices which are called components of holomorphic sectional curvature tensor. and {αα bb bb, αα } are the components of the covariant differential structure tensor of first and second type and {αα, αα } are the components of the Lee form on adjoint GG-structure space such that: dddd a + αα bb ωω bb = αα bb ωω bb + αα ωω bb, and dddd αα bb ωω bb = αα bb ωω bb + αα ωω bb, where, {ωω, ωω } are the components of mixture form, {ωω bb } are the components of Riema-nnian connection of metric gg. The other components of Riemannian curvature tensor RR can be obtained by the property of symmetry for RR. There are three special classes of almost Hermitian manifold depending on the components of the Riemannian curvature tensor. Their conditions are embodied in the following definition: Definition 2.7. [6] In the adjoined GG-structure space, an AAAA-manifold is a manifold of class: ) RR if and only if, RR = RR bbbbbb = RR bb cccc = 0; 2) RR 2 if and only if, RR = RR bbbbbb = 0; 3) RR 3 (RRRR-manifold) if and only if, RR bbbbbb = 0. It easy to see that RR RR 2 RR 3. Theorem 2.8 [5] In the adjoined GG-structure space, the components of Ricci tensor of VVVV-manifold are given by the following forms: ) rr = nn (αα 2 + αα bbbb + αα αα bb ); 2) rr bb = 3ΒΒ cccch ΒΒ cccch ΑΑ cccc bbbb + nn 2 (αα αα bb αα h αα h ) αα h 2 hδδ bb + (nn 2)αα bb, and the others are conjugate to the above components. Definition 2.9 [7] A Riemannian manifold is called an Einstein manifold, if the Ricci tensor satisfies the equation rr iiii = CCgg iiii, where, CC is an Einstein constant. Definition 2.0 [8] An AAAA -manifold has JJ-invariant Ricci tensor, if JJ rr = rr JJ. The following Lemma gives a fact about Ricci tensor in the adjoined GG-structure space. Lemma 2. [6] An AAAA-manifold has JJ-invariant Ricci tensor if and only if, we have rr bb = rr = 0. Definition 2.2 [3] Define two endomorphisms on ττ 0 rr (VV) as follows: i) Symmetric mapping Sym: ττ 0 rr (VV) ττ 0 rr (VV) by: ssssss(tt)(vv,, vv rr ) = tt vv rr! σσ(),, vv σσ(rr) σσ SS rr. ii) Antisymmetric mapping Alt: ττ 0 rr (VV) ττ 0 rr (VV) by: AAAAAA(tt)(vv,, vv rr ) = εε(σσ)tt vv rr! σσ(),, vv σσ(rr) σσ SS rr. The symbols ( ) and [ ] are usually used to denote the symmetric and antisymmetric respectively. 3. Main Results Theorem 3.. In the adjoined GG -structure space, the components of conharmonic tensor of VVVV -manifold are given by the following forms:

40 Habeeb M. Abood et al.: Conharmonically Flat Vaisman-Gray Manifold ) TT = 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc ; 2) TT bbbbbb = 2ΑΑ bbbbbb + (rr 2(nn ) bbbb δδ cc rr bbbb δδ dd ); 3) TT bb cccc = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) 4) TT bbbb dd = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + [ δδ cc bb] + rr cc [bb δδ dd ] ); (rr (nn ) dd (rr ( (nn ) (cc δδ dd) bb) ), and the others are conjugate to the above components. ) Put ii =, jj = bb, kk = cc and ll = dd, we have TT = RR 2(nn ) (rr gg bbbb rr bbbb gg + rr bbbb gg rr gg bbbb ) According to the equation (2.) we deduce that TT = RR = 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc. 2) Put ii =, jj = bb, kk = cc and ll = dd, we get TT bbbbbb = RR bbbbbb 2(nn ) (rr ddgg bbbb rr bbbb gg cc + rr bbbb gg dd rr cc gg bbbb ) = RR bbbbbb + 2(nn ) (rr bbbb gg cc rr bbbb gg dd ) = 2ΑΑ bbbbbb + (rr 2(nn ) bbbb δδ cc rr bbbb δδ dd ). 3) Put ii =, jj = bb, kk = cc and ll = dd, it follows that TT bb cccc = RR bb cccc 2(nn ) (rr ddgg bb cc rr bb ddgg cc + rr bb ccgg dd rr cc gg bb dd) = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) (rr 2(nn ) ddδδ bb cc rr bb ddδδ cc + rr bb ccδδ dd rr cc δδ bb dd ) = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) (rr 2(nn ) dd δδ bb cc rr bb dd δδ cc + rr bb cc δδ dd rr cc δδ bb dd ) = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) (rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd ). 4) Put ii =, jj = bb, kk = cc and ll = dd, we obtain TT bbbb dd = RR bbbb dd 2(nn ) (rr dd gg bbbb rr bbdd gg cc + rr bbbb gg dd rr cc gg bbdd ) = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + (rr 2(nn ) bbdd δδ cc + rr cc δδ dd bb ) = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + (rr 2(nn ) bb dd δδ cc + rr cc δδ dd bb ) = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + rr ( (nn ) (cc δδ dd) bb). Definition 3.2. In the adjoined GG-structure space, an almost Hermitian manifold is a manifold of class: TTRR if and only if, TT = TT bbbbbb = TT bb cccc = 0; TTRR 2 if and only if, TT = TT bbbbbb = 0; TTRR 3 (TTTTTT-manifold) if and only if, TT bbbbbb = 0. We call TTRR a conharmonic paraka hler manifold. Theorem 3.3. Let MM be a VVVV-manifold of class TTRR with JJ-invariant Ricci tensor, then MM is a manifold of class RR if and only if, MM is a manifold of flat Ricci tensor. To prove MM is a manifold of class RR, we must prove that RR = RR bbbbbb = RR bb cccc = 0. Let MM be a manifold of class TTRR, according to definition 3.2, we have According to theorem 3., we have TT = TT bbbbbb = TT bb cccc = 0 TT = 0 (3.)

American Journal of Mathematics and Statistics 207, 7(): 38-43 4 By using the equation (3.), we get 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc = 0 RR = 0. (3.2) TT bbbbbb = 0 According to theorems 2.6 and 3., we deduce RR bbbbbb + 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0 Since, MM has JJ-invariant Ricci tensor, then Also, by the equation (3.), we deduce TT bb cccc = 0 By using theorems 2.6 and 3., we have RR bb cccc (nn ) rr [ dd δδ bb] cc + rr [bb cc δδ ] dd = 0 Suppose that MM is a manifold of flat Ricci tensor, then Hence, according to the equations, (3.2), (3.3) and (3.4), we get Conversely, by using the equation (3.), we have By using theorem 3., it follows that RR bbbbbb = 0. (3.3) RR bb cccc = 0. (3.4) RR = RR bbbbbb = RR bb cccc = 0. TT = TT bbbbbb = TT bb cccc = 0 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc = 2ΑΑ bbbbbb + (rr 2(nn ) bbbb δδ cc rr bbbb δδ dd ) = 2 ΒΒ h ΒΒ hcccc + αα [ [cc rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd Let MM be a manifold of class RR, then, according to theorem 2.6 and definition 2.7, we obtain 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) + (nn ) rr [ dd δδ bb] cc + rr [bb cc δδ ] dd = 0 Symmetrizing by the indexes (bb, cc), we get rr 2(nn ) bbbb δδ cc (rr 2 bbbb δδ dd + rr cccc δδ dd ) + rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd Antisymmetrizing by the indexes (bb, cc), we have rr 2(nn ) bbbb δδ cc 2 (rr 2 bbbb δδ dd rr cccc δδ dd ) + (rr 2 cccc δδ dd rr bbbb δδ dd ) + rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd rr 2(nn ) bbbb δδ cc + rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd rr 2(nn ) bbbb δδ cc + rr 2(nn ) dd δδ bb cc rr bb dd δδ cc + rr bb cc δδ dd rr cc δδ bb dd Symmetrizing by the indexes (, bb), we deduce rr 2(nn ) bbbb δδ cc + 2(nn ) rr 2 dd δδ bb cc + rr bb dd δδ cc rr 2 dd bb δδ cc + rr dd δδ bb cc + rr 2 cc bb δδ dd + rr cc δδ bb dd (rr 2 cc δδ bb dd + rr bb cc δδ dd ) 2(nn ) rr bbbb δδ cc = 0 rr bbbb δδ cc = 0 Contracting by the indexes (, dd), we have rr bbbb δδ cc = 0 rr bbbb = 0 Since MM has JJ-invariant Ricci tensor, then rr iiii = 0. Theorem 3.4. Suppose that MM is VVVV-manifold of class TTRR 3 with JJ-invariant Ricci tensor, then MM is a manifold of class RR 3 if and only if, MM is a manifold of flat Ricci tensor. Suppose that MM is a manifold of class TTRR 3. According to definition 3.2., we have TT bbbbbb = 0. By using the Theorem 3., we deduce

42 Habeeb M. Abood et al.: Conharmonically Flat Vaisman-Gray Manifold RR bbbbbb + 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0 (3.5) Let MM be a manifold of class RR 3, then 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0. Symmetrizing by the indexes (bb, cc), we get (rr 2(nn ) bbbb δδ cc (rr 2 bbbb δδ dd +rr cccc δδ dd )) = 0. Antisymmetrizing by the indexes (bb, cc), we have (rr 2(nn ) bbbb δδ cc ((rr 2 bbbb δδ dd rr cccc δδ dd ) + (rr cccc δδ dd rr bbbb δδ dd ))) = 0. rr 2(nn ) bbbb δδ cc = 0. rr bbbb δδ cc = 0. Contracting by the indexes (, dd), it follows that rr bbbb δδ cc = 0. rr bbbb = 0. Since MM has JJ-invariant Ricci tensor, then rr iiii = 0. Conversely, by using the equation (3.5), we have RR bbbbbb + 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0 Suppose that MM is a manifold of flat Ricci tensor, then RR bbbbbb = 0. Therefore, MM is a manifold of class RR 3. The following theorem gives the necessary and sufficient condition in which an VVVV-manifold is an Einstein manifold. Theorem 3.5. Suppose that MM is conharmonically flat VVVV -manifold with JJ -invariant Ricci tenor. Then the necessary and sufficient condition that MM an Einstein manifold, is αα cc = kkδδ cc, where kk is a constant. Let MM be a conharmonically flat VVVV -manifold. According to the definition 2.4 and theorem 3., we have 2 ΒΒ h ΒΒ hcccc + αα [ [cc rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd Contracting by the indexes (bb, dd), it follows that 2 ΒΒ h ΒΒ hcccc + αα [ [cc δδ bb] bb] rr [ (nn ) bb δδ bb] cc + rr [bb cc δδ ] bb 2ΒΒ h ΒΒ hcccc + 2nnαα cc 2 rr (nn ) cc = 0. ΒΒ h ΒΒ hcccc + nnαα cc rr (nn ) cc = 0. Symmetrizing by the indexes (, bb), we deduce 2 (ΒΒ h ΒΒ hcccc + ΒΒ bbbb h ΒΒ hcccc ) + nnαα cc rr (nn ) cc = 0. Antisymmetrizing by the indexes (, bb), it follows that 2 ( 2 (ΒΒ h ΒΒ hcccc ΒΒ bbbb h ΒΒ hcccc ) + 2 (ΒΒbbbb h ΒΒ hcccc ΒΒ h ΒΒ hcccc )) + nnαα cc rr (nn ) cc = 0. nnαα cc rr (nn ) cc = 0. (3.6) nnαα cc = rr (nn ) cc. Let MM be an Einstein manifold, then nnαα cc = CC δδ (nn ) cc. αα cc = kkδδ cc. Conversely, by using the equation (3.6), we have nnαα cc rr (nn ) cc = 0. Since, αα cc = kkδδ cc, we deduce nnkkδδ cc = rr (nn ) cc. rr cc = kkkk(nn )δδ cc. rr cc = CCδδ cc, where, CC represent an Einstein constant. Since MM has JJ-invariant Ricci tensor. Therefore, MM is an Einstein manifold. 4. Conclusions The present work is devoted to study the flatness of conharmonic curvature tensor of Vaisman-Gray manifold. We found out an interesting application in theoretical physics. In particular, we found the necessary and sufficient condition that a conharmonically flat Vaisman-Gray manifold is an Einstein manifold. REFERENCES [] Gray A., Hervella L. M., Sixteen classes of almost Hermitian manifold and their linear invariants, Ann. Math. Pure and Appl., Vol. 23, No. 3, 35-58, 980. [2] Kozal J. L., Varicies Kahlerian-Notes, Sao Paolo, 957. [3] Kirichenko V.F., Some properties of tensors on K-spaces, Journal of Moscow state University, Math-Mach. department, Vol. 6, 78-85, 975. [4] Kirichenko V. F., K spaces of constant type, Seper. Math. J., Vol.7, No. 2, 282-289, 975. [5] Kirichenko V. F., K-spaces of constant holomorphic sectional curvature, Mathematical Notes, V.9, No.5, 805-84, 976. [6] Kirichenko V. F. Differential geometry of K spaces, problems f Geometry, V.8, 39-60, 977. [7] Ishi Y. On conharmonic transformation Tensor, N. S. 7, 73-80, 957. [8] Agaoka Y., Kim B. H., Lee H. J., Conharmonic Transformation of Twisted Produced Manifolds. Mem. Fac. Integrated Arts and Sci., Hiroshima Univ., ser. IV, Vol. 25, -20, Dec.999. [9] Agaoka Y., Kim B. H., Lee S. D., Conharmonic flat fibred Riemannian space, Mem. Fac. Integrated Arts and Sci., Hiroshima Univ., ser. IV, Vol. 26, 09-5, Dec.2000. [0] Siddiqui S. A., Ahsan Z., Conharmonic curvature tensor and the Space-time of General Relativity, Differential

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