P. Mutlu and Z. Şentürk ON WALKER TYPE IDENTITIES LOCALLY CONFORMAL KAEHLER SPACE FORMS

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1 Geo. Struc. on Rie. Man.-Bari Vol. 73/, , 5 59 P. Mutlu and Z. Şentürk ON WALKER TYPE IDENTITIES LOCALLY CONFORMAL KAEHLER SPACE FORMS Abstract. The notion of locally conforal Kaehler anifold l.c.k-anifold in Heritian Geoetry has been introduced by I. Vaisan in 976. In this work, we present results on l.c.k-space fors satisfying curvature identities naed Walker type identities.. Introduction Let M, g, J be a real = 2n-diensional Heritian anifold with the structure J, g, where J is the alost coplex structure and g is the Heritian etric. Then J 2 = Id, gjx,jy =gx,y, for any vector fields X and Y tangent to M. The fundaental 2-for Ω is defined by ΩX,Y =gjx,y = ΩY, X. The anifold M is called a locally conforal Kaehler anifold an l.c.k-anifold if each point x in M has an open neighborhood U with a positive differentiable function ρ : U R such that g = e 2ρ g U is a Kaehlerian etric on U. Especially, if we can take U = M, then the anifold M is said to be globally conforal Kaehler. A Heritian anifold whose etric is locally conforal to a Kaehler etric is called an l.c.k-anifold. I. Vaisan gives its characterization as follows [0] : A Heritian anifold M is an l.c.k-anifold if and only if there exists on M a global closed -for α such that dω = 2α Ω, where α is called the Lee for. A Heritian anifold M,g,J is an l.c.k-anifold if and only if A. where k J ij = β i g kj + β j g ki α i J kj + α j J ki, β j = α r J r j. An l.c.k-anifold M is called an l.c.k-space for if the holoorphic sectional curvature of the section {X,JX} at each point of M has a constant value. Let Mc be an 5

2 52 P. Mutlu and Z. Şentürk l.c.k-space for with constant holoorphic sectional curvature c, then the Rieannian curvature tensor R i jhk with respect to g ij has the for [8] R hi jk = c 4 g hkg ij g hj g ik + J hk J ij J hj J ik 2J hi J jk P hkg ij + P ij g hk P hj g ik P ik g hj A.2 4 P hk J ij + P ij J hk P hj J ik P ik J hj 2 P hi J jk 2 P jk J hi, where P ij = P ir J r j, A.3 P ij = i α j α i α j + α 2 2 g ij is hybrid, i.e.,p ir J r j + P jrj r i = 0, P ij = P ji and α denotes the length of Lee for. Contracting A.2 with g hk, we have A.4 S ij = 4 { + 2c + 3 trp}g ij P ij, where S denotes the Ricci tensor with respect to g. PROPOSITION. [9] If the tensor field P is hybrid and the trace of the tensor field P is constant in a 4-diensional l.c.k-space for Mc, then Mc is Einstein. THEOREM. [9] A real -diensional 4 l.c.k-space for Mc in which the tensor field P is hybrid and the trace of the tensor field P is constant is Einstein if and only if the tensor field P is proportional to g. 2. Preliinaries Let M, g be an n-diensional, n 3, sei-rieannian connected anifold of class C with Levi-Civita connection. The Ricci operator S is defined by gsx,y = SX,Y, where X, Y ΞM, ΞM being the Lie algebra of vector fields on M. We define the endoorphiss X A Y, R X,Y Z and CX,Y of ΞM by A.5 A.6 A.7 X A Y Z = AY,ZX AX,ZY, R X,Y Z = X Y Z Y X Z [X,Y ] Z, CX,Y Z = R X,Y Z n 2 X g SY +SX g Y n X g Y Z, respectively, where X, Y, Z ΞM, A is a syetric 0,2-tensor, the scalar curvature and [X,Y ] is the Lie bracket of vector fields X and Y. In particular we have X g Y =X Y.

3 On Walker Type Identities Locally Conforal Kaehler Space Fors 53 The Rieannian-Christoffel curvature tensor R, the Weyl conforal curvature tensor C and the 0,4-tensor G of M,g are defined by A.8 RX,X 2,X 3,X 4 = gr X,X 2 X 3,X 4, CX,X 2,X 3,X 4 = gcx,x 2 X 3,X 4, GX,X 2,X 3,X 4 = gx g X 2 X 3,X 4, respectively, where X,X 2,X 3,X 4 ΞM. A tensor B of type,3 on M is said to be a generalized curvature tensor if A.9 X,X 2,X 3 BX,X 2 X 3 = 0, BX,X 2 +BX 2,X = 0, BX,X 2,X 3,X 4 = BX 3,X 4,X,X 2, where BX,X 2,X 3,X 4 =gbx,x 2 X 3,X 4. For syetric 0,2-tensor E and F we define their Kulkarni-Noizu product E F by E FX,X 2,X 3,X 4 = EX,X 4 FX 2,X 3 +EX 2,X 3 FX,X 4 EX,X 3 FX 2,X 4 EX 2,X 4 FX,X 3. For a syetric 0,2-tensor E and a 0,k-tensor T, k 2, we define their Kulkarni- Noizu product E T by [3] E T X,X 2,X 3,X 4 ;Y 3,...,Y k = EX,X 4 T X 2,X 3,Y 3,...,Y k For syetric 0, 2-tensor E and F we have [6] + EX 2,X 3 T X,X 4,Y 3,...,Y k EX,X 3 T X 2,X 4,Y 3,...,Y k EX 2,X 4 T X,X 3,Y 3,...,Y k. A.0 QE,E F= QF,Ē, where Ē = 2E E. We also have [7] A. E QE,F= QF,Ē. For a 0,k-tensor field T, k, a 0,2-tensor field A and a generalized curvature tensor B on M,g, we define the tensors B T and QA,T by A.2 B T X,...,X k ;X,Y = T BX,Y X,X 2,...,X k... T X,X 2,...,X k,bx,y X k, A.3 QA,T X,...,X k ;X,Y = T X A Y X,X 2,...,X k... T X,X 2,...,X k,x A Y X k,

4 54 P. Mutlu and Z. Şentürk respectively, where X,Y,X,X 2,...,X k ΞM. Putting in the above forulas B = R or B = C, T = R or T = C or T = S, A = g or A = S, we obtain the tensors R R, R C, R S, C S, Q g, R, Q S, R, Q g, C, Q g, S and Q S, C respectively. Let M,g be covered by a syste of charts {W;x k }. We define by g ij, R hi jk, S ij, G hi jk = g hk g ij g hj g ik and A.4 C hi jk = R hi jk n 2 g hks ij g hj S ik + g ij S hk g ik S hj + n n 2 G hi jk, the local coponents of the etric tensor g, the Rieannian-Christoffel curvature tensor R, the Ricci tensor S, the tensor G and the Weyl tensor C, respectively. Further, we denote by S ij = S ir g r j and S j i = g jr S ir. The local coponents of the 0,6-tensors R T and Q g, T on M are the following: A.5 R T hi jkl = g rs T ri jk R shl + T hr jk R sil + T hirk R sjl + T hi jr R skl, A.6 Qg,T hi jkl = g h T lijk g i T hl jk g j T hilk g k T hi jl + g lh T i jk + g li T h jk + g lj T hik + g lk T hi j, where T hi jk are the local coponents of the tensor T. In this part we present soe considerations leading to the definition of Deszcz Syetric Pseudosyetric in the sense of Deszcz and Ricci-pseudosyetric anifolds. A sei-rieannian anifold M, g satisfying the condition R = 0 is said to be locally syetric. Locally syetric anifolds for a subclass of the class of anifolds characterized by the condition A.7 R R = 0. Sei-Rieannian anifolds fulfilling A.7 are called seisyetric. Here R R is a 0,6-tensor with coponents A.8 R R hi jkl = l R hi jk l R hi jk = g rs R ri jk R shl + R hr jk R sil + R hirk R sjl + R hi jr R skl. A ore general class of anifolds than the class of seisyetric anifolds is the class of Deszcz Syetric anifolds. A sei-rieannian anifold M, g is said to be Deszcz Syetric [2] if at every point of M the condition A.9 R R = L R Qg,R holds on the set U R = {x M R nn G 0 at x}, where L R is soe function on U R. There exist various exaples of Deszcz Syetric anifolds which are not seisyetric.

5 On Walker Type Identities Locally Conforal Kaehler Space Fors 55 A sei-rieannian anifold is said to be Ricci-seisyetric if we have R S = 0 on M. A sei-rieannian anifold M, g is said to be Ricci-pseudosyetric [2], [5] if at every point of M the condition A.20 R S = L S Qg,S holds on the set U S = {x M S n g 0 at x}, where L S is soe function on U S. The class of Ricci-pseudosyetric anifolds is an extension of the class of Ricciseisyetric anifolds as well as of the class of pseudosyetric anifolds. Every pseudosyetric anifold is Ricci-pseudosyetric. The converse stateent is not true. Evidently, every Ricci-seisyetric R S = 0 is Ricci-pseudosyetric. There exist various exaples of Ricci-pseudosyetric anifolds which are not pseudosyetric. 3. On Walker Type Identities Locally Conforal Kaehler Space Fors In this section, we present results on l.c.k-space fors satisfying curvature identities naed Walker type identities. LEMMA. [] For a syetric 0,2-tensor A and a generalized curvature tensor B on a sei-rieannian anifold M,g, n 3, we have A.2 QA,B hi jkl + QA,B jklhi + QA,B lhi jk = 0. It is well-known that the following identity A.22 R R hi jkl +R R jklhi +R R lhi jk = 0 holds on any sei-rieannian anifold. The equation A.22 is called the Walker type identity. On any sei-rieannian anifold M, g, n 4, the following three identities are equivalent to each other [4]: A.23 R C hi jkl +R C jklhi +R C lhi jk = 0, A.24 C R hi jkl +C R jklhi +C R lhi jk = 0 and A.25 R C C R hi jkl +R C C R jklhi +R C C R lhi jk = 0. The equations A.23 - A.25 are naed the Walker type identities. We also can consider the following Walker type identity A.26 C C hi jkl +C C jklhi +C C lhi jk = 0.

6 56 P. Mutlu and Z. Şentürk THEOREM 2. Let Mc be a 4-diensional l.c.k-space for, such that the tensor field P is hybrid and the trace of the tensor field P is constant. Then the Walker type identities A.23 - A.25 and A.26 hold on Mc. Proof. In view of A.5, we have A.27 R C hi jkl = g rs C ri jk R shl +C hr jk R sil +C hirk R sjl +C hi jr R skl, A.28 C R hi jkl = g rs R ri jk C shl + R hr jk C sil + R hirk C sjl + R hi jr C skl. Using A.4 in A.27 we obtain R C hi jkl = R R hi jkl [ R hkl S ij R jhl S ik + R jil S hk R kil S hj R hjl S ik + R ijl S hk + R khl S ij R ikl S hj A.29 + g ij Sk s R shl + g hk S s jr sil + g hk Si s R sjl + g ij Sh s R skl ] g ik S s jr shl g hj Sk s R sil g ik Sh s R sjl g hj Si s R skl [ + R hkl g ij R jhl g ik + R jil g hk ] R kil g hj + R ijl g hk R hjl g ik + R hkl g ij R ikl g hj = R R hi jkl [ g ij A khl + A hkl +g hk A jil + A ijl ] g ik A jhl + A hjl g hj A kil + A ikl, where A.30 A i jk = S s R si jk. Applying, in the sae way, A.4 in A.28 we get A.3 C R hi jkl = R R hi jkl QS,R hi jkl + Qg,R hi jkl g lha i jk g h A lijk g li A h jk + g i A lhjk + g lj A khi g j A lkhi g kl A jhi + g k A l jhi. Substituting A.4 into A.29 and A.3 we get A.32 R C = R R β T

7 On Walker Type Identities Locally Conforal Kaehler Space Fors 57 and A.33 α β trp C R = R R Qg,R β QP,R β T, where T and T are the 0,6-tensor fields whose local coponents are given by A.34 T hi jkl = g ij E khl + E hkl +g hk E jil + E ijl g ik E jhl + E hjl g hj E kil + E ikl, T hi jkl = g lh E i jk g h E lijk g li E h jk + g i E lhjk A.35 + g lj E khi g j E lkhi g kl E jhi + g k E l jhi, E hi jk = Ph sr si jk, α = 4 { + 2c + 3 trp} and β = Then by direct coputation, one obtains: T + T X,X 2,X 3,X 4,X,Y =0. X,X 2,X 3,X 4,X,Y Hence A.23 equivalently A.24, A.25 holds if and only if β T X,X 2,X 3,X 4,X,Y =0. X,X 2,X 3,X 4,X,Y In particular, if = 4, then β = 0, so A.23 holds. Further, we note that A.4 turns into C = R 2c+trP 4 G. This gives A.36 2c +trp C C = C R G=C R 4 2c +trp 2c +trp = R G R = R R Qg,R. 4 4 Now using A.2 and A.22 we coplete the proof. THEOREM 3. Let Mc be an -diensional > 4 l.c.k-space for. If the tensor field P is proportional to g and the trace of the tensor field P is constant, then the Walker type identities A.23 - A.25 and A.26 hold on Mc. Proof. In view of Theore., the -diensional l.c.k-space { for Mc is Einstein, so A.23 - A.25 hold on Mc. Substituting S = 4 + 2c }g trp into A.4, we have C = R 4 { + 2c + 6 } trp G.

8 58 P. Mutlu and Z. Şentürk This gives C C = C R 4 = C R = R 4 = R R 4 Using A.2 and A.22, we get the result. { + 2c + { + 2c + { + 2c + 6 } trp G 6 } trp G R } 6 trp Qg,R. LEMMA 2. Let Mc be an -diensional > 4 l.c.k-space for such that the tensor field P is hybrid. Then, we have R C hi jkl +R C jklhi +R C lhi jk A.37 = β g R P hi jkl +g R P jklhi +g R P lhi jk. Proof. Substituting A.4 into A.4, we obtain A.38 C = R β α β trp g P 2 G, where α = 4 { + 2c + 3 trp} and β = Fro A.38, we get A.39 Using A.22 the proof is copleted. R C = R R β g R P. 2 LEMMA 3. If one of the Walker type identities A.23 - A.25 holds on an -diensional > 4 l.c.k-space for Mc and the tensor field P is hybrid, then on Mc we have A.40 g R P hi jkl +g R P jklhi +g R P lhi jk = 0. Proof. Lea 2. copletes the proof. THEOREM 4. On any Ricci-pseudosyetric l.c.k-space for Mc, > 4 such that the tensor field P is hybrid, the Walker type identities A.23 - A.25 hold on U s M. Proof. In view of A.20 and A.4, -diensional Ricci-pseudosyetric > 4 l.c.k-space fors satisfy A.4 R P = L S Qg,P. Using A.4 and A.37, we obtain the following identity on U S

9 On Walker Type Identities Locally Conforal Kaehler Space Fors 59 R C hi jkl +R C jklhi +R C lhi jk = βl S g Qg,P hi jkl +g Qg,P jklhi +g Qg,P lhi jk. Making use of A. and A.2, we obtain on U S R C hi jkl +R C jklhi +R C lhi jk = βl S QP,G hi jkl + QP,G jklhi + QP,G lhi jk = 0. Hence A.23 equivalently A.24, A.25 holds on Mc. Acknowledgeents. The authors would like to thank the referee for his/her valuable coents. References [] DEPREZ J., DESZCZ R., VERSTRAELEN L., Exaples of pseudo-syetric conforally flat warped products, Chinese J. Math , [2] DESZCZ R., On pseudosyetric spaces, Bull. Soc. Math. Belg. Sér. A , 34. [3] DESZCZ R. AND GŁOGOWSKA M., Soe nonseisyetric Ricci-seisyetric warped product hypersurfaces, Publ. Inst. Math.Beograd N.S , [4] DESZCZ R., GŁOGOWSKA M., VERSTRAELEN L., On soe generalized Einstein etric conditions on hypersurfaces in sei-rieannian space fors, Colloq. Math , [5] DESZCZ R. AND HOTLOS M., Rearks on Rieannian anifolds satisfying a certain curvature condition iposed on the Ricci tensor, Prace. Nauk. Pol. Szczec. 989, [6] DESZCZ R., VERSTRAELEN L., VRANCKEN L., On the syetry of warped product spaceties, Gen. Relativity Gravitation 23 99, [7] DESZCZ R. AND VERSTRAELEN L., Hypersurfaces of sei-rieannian conforally flat anifolds, Geoetry and Topology of Subanifolds, III, World Sci., River Edge, NJ 99, [8] KASHIWADA T., Soe properties of locally conforal Kaehler anifolds, Hokkaido Math. J , [9] MATSUMOTO K., Locally conforal Kaehler anifolds and their subanifolds, Me. Secţ. Ştiinţ. Acad. Roânǎ Ser. IV , [0] VAISMAN I., On locally conforal alost Kaehler anifolds, Israel J. of Math , AMS Subject Classification: 53B20, 53B25, 53B30, 53B50, 53C25 Pegah MUTLU, Zerrin ŞENTÜRK, Matheatics Engineering Departent, Istanbul Technical University Maslak, TR-34469, Istanbul, TURKEY e-ail: sariaslani@itu.edu.tr, senturk@itu.edu.tr Lavoro pervenuto in redazione il e accettato il