C O M M U N I C AT I O N S I N M AT H E M AT I C S

Transcription

1 VOLUME 2/203 No. ISSN C O M M U N I C AT I O N S I N M AT H E M AT I C S Editor-in-Chief Olga Rossi, The University of Ostrava & La Trobe University, Melbourne Division Editors Ilka Agricola, Philipps-Universität Marburg Attila Bérczes, University of Debrecen Anthony Bloch, University of Michigan George Bluman, The University of British Columbia, Vancouver Yong-Xin Guo, Eastern Liaoning University, Dandong Haizhong Li, Tsinghua University, Beijing Vilém Novák, The University of Ostrava Štefan Porubský, Academy of Sciences of the Czech Republic, Prague Geoff Prince, La Trobe University, Melbourne Thomas Vetterlein, Johannes Kepler University Linz Technical Editors Jan Štěpnička, The University of Ostrava Jan Šustek, The University of Ostrava Available online at

2 Communications in Mathematics 2 203) Copyright c 203 The University of Ostrava 77 The conformal change of the metric of an almost Hermitian manifold applied to the antiholomorphic curvature tensor Mileva Prvanović Abstract. By using the technique of decomposition of a Hermitian vector space under the action of a unitary group, Ganchev 2 obtained a tensor which he named the Weyl component of the antiholomorphic curvature tensor. We show that the same tensor can be obtained by direct application of the conformal change of the metric to the antiholomorphic curvature tensor. Also, we find some other conformally curvature tensors and examine some relations between them. Introduction Let M, g, J) be an almost Hermitian manifold, dim M = 2n 4, with the complex structure J and the Hermitian metric g, i.e., J 2 = Id, gjx, JY ) = gx, Y ) for all X, Y T p M), where T p M) is the tangent vector space of M at p M. Then the fundamental 2-form is F X, Y ) = gjx, Y ) = F Y, X). We consider the tensors having the standard symmetries of the curvature tensors of a Riemannian manifold. In 7 the linear space RV ) of such tensors over a 2n-dimensional Hermitian vector space V was decomposed into irreducible components under the action of the unitary group. Furthermore, all conformally invariant subspaces of RV ) were found. In and 2 the holomorphic and antiholomorphic curvature tensors for an almost Hermitian manifold are introduced, and, using the same technique as in the paper 7, the generalized Bochner curvature tensor and the Weyl component of the antiholomorphic curvature tensor are obtained. We ask the question: Is it possible to get these tensors in the classical way, i.e., by direct application of the conformal change of this metric? 200 MSC: 53C56 Key words: Almost Hermitian manifold, antiholomorphic curvature tensor, conformal change of metric

3 78 Mileva Prvanović In 6, we examined conformally invariant tensors associated with holomorphic curvature tensor, and we found, among others, the generalized Bochner curvature tensor. In the present paper, we deal with the antiholomorphic curvature tensor and, in Section 3, we find the Weyl component of the antiholomorphic curvature tensor. In Sections 2 and 4, we determine some other conformally invariant tensors. In Section 5 we find some relations between obtained conformally invariant tensors, and in Section 6 we discuss the case of Kähler manifold. We denote by R the Riemannian curvature tensor. Then the first and the second Ricci tensors are defined as follows: ρx, Y ) = 2n i= Re i, X, Y, e i ), and ρx, Y ) = 2n i= Re i, X, JY, Je i ), where {e i } is an orthonormal basis of T p M). Finally, the first and the second scalar curvatures are = 2n i= ρe i, e i ), = 2n i= ρe i, e i ). In general, the second Ricci tensor is not symmetric, but it satisfies the condition ρjx, JY ) = ρy, X). ) To make some formulas clearer we use the following notations: π X, Y, Z, W ) = gx, W )gy, Z) gx, Z)gY, W ), π 2 X, Y, Z, W ) = F X, W )F Y, Z) F X, Z)F Y, W ) 2F X, Y )F Z, W ), ϕx, Y, Z, W ) = gx, W )ρy, Z) gy, Z)ρX, W ) gx, Z)ρY, W ) gy, W )ρx, Z), Jϕ)X, Y, Z, W ) = ϕjx, JY, JZ, JW ) = gx, W )ρjy, JZ) gy, Z)ρJX, JW ) gx, Z)ρJY, JW ) gy, W )ρjx, JZ), ψx, Y, Z, W ) = F X, W ) ρy, JZ) ρjy, Z) F Y, Z) ρx, JW ) ρjx, W ) F X, Z) ρy, JW ) ρjy, W ) F Y, W ) ρx, JZ) ρjx, Z) 2F X, Y ) ρz, JW ) ρjz, W ) 2F Z, W ) ρx, JY ) ρjx, Y ), ϕx, Y, Z, W ) = gx, W ) ρy, Z) ρz, Y ) gy, Z) ρx, W ) ρw, X) gx, Z) ρy, W ) ρw, Y ) gy, W ) ρx, Z) ρz, X), ψx, Y, Z, W ) = F X, W ) ρy, JZ) ρjy, Z) F Y, Z) ρx, JW ) ρjx, W ) F X, Z) ρy, JW ) ρjy, W ) F Y, W ) ρx, JZ) ρjx, Z) 2F X, Y ) ρz, JW ) ρjz, W ) 2F Z, W ) ρx, JY ) ρjx, Y ). 2) A 0, 4) tensor T X, Y, Z, W ) is said to be generalized curvature tensor if it satisfies

4 The conformal change of the metric of an almost Hermitian manifold 79 all algebraic properties of the Riemannian curvature tensor, i.e., T X, Y, Z, W ) = T Y, X, Z, W ) = T X, Y, W, Z) = T Z, W, X, Y ), T X, Y, Z, W ) T Y, Z, X, W ) T Z, X, Y, W ) = 0. All tensors in the relations 2) are generalized curvature tensors. Hence we can consider the corresponding first Ricci tensor and the second Ricci tensor of each of them, and denote them as ρt ) and ρt ), respectively. In particular, we have ρr) = ρ and ρr) = ρ. For the latter use, we note that ρπ )X, Y ) = 2n )gx, Y ), ρπ 2 )X, Y ) = 3gX, Y ), ρϕ)x, Y ) = 2n )ρx, Y ) gx, Y ), ρjϕ)x, Y ) = 2n )ρjx, JY ) gx, Y ), ρψ)x, Y ) = 6 ρx, Y ) ρjx, JY ), ρ ϕ)x, Y ) = 2n ) ρx, Y ) ρy, X) 2gX, Y ), ρ ψ ) X, Y ) = 6 ρx, Y ) ρy, X), ρπ )X, Y ) = gx, Y ), ρϕ)x, Y ) = ρx, Y ) ρjx, JY ), ρjϕ)x, Y ) = ρx, Y ) ρjx, JY ), ρπ 2 )X, Y ) = 2n )gx, Y ), ρψ)x, Y ) = 2n ) ρx, Y ) ρjx, JY ) 2gX, Y ), ρ ϕ)x, Y ) = 2 ρx, Y ) ρy, X), ρ ψ)x, Y ) = 2n ) ρx, Y ) ρy, X) 2X, Y ). 3) 2 Some conformally invariant tensors For an almost Hermitian manifold M, g, J), we consider the conformal change of metric ḡ = e 2f g, where f is a scalar function. If and are the Levi-Civita connections with respect to the metrics g and ḡ respectively, we have )X, Y ) = θx)y ) θy )X) gx, Y )U, for any vector fields X, Y T M), where θ = df, and U is the vector field such that gu, X) = θx). From now on, all geometric objects in M, ḡ, J) will be denoted by analogous letters as in M, g, J), but with bar. It is well known see e.g., 3) that the Riemannian curvature tensors R and R are related as follows e 2f RX, Y, Z, W ) = RX, Y, Z, W ) gx, W )σy, Z) gy, Z)σX, W ) gx, Z)σY, W ) gy, W )σx, Z) 4) where σx, Y ) = X θ)y ) θx)θy ) gx, Y )θu). 2

5 80 Mileva Prvanović We note that σx, Y ) = σy, X). The relation 4) implies ρy, Z) = ρy, Z) 2n )σy, Z) gy, Z)σ, where σ = 2n i= σe i, e i ). Therefore and Thus σy, Z) = 2n ) 2n ) σ = e2f 22n ), 5) ρy, Z) ρy, Z) 22n )ḡy, Z) σy, Z) σjy, JZ) = ρy, Z) ρjy, JZ) 2n ) ρy, Z) ρjy, JZ) 2n ) On the other hand, the relation 4) implies 22n ) gy, Z). 6) Z) 2n )2n )ḡy, gy, Z). 7) 2n )2n ) e 2f RX, Y, JZ, JW ) = RX, Y, JZ, JW ) F X, W )σy, JZ) wherefrom it follows or σy, Z) σjy, JZ) = 2 F Y, Z)σX, JW ) F X, Z)σY, JW ) F Y, W )σx, JZ), σy, Z) σjy, JZ) = ρy, Z) ρy, Z), 8) ρy, Z) ρz, Y ) ρy, Z) ρz, Y ) 9) 2 and σ = 2 e 2f ). 0) Comparing 7) and 9), as well as 5) and 0), we find ρy, ρy, Z) ρjy, JZ) Z) ρz, Y ) n = n ρy, ρy, Z) ρjy, JZ) Z) ρz, Y ) Z) n )2n )ḡy, gy, Z), n )2n ) )

6 The conformal change of the metric of an almost Hermitian manifold 8 and The relation ) shows that the tensor e 2f ) = 2n 2n. 2) V Y, Z) = n ρy, Z) ρjy, JZ) ρy, Z) ρz, Y ) 3) n )2n ) gy, Z) is conformally invariant, i.e., V Y, Z) = V Y, Z). Thus and therefore e 2f ḡx, W ) V Y, Z) = gx, W )V Y, Z) e 2f { ḡx, W ) V Y, Z) ḡy, Z) V X, W ) ḡx, Z) V Y, W ) ḡy, W ) V X, Z) } = gx, W )V Y, Z) gy, Z)V X, W ) gx, Z)V Y, W ) gy, W )V X, Z). This relation, in view of 3) and 2), can be rewritten in the form e 2f 2 ϕ J ϕ) ϕ n n )2n ) π = n ϕ Jϕ) 2 ϕ n )2n ) π. This means that the tensor W 3 = n ϕ Jϕ) 2 ϕ n )2n ) π 4) satisfies the condition e 2f W3 = W 3. 5) Since all the tensors ϕ, Jϕ, ϕ and π are generalized curvature tensors, the tensor 4) is also a generalized curvature tensor. In a similar way, we find that the tensors W 5 = satisfy the conditions W 4 = n ψ ψ 2 n )2n ) π 2 ) 2n π and W 6 = 2n )π 2 6) e 2f W4 = W 4, e 2f W5 = W 5 and e 2f W6 = W 6 7) respectively. Thus, we can state the following theorem. Theorem. For an almost Hermitian manifold we have. the tensor 3) is conformally invariant; 2. the generalized curvature tensors 6) satisfy the conditions 7) respectively.

7 82 Mileva Prvanović 3 The Weyl component of the antiholomorphic curvature tensor A 2-plane α T p M) is said to be holomorphic if Jα = α and antiholomorphic if Jα α. The sectional curvatures with respect to such 2-planes are holomorphic and antiholomorphic respectively. The holomorphic sectional curvatures can be examined using the holomorphic curvature tensor see e.g., 2, 4, 5). In 6 we found some conformally invariant tensors associated with holomorphic curvature tensor. To examine the antiholomorphic sectional curvatures, G. Ganchev, 2 introduced antiholomorphic curvature tensor which is AR)X, Y, Z, W ) = RX, Y, Z, W ) { F X, W ) ρy, JZ) F Y, Z) ρx, JW ) 2n 2 F X, Z) ρy, JW ) F Y, W ) ρx, JZ) } 2F X, Y ) ρz, JW ) 2F Z, W ) ρx, JY ) The corresponding Ricci tensor is ρar)x, Y ) 2n i= = ρx, Y ) 3 2n 2 2n 2)2n ) π 2X, Y, Z, W ). AR)e i, X, Y, e i ) ρx, Y ) ρy, X) and therefore the corresponding scalar curvature is AR) = 2n i= ρar)e i, e i ) = 8) 3 gx, Y ), 9) 2n 2)2n ) 3 2n, 20) while the second Ricci tensor vanishes. In this section we apply the conformal change of the metric to the tensor 8). Let A R be the antiholomorphic curvature tensor with respect to the metric ḡ. Then, using 4), 8), and 0), we get e 2f A R)X, Y, Z, W ) = AR)X, Y, Z, W ) gx, W )σy, Z) gy, Z)σX, W ) gx, Z)σY, W ) gy, W )σx, Z) { F X, W )σy, JZ) σjy, Z) 2n 2 F Y, Z)σX, JW ) σjx, W ) F X, Z)σY, JW ) σjy, W ) F Y, W )σx, JZ) σjx, Z) 2F X, Y )σz, JW ) σjz, W ) } 2F Z, W )σx, JY ) σjx, Y ) σ n )2n ) π 2X, Y, Z, W ). 2)

8 The conformal change of the metric of an almost Hermitian manifold 83 To determine σx, Y ), we put into 2) X = W = e i, sum up and obtain ρa R)Y, Z) ρar)y, Z) = 2n2 5 3 σy, Z) σjy, JZ) n n 2n2 3n 4 gy, Z)σ. n )2n ) It follows from 22) that e 2f A R) AR) = 8n2 ) 2n σ, or, in view of 20), σ = 2n {e 3 2f ) 8n 2 3 )}. 23) ) 2n 2n On the other hand, the relation 22) yields 4n )n 2 4)σY, Z) = 2n 2 5) ρa R)Y, Z) ρar)y, Z) 3 ρa R)JY, JZ) ρar)jy, JZ) 2n2 )2n 2 3n 4) gy, Z)σ n )2n ) wherefrom, using 9) and 23) and supposing n > 2, we find σy, Z) = 2n 2 4n )n 2 5) ρy, Z) 3 ρjy, JZ) 4) 3 4n 2 ρy, Z) ρz, Y ) 4) 6n 2 )n 2 2n 2 3n 4) 9n ḡy, Z) 4) { 2n 2 4n )n 2 5)ρY, Z) 3ρJY, JZ) 4) 3 ρy, Z) 4n 2 ρz, Y ) 4) 2n 2 6n 2 )n 2 3n 4) 9n gy, Z)}. 4) Finally, substituting 23) and 24) into 2) and using the notations 2), we get e {A 2f R 4n 2 2n2 5 4) n ϕ 3 n J ϕ ψ 3 ϕ ) ψ n 2n2 3n 4) 9n 3 2n 2 π ) 2n 2 n 7n 4 } ) π 2 ) 2n = AR 4n 2 2n2 5 4) n ϕ 3 n Jϕ ψ 3 ϕ ) 25) ψ n 2n2 3n 4) 9n 3 2n 2 π ) 2n 2 n 7n 4 )π 2. ) 2n 22) 24)

9 84 Mileva Prvanović Thus, putting W = AR 4n 2 2n2 5 4) n ϕ 3 n Jϕ ψ 3 ϕ ) ψ n 2n2 3n 4) 9n 3 2n 2 π ) 2n 2 n 7n 4 26) )π 2 ) 2n we have e 2f W = W. In 2 the tensor 26) is obtained applying the technique of a decomposition of the Hermitian vector space under an action of a unitary group, is denoted by AR) W and is called the Weyl component of the antiholomorphic curvature tensor. It is here obtained as a result of a direct application of the conformal change of the metric to the antiholomorphic curvature tensor. It can be proved, using the relation 3) that ρw ) = ρw ) = 0, and this explains the name Weyl component... for the tensor 26) 4 The second conformally invariant tensor associated with the antiholomorphic curvature tensor Putting into 7) JZ instead of Z, we find σy, JZ) σjy, Z) = ρy, JZ) ρjy, Z) 2n ) 2n )2n ) F Y, Z) ρy, JZ) ρjy, Z) 2n ) F Y, Z). 2n )2n ) Substituting this relation as well as the relations 5) and 6) into 2), and using the notation 2), we obtain or where e 2f {A R = AR 2n ) ϕ 4n 2 ) ψ 2n )2n ) 2n ) ϕ 4n 2 ) ψ W 2 = AR 2n )2n ) π π 3n ) } n )2n ) π 2 3n ) n )2n ) π 2 27) e 2f W2 = W 2, 28) 2n ) ϕ 2n )2n ) 4n 2 ) ψ π 3n n )2n ) π 2 ). 29)

10 The conformal change of the metric of an almost Hermitian manifold 85 We say that 29) is the second conformally invariant in the sense of 28)) curvature tensor of an almost Hermitian manifold associated with the antiholomorphic curvature tensor. In the case n > 2, the relation 27) can also be obtained from 25), using ) and 2). Namely, we have, according to ) ρy, Z) ρz, Y ) = ρy, Z) ρjy, JZ) n ) n )2n Thus e 2f ḡx, W ) such that ρy, Z) ρz, Y ) n ) Z) )ḡy, ρy, Z) ρjy, JZ) gy, Z). n )2n ) ρy, Z) ρz, Y ) = e 2f { n )ḡx, W ) ρy, Z) ρjy, JZ) } n )2n )ḡx, W )ḡy, Z) gx, W ) ρy, Z) ρz, Y ) n ) gx, W ) ρy, Z) ρjy, JZ) gx, W )gy, Z), n)2n) { } 2f e ϕ = e 2f 2 ϕ J ϕ) n n )2n ) π ϕ 2 ϕ Jϕ) n n )2n ) π. We get, in a similar way, { } e 2f ψ = e 2f n ψ 2 n )2n ) π 2 Therefore 3e 2f ϕ ψ n ψ 2 n )2n ) π 2. ψ ) { 3 = e 2f ϕ J ϕ) 3 n n 6 n )2n ) ψ 3 ϕ n 6 n )2n ) ψ n 2 π n π 2 ) 3 ) } ϕ Jϕ) 3ψ n π ) n π 2. n 2

11 86 Mileva Prvanović Substituting this into 25), and using 2), we get just 27). Thus we can state a theorem. Theorem 2. For n > 2 the relation 27) is a consequence of 25), ) and 2). The first Ricci tensor of the tensor W 2 is 3 { ρw 2 )Y, Z) = 2n ) n ρy, Z) ρjy, JZ) ρy, Z) ρz, Y ) 3n } 2n n )2n ) gy, Z), while for the second Ricci tensor we have ρw 2 ) = 0. 5 The relations between some conformally invariant tensors 5. According to 4) and 6), we have ϕ = W 3 2 ϕ Jϕ) n n )2n ) π, ψ = W 4 n ψ 2 n )2n ) π 2. Substituting this into 26), we get 4n 2 4)W = 4n 2 4) AR 2n 2 3n 4 2n 2 ) But, in view of 29), we have ϕ 2n ) ψ 6 n )2n ) 3n 2n 2 ) 6 n 2 )2n ) 4n 2 ) ) ) 3W 3 3 n W 4 9n π 2n 2 ) 37n 4) 2n 2 )2n ) π 2. AR 2n ) ϕ 4n 2 ) ψ = W 2 2n )2n ) π 3n 2n 2 )4n 2 ) π 2, because of which and using 6) the preceding relation can be rewritten in the form 3 W AR) W = W 2 4n 2 W 3 ) 4) n W 4 3 8n 2 )n 2 3nW 5 7n 4 30) 4) 2n W 6. Thus, we can state the following theorem. Theorem 3. The Weyl component of the antiholomorphic curvature tensor can be expressed as a linear combination of the tensors W 2, W 3, W 4, W 5 and W 6 such that 30) holds. Each of the tensors W is a generalized curvature tensor and each satisfies the condition of the type e 2f W = W.

12 The conformal change of the metric of an almost Hermitian manifold It is well known see e.g., 3) that the Weyl conformal curvature tensor for the Riemannian manifold M, g), dim M = 2n, is CX, Y, Z, W ) = RX, Y, Z, W ) gx, W )ρy, Z) 2n ) gy, Z)ρX, W ) gx, Z)ρY, W ) gy, W )ρx, Z) 2n )2n ) π X, Y, Z, W ) and that satisfies the condition e 2f CX, Y, Z, W ) = CX, Y, Z, W ). Now, let us apply 8) to the tensor 3), instead to the tensor R. In such a way we obtain AC)X, Y, Z, W ) = CX, Y, Z, W ) F X, W ) ρc)y, JZ) 2n ) F Y, Z) ρc)x, JW ) F X, Z) ρc)y, JW ) F Y, W ) ρc)x, JZ) 2F X, Y ) ρc)z, JW ) 2F Z, W ) ρc)x, JY ) C) 2n 2)2n ) π 2X, Y, Z, W ). The first Ricci tensor of the tensor 3) vanishes. But, for the second Ricci tensor we have ρc)y, Z) = 2n i= Ce i, Y, JZ, Je i ) = ρy, Z) ρy, Z) ρjy, JZ) 2n ) gy, Z), 2n )2n ) wherefrom it follows C) = 2n. Substituting this and 3) into 32) and using 8), we get AC) = AR 2n ) ϕ ψ 2n ) 2n π 3n n )2n ) π 2 such that, and in view of 29), we can state the following theorem. ), Theorem 4. The second conformally invariant curvature tensor, associated with the antiholomorphic curvature tensor, satisfies the relation W 2 = AC), while ) can be expressed in the form ρ C)Y, Z) ρ C)Z, Y ) = ρc)y, Z) ρc)z, Y ). 3) 32)

13 88 Mileva Prvanović 5.3 We note that, starting from 4), and using 3), we have ρw 3 )Y, Z) = wherefrom it follows Substituting this into we get 2n i= { = 2 n W 3 e i, Y, JZ, Je i ) ρy, ρy, Z) ρjy, JZ) Z) ρz, Y ) } n )2n ) gy, Z) = 2V Y, Z), W 3 ) = 4 AW 3 )X, Y, Z, W ) = W 3 X, Y, Z, W ) 2n ). { F X, W ) ρw 3 )Y, JZ) 2n ) F Y, Z) ρw 3 )X, JW ) F X, Z) ρw 3 )Y, JW ) F Y, W ) ρw 3 )Y, JZ) 2F X, Y ) ρw 3 )Z, JW ) 2F Z, W ) ρw 3 )X, JY ) } W 3 ) 2n )2n ) π 2X, Y, Z, W ), AW 3 ) = W 3 n W 2 4 n )2n ) W 6. 33) As for the tensor W 4, we have { } ρw 4 )Y, Z) = 2n ) n ρy, Z) ρjy, JZ) ρy, Z) ρz, Y ) 2 2 n )2n ) gy, Z), wherefrom it follows Thus W 4 ) = 42n ) 2n AW 4 ) = W 4 n ψ ψ 2 n )2n ) π 2 = 0. In a similar way we find AW 5 ) = 2n. ) π π ) 2, AW 6 ) = 0. 2n Summing up the preceding results, we can state the following theorem.

14 The conformal change of the metric of an almost Hermitian manifold 89 Theorem 5. Applying 8) to the tensors W, we get AW ) AAR W ) = AR) W, AW 2 ) = W 2, AW 3 ) = W 3 n W 2 4 n )2n ) W 6, AW 4 ) = 0, AW 5 ) = W 5 2n W 6, AW 6 ) = 0. 6 Kähler spaces In the case of Kähler spaces, we have ρ = ρ, =, ρjx, JY ) = ρx, Y ), 34) and therefore, the conformally invariant tensor 3) has the form Also, putting V Y, Z) = 2n 2) n ρy, Z) gy, Z). n )2n ) we get ψ 0 X, Y, Z, W ) = F X, W )ρy, JZ) F Y, Z)ρX, JW ) F X, Z)ρY, JW ) F Y, W )ρx, JZ) 2F X, Y )ρz, JW ) 2F Z, W )ρx, JY ), ψ = ψ = 2ψ 0. 35) Thus, for Kähler manifolds, the antiholomorphic curvature tensor is AR = R 2n ) ψ 0 2n )2n ) π 2. 36) Using 34), 35) and 36), we find that, for Kähler manifolds, the relation 26) becomes or, explicitly, W AR) W = R 2n 2) ϕ ψ 0) 4n )n 2) π π 2 ), W )X, Y, Z, W ) AR) W X, Y, Z, W ) = RX, Y, Z, W ) { gx, W )ρy, Z) gy, Z)ρX, W ) gx, Z)ρY, W ) 2n ) gy, W )ρx, Z) F X, W )ρy, JZ) F Y, Z)ρX, JW ) F X, Z)ρY, JW ) } F Y, W )ρx, JZ) 2F X, Y )ρz, JW ) 2F Z, W )ρx, JY ) 4n )n 2) π X, Y, Z, W ) π 2 X, Y, Z, W ). But this is the Bochner curvature tensor of a Kähler manifold. Thus, we can state the following theorem.

15 90 Mileva Prvanović Theorem 6. For a Kähler manifold, the Weyl component of the antiholomorphic curvature tensor, AR) W, is Bochner curvature tensor. In a similar way we find that, in the case of Kähler manifolds, References W 2 = R 2n ) ϕ n 2 2n 2 ) ψ 0 π 2n )2n ) 2n 2 6n 2n 2 )4n 2 ) π 2, 2n 2) W 3 = n ϕ 2 n )2n ) π 2n ), W 5 = 2n π, 2n 2) W 4 = n ψ 2 0 n )2n ) π 2n ) 2, W 6 = 2n π 2. G. Ganchev: Almost Hermitian manifolds similar to the complex space forms. C. R. Acad. Bulg. Sci ) G. Ganchev: Bochner curvature tensors in almost Hermitian manifolds. PLISKA, Stud. Math. Bulg ) J. Mikeš, V. Kiosak, A. Vanžurová: Geodesic Mappings of Manifolds with Affine Connection. Palacký University, Faculty of Science, Olomouc 2008). 4 M. Prvanović: On the cutvature tensor of Kähler type in an almost Hermitian and almost anti-hermitian manifolds. Mat. Vasnik ) M. Prvanović: Some applications of the holomorphic curvature tensor. Tensor, N.S ) M. Prvanović: Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor. J. Geom ) F. Tricerrri, L. Vanhecke: Curvature tensors on almost Hermitian manifolds. Trans. Amer. Math. Soc ) Author s address: Mileva Prvanović: Matematički institut SANU, Kneza Mihaila 36, 000 Beograd, p.p. 367, Serbia Received: 6 January, 203 Accepted for publication: 7 April, 203 Communicated by: Olga Rossi