Bulletin T.CXLI de l Académie serbe des sciences et des arts 2010 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 35

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1 Bulletin T.CXLI de l Académie sere des sciences et des arts 2010 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 35 SOME PROPERTIES OF THE LOCALLY CONFORMAL KÄHLER MANIFOLD MILEVA PRVANOVIĆ (Presented at the 1st Meeting, held on Feruary 29, 2010) A s t r a c t. We determine the holomorphic curvature tensor of locally conformal Kähler manifold and find the expression of the Riemannian curvature tensor of such manifold if it is of pointwise constant holomorphic sectional curvature. We examine the geodesic mapping using the holomorphic curvature tensor and apply the otained results to the locally conformal Kähler manifolds. AMS Mathematics Suject Classification (2000): 53C56 Key Words: locally conformal Kähler manifold, holomorphic curvature tensor, geodesic mapping 1. The locally conformal Kähler manifolds Let (M, g, J) e a real 2n-dimensional almost Hermitian manifold, where J is almost complex structure and g is Hermitian metric. Then J 2 = Id., g(jx, JY ) = g(x, Y ), for any vector fields X, Y tangent to M. The fundamental 2-form F (X, Y ) is F (X, Y ) = g(jx, Y ) = F (Y, X).

2 10 Mileva Prvanović (M, g, J) is a Kähler manifold if J = 0, where denotes the covariant differentiation with respect to Levi-Civita connection. The manifold (M, g, J) is called locally conformal Kähler (f. l.c.kähler) manifold if each point p M has an open neighorhood U with a differentiale function σ : U R 2n such that ġ = e 2σ g U (1.1) is a Kähler metric on U. The necessary and the sufficient condition for (M, g, J) to e l.c. Kähler manifold is that it admits a gloal 1-form α satisfying the condition [2]: ( Z F )(X, Y ) = α(jx)g(z, Y ) α(jy )g(z, X) α(x)f (Z, Y )+α(y )F (Z, X). (1.2) With respect to the local coordinates, (1.2) is s F ij = α i F sj + α j F si α a F ai g sj + α a F aj g si, i α j = j α i, α i = α p g pi, (1.3) and the indices run over the range {1, 2,..., 2n}. From (1.3) we otain r s F ij s r F ij = = P rs J a j g si P ra J a i g sj P sa J a j g ri + P sa J a i g rj P rj F si + P ri F sj + P sj F ri P si F rj, where P ri = r α i α r α i α pα p g ri. We note that P ri = P ir. Using the Ricci identity, we get R sria J a j + R srja J a i = = P ri F sj + P sj F ri P rj F si P si F rj (1.) P ra J a i g sj P sa J a j g ri + P ra J a j g si + P sa J a i g rj,

3 Some properties of the locally conformal Kähler manifold 11 or R sra J a j J h = R srjh +P rh g sj P rj g sh + P sj g rh P sh g rj (1.5) +P ra J a hf sj P ra J a j F sh + P sa J a j F rh P sa J a hf rj. Transecting (1.) with g ir and denoting y ρ the Ricci tensor, we get ρ sa J a j + R srja J a i g ir = = (2n 3)P sa J a j P ja J a s + (P a g a )F sj, from which, taking the symmetric part, we otain [3]: ρ ia J a j + ρ ja J a i = 2(n 1)(P ja J a i + P ia J a j ). (1.6) 2. Holomorphic curvature tensor A 2-plane π in T p (M), p M, is said to e holomorphic if Jπ = π. The manifold M has pointwise constant holomorphic sectional curvature if the sectional curvature relative to π does not depend on the holomorphic 2-plane π in T p (M). The curvature tensor of Kähler manifold satisfies the condition R(X, Y, JZ, JW ) = R(X, Y, Z, W ). (2.1) Using it, in [] (Proposition 7.3, p.167) is determined the curvature tensor of Kähler manifold of constant holomorphic sectional curvature. But, if J 0, (2.1) does not hold. Nevertheless, there exist for any almost Hermitian manifold, the algeraic curvature tensor, satisfying the condition of type (2.1). It is ([],[7]): (HR)(X, Y, Z, W ) = = 1 {3 [R(X, Y, Z, W ) + R(JX, JY, Z, W ) + R(X, Y, JZ, JW ) 16 +R(JX, JY, JZ, JW )] R(X, Z, JW, JY ) R(JX, JZ, W, Y ) (2.2) R(X, W, JY, JZ) R(JX, JW, Y, Z) + R(JX, Z, JW, Y ) +R(X, JZ, W, JY ) + R(JX, W, Y, JZ) + R(X, JW, JY, Z)}.

4 12 Mileva Prvanović Namely, it is easy to see that (HR)(X, Y, Z, W ) = (HR)(Y, X, Z, W ) = = (HR)(X, Y, W, Z) = (HR)(Z, W, X, Y ), (2.3) (HR)(X, Y, Z, W ) + (HR)(X, Z, W, Y ) + (HR)(X, W, Y, Z) = 0, as well as (HR)(X, Y, JZ, JW ) = (HR)(X, Y, Z, W ), (2.) (HR)(X, JX, JX, X) = R(X, JX, JX, X). (2.5) The tensor (2.2) is said to e the holomorphic curvature tensor. The relation (2.5) shows that the holomorphic sectional curvatures relative to R and HR are the same. Thus, applying the way of [] to the tensor HR, we can state Proposition 2.1 If an almost Hermitian manifold (M, g, J) is of pointwise constant holomorphic sctional curvature H(p), p M, then the holomorphic curvature tensor is (HR)(X, Y, Z, W ) = H(p) [g(x, W )g(y, Z) g(x, Z)g(Y, W ) +F (X, W )F (Y, Z) F (X, Z)F (Y, W ) 2F (X, Y )F (Z, W )]. (2.6) Now, we shall determine the holomorphic curvature tensor for a l.c. Kähler manifold. With respect to the local coordinates, (2.2) reads (HR) ijhk = = 1 { [ ] 3 R ijhk + R ahk Ji a Jj + R ija J a 16 hjk + R acd Ji a Jj JhJ c k d R iha Jk a Jj R akj Ji a Jh R ika Jj a Jh R ajh Ji a Jk } +R ahj Ji a Jk + R iak JhJ a j + R akj Ji a Jh + R iah Jk a Jj. (2.7)

5 Some properties of the locally conformal Kähler manifold 13 In view of (1.5), we find [ ] 3 R ijhk + R ahk Ji a Jj + R ija JhJ a k + R acd Ji a Jj JhJ c k d = 3 [R ijhk +g ih (3P jk P a J a j J k) g ik (3P jh P a J a j J h) +g jk (3P ih P a J a i J h) g jh (3P ik P a J a i J k) (2.8) +F ih (P ja Jk a P ka Ji a ) F ik (P ja Jh a P ha Jj a ) +F jk (P ia Jh a P ha Ji a ) F jh (P ia Jk a P ka Ji a )]. Using (1.5), we also otain (R iha Jk a Jj + R akj Ji a Jh + R ika Jj a Jh + R ajh Ji a Jk) = 2R ijhk 2(g ik P jh + g jh P ik g ih P jk g jk P ih ) F ik (P ha J a j P ja J a h) F jh (P ka J a i P ia J a k ) (2.9) F ih (P ja Jk a P ka Jj a ) F jk (P ia Jh a P ha Ji a ) 2F ij (P ka Jh a P ha Jk a ) 2F hk (P ja Ji a P ia Jj a ). Using (1.), as well as the relation R akht R ahkt = R khat, we find R ahj Ji a Jk + R akj Ji a Jh = = R kha Ji a Jj P hj g ik + P ka Ji a F hj + P hk g ij P ja Ji a F hk P ha Jj a F ik P a Ji a Jkg jh + P ha Jk a F ij + P a Ji a Jj g hk +P kj g ih P ha Ji a F kj P hk g ij + P ja Ji a F kh +P ka Jj a F ih + P a Ji a Jhg jk P ka JhF a ij P a Ji a Jj g hk,

6 1 Mileva Prvanović such that, applying (1.5), we have R ahj J a i J k + R akj J a i J h = = R ijhk + 2g ih P jk 2g ik P jh +g jk (P ih + P a J a i J h) g jh (P ik + P a J a i J k) (2.10) 2P ja J a i F hk + (P ha J a k P ka J a h)f ij. In a similar way, we otain R iak J a hj j + R iah J a k J j = = R ijhk 2P ik g jh + g ih (P jk + P a J a j J k) g ik (P jh + P a J a j J h) + 2P ih g jk (2.11) +2P ia J a j F hk + (P ha J a k P ka J a h)f ij. Sustituting (2.8)-(2.11) into (2.7), we get (HR) ijhk = R ijhk + 1 { g ih (7P jk P a Jj a J 8 k) g ik (7P jh P a Jj a Jh) +g jk (7P ih P a J a i J h) g jh (7P ik P a J a i J k) +F ih (P ja J a k P ka J a j ) F ik (P ja J a h P ha J a j ) (2.12) +F jk (P ia Jh a P ha Ji a ) F jh (P ia Jk a P ka Ji a ) } +2F ij (P ha Jk a P ka Jh) a + 2F hk (P ia Jj a P ja Ji a ). Thus, we can state Theorem 2.1 The holomorphics curvature tensor of l.c.kähler manifold has the form (2.12). In view of Proposition (2.1), l.c.kähler manifold has pointwise constant

7 Some properties of the locally conformal Kähler manifold 15 holomorphic sectional curvature if its curvature tensor has the form R ijhk = H(p) [g ik g jh g ih g jk + F ik F jh F ih F jk 2F ij F hk ] + 1 { g ik (7P jh P a Jj a J 8 h) g ih (7P jk P a Jj a Jk) +g jh (7P ik P a Ji a Jk) g jk (7P ih P a Ji a Jh) +F ik (P ja J a h P ha J a j ) F ih (P ja J a k P ka J a j ) +F jh (P ia Jk a P ka Ji a ) F jk (P ia Jh a P ha Ji a ) } 2F ij (P ha Jk a P ka Jh) a 2F hk (P ia Jj a P ja Ji a ). (2.13) Conversely, if (2.13) holds, then the holomorphic curvature tensor has the form (2.6). Thus, we can state Theorem 2.2 L.c. Kähler manifold has pointwise constant holomorphic sectional curvature if and only if its curvature tensor can e expressed in the form (2.13). If the tensor P ij is hyrid, i.e. if P a J a i J j = P ij, P ia J a j = P ja J a i, the relations (2.12) and (2.13) reduce, respectively to (HR) ijhk = R ijhk + 3 (P jkg ih + P ih g jk P jh g ik P ik g jh ) + 1 (F ihp ja J a k + F jk P ia J a h F ik P ja J a h F jh P ia J a k (2.1) and R ijhk = H(p) +2F hk P ia J a j + 2F ij P ha J a k ) [g ik g jh g ih g jk + F ik F jh F ih F jk 2F ij F hk ] + 3 (g ikp jh + g jh P ik g ih P jk g jk P ih ) + 1 (F ikp ja J a h + F jh P ia J a k F ih P ja J a k F jk P ia J a h 2F ij P ha J a k 2F hk P ia J a j ). (2.15)

8 16 Mileva Prvanović The relation (2.15) is just the form of the curvature tensor of l.c.kähler manifold otained in [3] and [5]. According (1.6), the tensor P ij is hyrid if and only if the Ricci tensor ρ ij is hyrid. Thus, we can state ([3], [5]): Corrollary 2.1. L.c.Kähler manifold whose Ricci tensor is hyrid is of pointwise constant holomorphic sectional curvature if and only if its curvature tensor has the form (2.15). 3. The expression of the holomorphic curvature tensor which does not include the tensor P ij We shall show that, if n > 2, the tensors HR and R in the relations (2.12)-(2.15) can e written in the form which include the second Ricci tensor instead of the tensor P ij. The second Ricci tensor is defined as follows ρ * jh = R ija J a hj kg ik. It has, for any almost Hermitian manifold, the property ρ * aj a i J j = ρ * ji. In general, it is not symmetric, ut for l.c.kähler manifold, it is. Namely, the relation (1.5) yields (2n 3)P jh P a J a j J h = ρ jh ρ * jh P g jh, (3.1) where and P = P a g a = κ κ* (n 1), κ = ρ a g a and κ * = ρ * ag a are the first and the second scalar curvatures. From (3.1) immediately follows ρ * jh = ρ * hj. Thus, for any l.c.kähler manifold, we have ρ * jh = ρ * hj, ρ * aj a j J h = ρ * jh.

9 Some properties of the locally conformal Kähler manifold 17 The relation (3.1), together with P jh + (2n 3)P a J a j J h = ρ a J a j J h ρ * jh P g jh, implies (n 1)(n 2)P jh = (2n 3)ρ jh +ρ a J a j J h 2(n 1) ρ * jh 2(n 1)P g jh, (3.2) such that, if n > 2, we have 1 { P jh = (2n 3)ρ jh +ρ a Jj a J (n 1)(n 2) h 2(n 1) ρ * } jh 2(n 1)P g jh. (3.3) Sustituting (3.3) into (2.12) and (2.13), we get, respectively, (HR) ijhk = R ijhk + P (n 2) {3(g ikg jh g ih g jk ) (F ik F jh F ih F jk 2F ij F hk )} 7n (n 1)(n 2) (g ihρ jk + g jk ρ ih g ik ρ jh g jh ρ ik ) n 5 16(n 1)(n 2) (g ihρ a Jj a Jk + g jk ρ a Ji a Jh g ik ρ a Jj a Jh g jh ρ a Ji a Jk) 3 8(n 2) (g ik ρ * jk + g jk ρ * ih g ik ρ * jh g jh ρ * ik) 1 { + F ih (ρ ja Jk a ρ ka Jj a 2 ρ 16(n 2) * jajk a ) +F jk (ρ ia Jh a ρ ha Ji a 2 ρ * iajh) a F ik (ρ ja J a h ρ ha J a j 2 ρ * jaj a h) F jh (ρ ia J a k ρ ka J a i 2 ρ * iaj a k ) +2F ij (ρ ha Jk a ρ ka Jh a 2 ρ * hajk a ) +2F hk (ρ ia Jj a ρ ja Ji a 2 ρ * } iajj a ) (3.) and

10 18 Mileva Prvanović R ijhk = 1 (H 3P n 2 )(g ikg jh g ih g jk ) + 1 (H + P n 2 )(F ikf jh F ih F jk 2F ij F hk ) 7n (n 1)(n 2) (g ikρ jh + g jh ρ ik g ih ρ jk g jk ρ ih ) n 5 16(n 1)(n 2) (g ikρ a Jj a Jh + g jh ρ a Ji a Jk g ih ρ a Jj a Jk g jk ρ a Ji a Jh) 3 8(n 2) (g ik ρ * jh + g jh ρ * ik g ih ρ * jk g jk ρ * ih) 1 { + F ik (ρ ja Jh a ρ ha Jj a 2 ρ 16(n 2) * jajh) a +F jh (ρ ia J a k ρ ka J a i 2 ρ * iaj a k ) F ih (ρ ja J a k ρ ka J a j 2 ρ * jaj a k ) F jk (ρ ia J a h ρ ha J a i 2 ρ * iaj a h) (3.5) Thus, we can state 2F ij (ρ ha Jk a ρ ka Jh a 2 ρ * hajk a ) 2F hk (ρ ia Jj a ρ ja Ji a 2 ρ * } iajj a ). Theorem 3.1. If n > 2, the holomorphic curvature tensor of a l.c.kähler manifold can e expressed in the form (3.). If such a manifold has pointwise constant holomorphic sectional curvature, its Riemannian curvature tensor can e expressed in the form (3.5). Remark. If n = 2, we can not eliminate the tensor P ij from the equations (2.12)-(2.15). On the other hand, according (3.2), we have ρ jh + ρ a J a j J h = 2 ρ * jh + 2P g jh, and, if ρ jh is hyrid, ρ jh ρ * jh = P g jh.

11 Some properties of the locally conformal Kähler manifold 19. Geodesic mapping The diffeomorphism transforming all geodesics of the Riemannian manifold ( M, ḡ) onto the geodesics of the Riemannian manifold (M, g) is said to e the geodesic mapping. If we consider the manifolds M and M with respect to the coordinate system which is common for M and M, then the necessary and the sufficient condition for geodesic mapping in [6] Γ t ij = Γ t ij + δ t iψ j + δ t jψ i. (.1) where Γ and Γ are the Levi-Civita connections of the metrics ḡ and g respectively, and ψ i is the gradient vector field. As for the curvature tensors R and R, they are related as follows where R t jhk = R t jhk + δ t kψ jh δ t hψ jk (.2) ψ jh = h ψ j ψ j ψ h. Because ψ j is a gradient, we have ψ jh = ψ hj. Now, let us consider the geodesic mapping f : ( M, ḡ, J) (M, g, J) of an almost Hermitian manifold ( M, ḡ, J) onto an almost Hermitian manifold (M, g, J). For ( M, ḡ, J), the holomorphic curvature tensor is (H R) t jhk = 1 16 { [ 3 Rt jhk + R t jajhj a k R a hkjaj t j R ] a cdjaj t j JhJ c k d R t hajk a Jj + R a kjjaj t h R t kajj a Jh + R a jhjaj t k R a hjjaj t k + R t akjhj a j R a kjjaj t h + R } t ahjk a Jj. Sustituting (.2), we find 8(H R) t jhk = 8(HR) t jhk δ t kq jh δ t hq jk + J t kq ja J a h J t hq ja J a k 2J t jq ha J a k, (.3) where Q jh = ψ jh + ψ a J a j J h.

12 20 Mileva Prvanović Contracting (.3) with respect to t and k, we get Q jh = [ ] ρ(h R)jh ρ(hr) jh, (.) n + 1 where ρ(h R) and ρ(hr) are the Ricci tensors of H R and HR respectively. Sustituting (.) into (.3), we find (H W ) t jhk = (HW ) t jhk, where (HW ) t jhk = (HR) t jhk 1 { δ t 2(n + 1) kρ(hr) jh δhρ(hr) t jk +J t kρ(hr) ja J a h J t hρ(hr) ja J a k 2J t jρ(hr) ha J a k }, (.5) and (H W ) t jhk is constructed in the same way, ut with respect to the tensor H R. Thus we can state Theorem.1. For an almost Hermitian manifold, the tensor (.5) is invariant with respect to the geodesic mapping (.1), preserving the complex structure. Now, let us suppose that HW = 0. Then (HR) ijhk = 1 2(n + 1) [g ikρ(hr) jh g ih ρ(hr) jk F ik ρ(hr) ja J a h + F ih ρ(hr) ja J a k + 2F ij ρ(hr) ha J a k ], (.6) from which, transvecting with g jh and taking into account that, as a consequence of (2.), the Ricci tensor ρ(hr) is hyrid, we get ρ(hr) ik = κ(hr) 2n g ik, where κ(hr) = ρ(hr) jh g jh. Sustituting this into (.6), we otain (HR) ijhk = κ(hr) n(n + 1) (g ikg jh g ih g jk + F ik F jh F ih F jk 2F ij F hk ).

13 Some properties of the locally conformal Kähler manifold 21 But, this is just the equation (2.6). Conversely, if (2.6) holds, the tensor (.5) vanishes. Thus, we can state Theorem.2. For an almost Hermitian manifold (M, g, J) the tensor (.5) vanishes if and only if (M, g, J) is of pointwise constant holomorphic sectional curvature. The l.c.kähler manifold can serve as an example. Indeed, for such a manifold the holomorphic curvature tensor has the form (2.12). Therefore ρ(hr) ik = ρ ik 3 P g ik 7n 10 P ik + n + 2 P a yi a y k. Sustituting this and (2.12) into (.5), we find that for l.c.kähler manifold, the tensor (HW ) ijhk is (HW ) ijhk = R ijhk + 1 { g ih (7P jk P a Jj a J 8 k) g ik (7P jh P a Jj a Jh) g jk (7P ih P a Ji a Jh) g jh (7P ik P a Ji a Jk) +F ih (P ja J a k P ka J a j ) F ik (P ja J a h P ha J a j ) +F jk (P ia Jh a P ha Ji a ) F jh (P ia Jk a P ka Ji a ) } +2F ij (P ha Jk a P ka Jh) a + 2F hk (P ia Jj a P ja Ji a ) { g ik (ρ jh 3 P g 7n 10 jh 1 2(n + 1) g ih (ρ jk 3 P g 7n 10 jk F ik (ρ ja 3 P g 7n 10 ja +F ih (ρ ja 3 P g 7n 10 ja +2F ij (ρ ha 3 P g 7n 10 ha If this tensor vanishes, then P jh + n + 2 P a Jj a J h) P jk + n + 2 P a Jj a J k) P ja + n + 2 P st Jj s J t a)jh a P ja + n + 2 P st J s j J t a)j a k P ha + n + 2 P st y s hya)y t k a (2n + 3)ρ ik 3ρ a Ji a Jk = 7n2 + 2n 12 P ik n2 + 1n 12 P a Ji a Jk 2 2 [ ] 3(n 2) κ P g ik. 2 }. (.7)

14 22 Mileva Prvanović This relation, together with yields 3ρ ik + (2n + 3)ρ a Ji a Jk = n2 + 1n 12 P ik 2 [ ] + 7n2 + 2n 12 P a Ji a Jk 3(n 2) + κ P g ik, 2 2 ρ ik 3 P g ik 7n 10 P ik + n + 2 P a Ji a Jk = 1 2n [κ 3(n 1)P ] g ik, and the relation (.7), if HW = 0, gives κ 3(n 1)P R ijhk = (g ik g jh g ih g jk + F ik F jh F ih F jk 2F ij F hk ) n(n + 1) 1 { g ik (7P jh P a Jj a J 8 h) + g jh (7P ik P a Ji a Jk) g ih (7P jk P a Jj a Jk) g jk (7P ih P a Ji a Jh) +F ik (P ja J a h P ha J a j ) + F jh (P ia J a k P ka J a i ) F ih (P ja Jk a P ka Jj a ) F jk (P ia Jh a P ha Ji a ) } 2F ij (P ha Jk a P ka Jh) a 2F hk (P ia Jj a P ja Ji a ). But, this is just the relation (2.13), where H(p) = κ 3(n 1)P n(n + 1) = κ + 3 κ* n(n + 1). Thus, and also as a consequence of theorems.1 and.2, we have Theorem.3 Let (M, g, J) e l.c.kähler manifold. Then the tensor (.7) is invariant with respect to the geodesic mapping. The tensor (.7) vanishes if and only if (2.13) holds. 5. The remark on Kähler manifolds A curve x i = x i (t) of an almost Hermitian manifold (M, g, J) having the property that the holomorphic planes determined y its tangent vectors

15 Some properties of the locally conformal Kähler manifold 23 are parallel along the curve itself is called a holomorphically planer curve. The manifold (M, ḡ, J) and (M, g, J) are said to e holomorphically projectively related if they have all holomorphically planer curves in common, and the corresponding invariant tensor is holomorphically projective curvature tensor. In the case of Kähler manifolds, this tensor is [8]: R i jhk 1 2(n + 1) ( δ t kρ jh δ t hρ jk + J t kρ ja J a h J t hρ ja J a k 2J t jρ ha J a k On the other hand, if (M, g, J) is a Kähler manifold, then (HR) t jhk = R t jhk, ρ(hr) jh = ρ jh, ) (5.1) such that (.5) reduces just to the tensor (5.1). Thus, in the case of Kähler manifolds, the tensor (5.1) can e otained considering holomorphically planer curves or applying geodesic mapping to the holomorphic curvature tensor (2.2). This does not hold for l.c.kähler manifolds. REFERENCES [1] G. G a n c h e v, On Bochner curvature tensors in almost Hermitian manifolds, Pliska, Studia math. Bulgarica., 9 (1987), [2] A. G r a y, L. H e r v e l l a, The sixteen classes of almost Hermitian manifolds and their linear invariants, Annali di Math. pura ed applicata, (IV), CXXIII, 1980, [3] T. K a s h i w a d a, Some properties of locally conformal Kähler manifold, Hokkaido Math. J., 8 (1979), [] S. K o a y a s h i, K. N o m i z u, Fondation of differential gormetry,vol. II, Interscience Pulishers, New York - London, 1969 [5] K. M a t s u m o t o, Locally conformal Kähler manifolds and their sumanifolds, Mem. Secţ. Stiinţ. Acad. Românâ, Ser. IV, 1, 1991, No. 2, 7 9. [6] J. M i k e š, V. K i o s a k, A. V a n ž u r o v a, Geodesic mappings of manifolds with affine connection, Palasky Univ. Olomouc, Faculty of Science, [7] M. P r v a n o v i ć, On the curvature tensors of Kähler type in an almost Hermitian and almost para-hermitian manifold, Matematički vesnik, 50 (1998), [8] K. Y a n o, Differential geometry on complex and almost complex spaces, Pergamon Press, Oxford, Mathematical Institute SANU Kneza Mihaila Belgrade Seria