ESI The Erwin Schrodinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien, Austria Compatible Almost Complex Structures on Quaternion{Kahler Manifolds D.V. Alekseevsky S. Marchiafava M. Pontecorvo Vienna, Preprint ESI 419 (1997) Supported by Federal Ministry of Science and Research, Austria Available via http://www.esi.ac.at January 10, 1997 COMPATIBLE ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS D.V. ALEKSEEVSKY1 - S. MARCHIAFAVA2 - M. PONTECORVO3 Sophus Lie Center (Moscow)1 - Universita' di Roma I2 - Universita di Roma Tre3 Abstract. Let (M 4n ; g; Q) be a quaternion-Kahler manifold with reduced scalar curvature = K=4n(n +2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let = ? J be the Lee form of J . We prove the following local results: 1) if J is conformally symplectic then it is parallel and = 0; 2) if J is cosymplectic then 0 and = 0 if and only if J is parallel; 3) if J is integrable then d is Q-Hermitian and harmonic; 4) if J is conformally balanced but not symplectic then there exists an associated non-zero Killing vector eld. We prove also that any closed self-dual 2-form ! 2 2+ = g Q 2 is parallel. When M 4n is compact our main global results are the following: 1) if > 0 and (M 4n ; g) is dierent from the Grassmannian G2 (C m ) of 2-planes of C m then there exists no compatible almost complex structure J ; 2) if the rst Chern class c1 (J ) c1 (T M ) = 0 for the almost complex structure J on the tangent bundle T M then = 0; 3) if 6= 0 then there exists no compatible complex structure J ; 4) if = 0 a compatible complex structure J is parallel. The last two results have been proved in [P2] by twistor methods. In x5 and x6 we compute the Laplacian and the (real and quaternionic) Weyl tensors acting on self-dual 2-forms, by relating to the work of Gauduchon in [G1]. 1. Introduction and main results In a previous paper ([AMP]) we studied several problems concerning almost complex structures which are dened on an almost quaternionic manifold (M 4n ; Q) and which are compatible with the almost quaternionic structure Q. We recall that an almost quaternionic structure Q on M 4n is a rank-3 subbundle Q End(TM ) which is locally spanned by almost hypercomplex structures H = (J ); such a locally dened triple H = (J), where J2 = ?id and J1J2 = ?J2J1 = J3, is called an admissible 4n basis of Q. An almost P complex structure J on M is compatible with Q if it can be written as J = cJ with respect to any admissible basis H = (J) for suitable 1991 Mathematics Subject Classication. 53C10 - 32C10. Work done under the program of G.N.S.A.G.A. of C.N.R. and partially supported by M.U.R.S.T (Italy) and E.S.I. (Vienna). Typeset by AMS-TEX 1 2 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO functions c. By referring to the twistor bration t : Z ?! M 4n one can say equivalently that J is a section of the twistor space Z , J : M ?! Z , see [S3, pag. 130] and [AMP]. In this work we will consider the stronger notion of a quaternion-Kahler manifold (M 4n ; g; Q) . This means that g is a Riemannian metric on M which is Q-Hermitian (i.e. any endomorphism of Q is g-skew-symmetric) and, moreover, that the LeviCivita connection of g preserves Q. This is equivalent to the statement that the holonomy group of g is contained in Sp(n) Sp(1). Notice that every oriented Riemannian 4-manifold satises the above conditions. For this reason we will assume that the quaternionic dimension n 2 and we will systematically treat the 4-dimensional case in separate statements. For the basic properties of quaternion-Kahler manifolds we will refer to [S1] and [S2]. Let us just recall here that any quaternion-Kahler manifold (M 4n ; g; Q) with n 2 is automatically Einstein. When the scalar curvature K of g is positive a complete quaternion-Kahler manifold is always compact and simply connected; the only known examples are the symmetric spaces of Wolf [W]. When K = 0 the bundle Q is locally parallelizable so that the universal covering of M is a hyperKahler manifold. We will assume that (M 4n ; g; Q) , n 2, is a quaternion-Kahler manifold and discuss various properties of a compatible almost complex structure J , under several assumptions (see denitions 2.2) both from local and global point of view. Our main results about existence of local compatible almost complex structures can be summarized as follows. Let (M 4n ; g; Q) be a quaternion-Kahler manifold with reduced scalar curvature and J an almost complex structure which is compatible with the quaternionic structure Q on M 4n . Let F = g J , = ?F J be the Kahler form and , respectively, the Lee form of J . Then: 1) if J is locally conformally symplectic (that is dF = ^ F for some 1-form ) then it is parallel and = 0. 2) if J is cosymplectic (that is = 0) then 0 and = 0 if and only if J is parallel. 3) if J is integrable then the dierential d of the Lee form is Q-Hermitian and harmonic. 4) if J is conformally balanced (i.e. J is integrable and is exact) but not symplectic then there exists a non-zero Killing vector eld associated to J . The proofs of the statements will be given through Theorems 2.4, 3.1, 3.2 and Propositions 4.6, 4.11. A further local result concerns self-dual forms on M 4n , that is sections of the 3dimensional subbundle 2+ = g Q of 2 which corresponds to Q by means of the metric g. In Theorem 3.1 we prove that any closed self-dual 2-form ! is parallel . Our main global results can be stated as follows. ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 3 Let (M 4n ; g; Q) be compact, then 1) if > 0 and (M 4n ; g) is dierent from the Grassmannian G2(C m ) of 2-planes in C m , then there exists no compatible almost complex structure J . 2) if the rst Chern class c1(J ) c1(TJ(1;0)M ) = 0 for some compatible almost complex structure J , then = 0. 3) if 6= 0 there exists no compatible complex structure J . 4) if = 0 a compatible complex structure J is parallel. The last two results have been proved in [P2] by twistor methods. The proof of these statements will be given throughout Proposition 3.7 and its Corollary 3.8, Theorem 4.3, Proposition 4.6 and Theorem 3.10 where we establish the relationship between the rst Chern classes c1(J ) of (TM; J ) and the rst Chern class c1(J ?) of the complex line bundle J ? orthogonal to J in Q. In paragraphs 5 and 6 we compute the Laplacian and the (real and quaternionic) Weyl tensors acting on self-dual 2-forms. This part is very much related to work of Gauduchon [G1]. 2. Preliminary results Notations and denitions. Let (M n ; g; Q) be a quaternion-Kahler manifold, with n 2. We recall that the Riemannian metric g is Hermitian with respect 4 to Q, that is g(JX; Y ) + g(X; JY ) = 0 for all X; Y 2 TM 4n and any J 2 Q, and also it is Einstein, that is Ric(g) = 4Kn g where K is the scalar curvature. We dene the bundle 2+ ?! M of self-dual 2-forms as the 3-dimensional bundle associated to Q by the metric g, that is 2+jx = (g Q)x ; x 2 M . If H = (J) is an admissible basis of Q the Kahler forms F of the almost Hermitian structures (J; g) F = g J = g(J; ) ( = 1; 2; 3) represent a (local) frame for 2+. The notation is consistent with the one used for an oriented 4-dimensional manifold (see also [AMP, section 5]). The Levi-Civita connection r rg of g is a quaternionic connection, that is for any admissible basis H = (J): rJ = ! J ? ! J (2:1:1) where the ! , = 1; 2; 3 are 1-forms and (; ; ) is a cyclic permutation of (1,2,3). Hence dF = ! ^ F ? ! ^ F (2:1:2) 4 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO since rF = g(rJ; ). We recall that the following fundamental identities hold: d! + ! ^ ! = ?F ( = 1; 2; 3) (2:2) where = K=4n(n + 2) is the reduced scalar curvature of g. Moreover F = 21n Tr[J R(X; Y )] ( = 1; 2; 3) (2:3) where R is the curvature tensor of r (see [AM2]). We recall also that on M 4n there is a globally dened fundamental 4-form , = X F ^ F which is parallel and non degenerate, see [Be, pag.419]. Remark 2.1. For n = 1, an oriented Riemannian 4-manifold (M 4 ; g) always satises (2.1.1), (2.1.2). The fundamental identities (2.2) hold if g is anti-self-dual and Einstein. M Now we dene dierent types of compatible almost complex structures J 2 ?(Q) on a quaternion-Kahler manifold (M 4n ; g; Q) . Denitions 2.2. Let J be an almost complex structure on the Riemannian manifold (M 4n ; g). Then J is said to be: (1) parallel if rJ = 0 where r is the Levi-Civita connection of g. Moreover let assume that g is J -Hermitian. Then J is called to be (2) symplectic if the Kahler form F = g J is closed, dF = 0; (3) locally conformally symplectic if dF = ^ F for some 1-form ; (4) cosymplectic if F is coclosed, F = 0 . Remark 2.3. If n = 1 dF = ^ F always holds (see also (2.5) below); for n 2 the same equality does not hold in general and it implies that is closed. M Parallel complex structures on a quaternion-Kahler manifold. The following theorem describes a parallel complex structure on a quaternion-Kahler manifold. Theorem 2.4. Let (M 4n ; g; Q) be a simply connected complete irreducible quaternion-Kahler manifold with reduced scalar curvature . Let J be a parallel complex structure on M 4n. P P (1) If J is compatible then = 0, and J = 3=1 a J where a = const; a2 = 1 and H = (J) is a parallel hypercomplex structure, (2) otherwise (M 4n ; g) is isometric to the quaternionic symmetric space G2 (C m ) or to its dual non compact symmetric space, and J is the unique parallel complex structure. ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 5 Proof. Remark that the parallel complex structures on M 4n correspond to the complex structures in the tangent space of a point x which are invariant under the holonomy group Hx. If = 0 the manifold (M 4n ; g; Q) has the holonomy group Sp(n) and the rst statement holds. If 6= 0 then either the holonomy group Hx = Sp(n) Sp(1) or the manifold is locally symmetric (see [Be]): the rst case cannot occur since there is no Sp(n) Sp(1) -invariant complex structure , and investigation of all holonomy groups of symmetric quaternion-Kahler manifolds gives the second statement. Some local formulas. Now we derive some formulas which will be useful for studying cosymplectic and integrable almost complex structures. For a tensor T = (Tijk ), where Tijk are the components of T with respect to an orthonormal basis, we denote kT k2 = X i;j; ;k (Tijk )2 (In the terminology of Gauduchon k k is the tensorial norm j jT ; [G1, p. 5]). For example, kFk2 = 4n ( = 1; 2; 3) From now on we will assume that J is a compatible almost complex structure on (M 4n ; g; Q) with Kahler form F , H = (J = J1; J2; J3) is a local admissible basis and ! are the corresponding 1-forms. We dene the Lee form of J by = ?(F ) J (2:4) where is the codierential. It is known that d(F 2n?1) = ^ F 2n?1 (2:5) Proposition 2.5. For a compatible almost complex structure J on a quaternionKahler manifold the following formulas hold: krF k2 = 4n(k!2k2 + k!3k2 ) (2:6) kdF k2 = 12(n ? 1)(k!2 k2 + k!3 k2) + 6k!2 J2 + !3 J3k2 (2:7) 1 kdF k2 ? 1 krF k2 = ?k! J ? ! J k2 2 2 3 3 6 2 = ! 2 J 2 + ! 3 J3 Moreover, by denoting h; i = g(; ) to simplify, = 4n ? 2h!2 J2; !3 J3i (2:8) (2:9) (2:10) 6 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO and 1 kdF k2 ? 1 krF k2 + kk2 + 2 = 8n 6 2 (2:11) Remarks 2.6. 1) The inequality kdF k2 3krF k2 which comes from (2.8) was previously established in [G1]. The equality holds if and only if J is integrable (see also Lemma 4.1 below). 2) For n = 1, the (2.7) still holds: kdF k2 = 6k!2 J2 + !3 J3k2. 3) For n = 1 the (2.9) still holds since dF1 = ^ F1 and, as it is easy to see, the identity ? ^ F1 + !3 ^ F2 ? !2 ^ F3 = 0 holds if and only if ? J1 + !3 J2 ? !2 J3 = 0. The (2.10), (2.11) hold under the hypothesis that the metric g is anti-self-dual and Einstein. M Proof. Since rF = g(rJ1; ) the (2.1.1) gives rF = !3 J2 ? !2 J3. We compute krF k2, by using an orthonormal frame (E1 ; ; E4n). By denoting h; i = g(; ), we have X krF k2 = [(rEi F )(Ej ; Ek )]2 = = i;j;k X i;j;k X i;j;k [!3(Ei )hJ2Ej ; Ek i ? !2(Ei )hJ3 Ej ; Ek i]2 [!32(Ei)hJ2 Ej ; Ek i2 + !22(Ei )hJ3Ej ; Ek i2 ? 2!2(Ei )!3(Ei )hJ2 Ej ; Ek ihJ3 Ej ; Ek i] = 4n(k!2k2 + k!3k2 ) P hJ E ; E i2 = kJ E k2 = kE k2 and For last identity we used j j P hJ E ; E ihJ E ; E i = hkJ E2 ;jJ Ek i = 0. 2Hence the (2.6) is proved. 2 j k 3 j k 2 j 3 j k Now we prove (2.7). We have kdF k2 = = = X i;j;k X i;j;k X i;j;k dF (Ei; Ej ; Ek )2 [(rEi F )(Ej ; Ek ) + (rEk F )(Ei ; Ej ) + (rEj F )(Ek ; Ei)]2 [!3(Ei)hJ2 Ej ; Ek i + !3(Ek )hJ2Ei ; Ej i + !3(Ej )hJ2 Ek ; Ei i ? !2(Ei )hJ3Ej ; Ek i ? !2(Ek )hJ3Ei ; Ej i ? !2(Ej )hJ3 Ek ; Ej i]2 By computing the square of the expression in brackets, after a long but straightforward calculation, we get kdF k2 = (12n ? 6)(k!2 k2 + k!3k2) + 12h!2 J2; !3 J3i ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 7 This implies (2.7). The (2.8) follows directly from (2.6), (2.7). For a vector eld X and an orthonormal frame Ei; i = 1; : : : ; 4n, we have (F )(JX ) = ? =? X X i h(rEi J1)Ei ; J1X i = ? !3 (Ei)hJ2 Ei ; J1X i + i that is X i X i h!3(Ei )J2 Ei ? !2(Ei )J3Ei ; J1X i !2 (Ei)hJ3 Ei ; J1X i = ?!3(J3X ) ? !2 (J2X ) F = !3 J2 ? !2 J3 Hence (2.9) follows. Now we compute . We have rX = (rX !2) J2 + (rX !3 ) J3 + !2 (!1 (X )J3 ? !3 (X )J1 ) + !3 (!2 (X )J1 ? !1(X )J2 ) Hence = ? X i [(rEi !2 )(J2Ei ) + (rEi !3 )(J3Ei ) ? !1(Ei )(!2 J2 + !3 J3)(J1 Ei) + 2!2 (Ei)!3 (J1Ei )] P (2:12) P since i ?!3(Ei )!2(J1 Ei) = i !3 (J1Ei)!2 (Ei ). Now let take into account that (rX !2 )Y ? (rY !2 )X = d!2(X; Y ) and hence, putting X = Ei and Y = J2Ei and summing up over i, X i that is [(rEi!2)(J2 Ei ) ? (rJ2Ei !2)(Ei )] = 2 Analogously 2 X X i i (rEi !2 )(J2Ei ) = X X i i (rEi !3 )(J3Ei ) = X i d!2(Ei ; J2Ei) d!2(Ei ; J2Ei) d!3(Ei ; J3Ei) By the second of integrability conditions (2.2) one has X i that is X i d!2 (Ei; J2 Ei) = d!2(Ei ; J2Ei ) = X i X i [?(!3 ^ !1 )(Ei ; J2Ei) + hEi ; J22Eii] [?!3(Ei )!1 (J2Ei) + !1(Ei )!3(J2 Ei)] ? 4n 8 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO Analogously, one has X i d!3(Ei ; J3Ei ) = Hence 2 = ? X i X i [?!1(Ei )!2 (J3Ei) + !2(Ei )!1(J3 Ei)] ? 4n [?!3(Ei )!1(J2 Ei) + !1 (Ei )!3(J2 Ei) ? !1(Ei )!2 (J3Ei ) + !2(Ei )!1 (J3Ei) ? 8n ? 2!1(Ei )(!2 J2 + !3 J3)(J1 Ei) + 4!2 (Ei)!3 (J1Ei )] and (2.10) follows immediately. Since by (2.8), (2.9) one has 61 kdF k2 ? 21 krF k2 = ?k!2 J2 ? !3 J3k2 and kk2 = k!2 J2 + !3 J3k2 it follows 1 kdF k2 ? 1 krF k2 + kk2 = 4h! J ; ! J i 2 2 3 3 6 2 and, by (2.10), we get (2.11). 3. Compatible almost complex and almost hypercomplex structures (Conformally) symplectic structures. Theorem 3.1. Let (M n ; g; Q) be a quaternion-Kahler manifold with n 2. Then, 4 (1) every compatible locally conformally symplectic almost complex structure J is parallel. (2) every closed self-dual 2-form ! is parallel In both cases = 0. Proof. If J is a compatible locally conformally symplectic almost complex structure then dF = ^ F . Choose an admissible basis H = (J = J1; J2; J3) of Q and denote by F, = 1; 2; 3, the corresponding Kahler forms. Then by (2.1.2) dF = !3 ^ F2 ? !2 ^ F3 , hence ^ F ? !3 ^ F2 + !2 ^ F3 = 0. By [ABM, Lemma 1 p.125] it follows that = !2 = !3 = 0 so that J is parallel and = 0. This proves the rst statement. To prove the second statement we put M = fp 2 M j! 6= 0g. On M we can write ! = fF where f is a never zero smooth function and F is the Kahler form of a compatible almost complex structure . Without loss of generality, we may assume that f > 0. Then we have 0 = d! = df ^ F + fdF and dF = ?d log f ^ F . Now the rst statement implies that f is a constant and F and ! are parallel on M and, hence, on M . ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 9 Cosymplectic almost complex structures. Theorem 3.2. Let J be a compatible cosymplectic almost complex structure on the quaternion-Kahler manifold (M n ; g; Q) , that is = 0. Then 0 and k! k = k! k = const = ?2n . Moreover = 0 if and only if J is parallel. Proof. From (2.9) = 0 implies ! J = ?! J and in particular k! k = k! k. 4 3 2 2 Then, by (2.10) we have 3 3 2 2 2 2 3 0 = 4n + 2k!2k2 and conclusion is immediate By combining the expression (2.9) of given in Proposition 2.5 with a result of [AMP, proposition 4.1] it is easy to deduce the following Proposition which for n = 1 was proved by S. Salamon in [S2, Prop. 1]. Proposition 3.3. Let (M 4n ; g; Q) be a quaternion-Kahler manifold and H = (J ) an admissible basis of Q: (1) If J1; J2 are cosymplectic then J3 is integrable; (2) If J1 is cosymplectic and J2 is integrable then J3 is cosymplectic; (3) If J1; J2 are integrable then J3 is integrable. Proof. We prove (1) as follows. Assume J1; J2 cosymplectic: then by (2.9) !2 J2 + !3 J3 = !3 J3 + !1 J1 = 0 and hence !1 J1 = !2 J2 which is equivalent to the integrability of J3 by proposition 4.1 of [AMP] (see also Lemma 4.1 below). To prove (2) we remark that by hypothesis one has !2 J2 + !3 J3 = 0 and !1 J1 = !3 J3. It implies !1 J1 + !2 J2 = 0, that is J3 is cosymplectic. (3) was proved in [AMP, Remark 2.6]. Now we associate to an almost complex structure J compatible with Q a globally dened closed 2-form J by the formula J = d!1 (3.1) for any admissible basis H = (J1 = J; J2; J3). If H 0 = (J10 = J; J20 ; J30 ) is another such admissible basis and (!10 ; !20 ; !30 ) are corresponding 1-forms, then J20 = cos'J2 + sen'J3 ; J30 = ?sen'J2 + cos'J3 which implies !10 = !1 + d': This shows that J is globally dened and depends only on J . Proposition 3.4. Let J be a compatible almost complex structure. Assume that = 0, that is J is cosymplectic. If < 0, then the 2-form J is a symplectic 10 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO form which tames J , that is g0 := J (; J ) is a Riemannian metric (which is quasiKahlerian with respect to J ). Proof. Since by hypothesis !3 = !2 J1, for = 1 the fundamental identity (2.2) gives d!1 = ?!2 ^ (!2 J ) ? F hence d!1(; J ) = !2 !2 + (!2 J ) (!2 J ) ? g and the conclusion follows immediately. Compatible almost complex and almost hypercomplex structures on a compact quaternion-Kahler manifold. We will now relate the 2-form J to the complex line bundle J ? orthogonal to J in Q. Proposition 3.5. Let (M 4n ; g; Q) be a quaternion-Kahler manifold and J a compatible almost complex structure. Then the linear connection r0 on the bundle Q which is dened by r0X = rX ? 21 J rX J rX ? 12 [!2(X )J2 + !3(X )J3 ] is Riemannian and preserves J (r0 is the rst canonical connection of the almost Hermitian structure (g; J ), [G3, pag. 31]). The 2-form J is the curvature form of the Riemannian connection induced by r0 on the complex line bundle J ? orthogonal to J in Q. When M 4n is compact the 2-form 21 J represents the Chern class c1 of J ? and the following conditions are equivalent: (1) The cohomology class c1(J ?) = [ 21 J ] 2 H 2 (M; Z) vanishes. (2) The rst Chern class c1(J ) c1(TM ) of the tangent bundle TM with respect to the almost complex structure J vanishes. (3) There exists a global admissible basis H = (J1 = J; J2; J3) of Q on M 4n. Proof. Let X; Y be any two vector elds on M . Since (rJ )J + J (rJ ) = 0, it is easy to see that g((rX J )Y; JZ ) + g(JY; (rX J )Z ) = 0 Hence r0 is Riemannian. Moreover, (r0X J )(Y ) = (rX J )(Y ) ? 21 (J rX J )(JY ) + 21 J 2(rX J )(Y ) = (rX J )(Y ) + 21 (J 2rX J )(Y ) + 21 (J 2 rX J )(Y ) = 0 that is r0 leaves J invariant. Hence we can think r0 as a connection for the 2-plane bundle J ? orthogonal complement to the line-bundle RJ in Q. For any admissible ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 11 basis H = (J1 = J; J2; J3) we have a local orthonormal frame (J2; J3) for J ? and the following identities hold r0X J2 = !1 (X )J3 ; r0X J3 = ?!1(X )J2 (3.2) This shows that the connection r0 preserves the complex structure I on J ? , induced by J : I (J2) = J3 ; I (J3 ) = ?J2: Then the matrix 1-form of the connection r0 with respect to the local frame (J2 ; J3) and matrix 2-form of the curvature are given by ! = !01 ?0!1 1 ; = d!1 ?d! 0 : 0 Hence, the 2-form J = 21 d!1 is the Chern form of r0, as to be proved. To prove that conditions (1) and (3) are equivalent we observe that, from the classication of S 1-bundles over M 4n by elements of H 2 (M; Z), one has c1(J ?) = 0 if and only if J ? is trivial or, equivalently, if and only if there exists an admissible basis H = (J1 = J; J2; J3 ) of Q = RJ J ? globally dened on M 4n . The equivalence of (1) and (2) will follows immediately from (3.6) of Theorem 3.10 below. Before giving an application of the last result we recall the following result which was proved in [AM2]. Theorem 3.6 ([AM2]). Let (M 4n ; g; Q) , n > 1, be a quaternion-Kahler manifold with the reduced scalar curvature . Assume that there exists a (globally dened) almost hypercomplex structure H = (J) which generates Q. Then either (1) = 0, that is (M 4n ; g) is a locally hyper-Kahler manifold. or P (2) the fundamental 4-form = F ^ F is exact. For a compact M 4n only (1) is possible. Proof. By identities (2.2) one has 2 = X = d( (d! + ! ^ ! ) ^ (d! + ! ^ ! ) X ! ^ d! + 2!1 ^ !2 ^ !3) wherePthe 1-forms ! are globally dened. Hence, if 6= 0 the fundamental 4-form = F ^ F is exact. The last case cannot occur if M 4n is compact, since the form is parallel and, hence, harmonic. 12 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO Proposition 3.7. Let (M n ; g; Q) be a compact quaternion-Kahler manifold with a compatible almost complex structure J . If the cohomology class [ J ] = 0 2 4 1 2 H (M; Z) then = 0. 2 Proof. If the Chern class c1(J ?) = [ 21 J ] vanishes then, by proposition 3.5, J ? admits a nowhere vanishing section and Q admits a global admissible basis - i.e. there exists a global compatible almost hypercomplex structure. Then we can apply theorem 3.6. Theorem 3.8. Let (M 4n ; g; Q) be a compact quaternion-Kahler manifold with > 0. Assume that it is not homothetic to the standard Grassmannian G2 (C m ). Then there is no compatible almost complex structure J on (M 4n ; g; Q) . Proof. By a result of C. LeBrun and S. Salamon [LS] we know that a compact quaternion-Kahler manifold (M 4n ; g; Q) with > 0, dierent from the complex Grassmannian G2 (C m ), has second Betti number b2(M ) = 0 . Since furthermore M must be simply connected , we conclude that also H 2 (M; Z) = 0 [LS, proof of 0.2 p. 123]. The result follows now from Proposition 3.7. Remark 3.9. At present we do not know if there exists any compatible almost complex structure J on G2 (C m ). M Now we prove a fundamental identity between the Chern class c1(J ?) and the rst Chern class c1(J ) of the tangent bundle TM 4n endowed with the compatible almost complex structure J . We rst remark that the curvature tensors R, R0 of the connections r, r0 are related by R0XY = RXY ? 21 (!2 ^ !3 )(X; Y )J ? 21 [(d!2 + !3 ^ !1)(X; Y )J2 + (d!3 + !1 ^ !2)(X; Y )J3 ] that is, by fundamental identities (2.2), X R0XY = RXY + 2 F(X; Y )J + 12 d!1(X; Y )J (3.3) Let now recall that the rst Chern form 1 of the almost complex structure J (on TM ) with respect to the connection r0 is given by (see for ex. [GBNV]) 21(X; Y ) = ? 21 Trace(R0XY J ) (3.4) Theorem 3.10. The Chern 2-forms for J ?, (TM; J ) respectively are related by 1 = n 21 1 (3.5) ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS Hence 13 c1(J ) = nc1(J ?) (3.6) X 0 hRXY Es; JEsi 21(X; Y ) = 12 (3.7) Proof. By (3.4) we have s=1;::;4n where X; Y 2 TpM; p 2 M 4n and fE1; E2; : : : ; E4ng is an orthonormal basis of Tp M . By using (3.3) we have X hR0X;Y Es; JEsi = s=1;::;4n + 2 X X =1;2;3 s=1;::;4n hR0X;Y Es; JEsi = s=1;::;4n hRX;Y Es; JEsi s=1;::;4n F(X; Y )hJ Es; JEsi + 12 that is X X X X s=1;::;4n d!1(X; Y )(JEs ; JEs) hRXY Es; JEsi +2nF (X; Y )+2nd!1(X; Y ) (3.8) s=1;::;4n On the other hand we get the following. By curvature identity, X hRX;Y EsJEsi = ? s=1;::;4n X hRJEs ;X Es; Y i ? s=1;::;4n X hRY;JEs Es; X i s=1;::;4n Moreover, by using the known identity for the curvature tensor of a quaternionKahler manifold (see for ex [Be, pag. 403]; also [M, pag. 423]), hRJZ T U; V i = ?hRZJT U; V i + (?hJ3U; V ihJ2 Z; T i + hJ2U; V ihJ3 Z; T i) where U; V; Z; T 2 TpM 4n , we get X hRX;Y EsJEsi s=1;::;4n = + X hREs JX Es; Y i + (hJ3Es; Y ihJ2 Es; X i ? hJ2Es; Y ihJ3 Es; X i) s=1;::;4n X hRJY Es Es; X i) ? (h?J3 Es; X ihJ2 Y; Esi + hJ2Es; X ihJ3 Y; Esi) s=1;::;4n and, by taking into account the identity hRUV Z; T i = hRJUJV JZ; JT i 14 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO (see [M, pag. 423]), it results X hRXY Es; JEsi s=1;::;4n = = X hREs ;JX Es; Y i + 2 hY; JX i + s=1;::;4n X s=1;::;4n s=1;::;4n X hREs JX Es; Y i ? Hence X hRJY Es Es; X i) ? 2 hX; JY i s=1;::;4n hRY;JEs JEs; JX i + 4 hJX; Y i hRX;Y Es; JEsi = ?2Ric(Y; JX ) + 4 hJX; Y i s=1;::;4n that is X = ?2(n + 2) hY; JX i + 4 hJX; Y i X hRX;Y Es; JEsi = ?2n hJX; Y i s=1;::;4n (3.9) Hence, by substitution in (3.8), we get X hR0X;Y Es; JEsi = ?2nF (X; Y ) + 2nF (X; Y ) + 2nd!1 (X; Y ) s=1;::;4n = 2nd!1(X; Y ) and, nally, that is (3.5) and (3.6) hold. 21 = nd!1 4. Compatible complex structures A quaternion-Kahler manifold M 4n locally admits many compatible almost complex structures which are integrable, see [S3, pag. 130] and [AMP]. Any such structure J denes a local section J : U M 4n ?! Z , U 3 x 7! Jx 2 Z of the twistor bration t : Z ?! M 4n such that the image X = J (U ) is a complex submanifold of Z . Conversely, any complex submanifold X Z such that the projection t : X ?! M is a dieomorphism denes a compatible complex structure on U = t(X ) M 4n . Now we rene some of the formulas of x2 in the special case when J = J1 is an integrable almost complex structure on the quaternion-Kahler manifold (M 4n ; g; Q) . Let us rst recall the following result. Lemma 4.1 ([AMP]). A compatible almost complex structure J on (M 4n ; g; Q) is integrable if and only if !2 J 2 = !3 J 3 for an admissible basis (J = J1; J2 ; J3) with connection 1-forms (!1 ; !2; !3 ). ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 15 Proposition 4.2. Assume that the compatible almost complex structure J on M with Kahler form F and Lee form is integrable. Then one has krF k2 = 2nkk2 ; kdF k2 = 6nkk2 ; kF k2 = kk2 and (4.1) kk2 + 2 = 8n (4.2) 4 = ? 21 d(kk2 ) + d (4.3) Proof. The three identities (4.1) and (4.2) follow from (2.5), (2.7), (2.9) and from (2.10) respectively, by considering that for J = J1 one has = 2!2 J2 = 2!3 J3. The (4.3) is obtained by dierentiating (4.2). As a consequence of our previous results we deduce the following theorem obtained in [P2] by twistor methods. Theorem 4.3. On a compact quaternion-Kahler manifold (M 4n ; g; Q) every compatible complex structure is necessarily parallel. Proof. If the scalar curvature is non-positive, by integrating (4.2) on M 4n we get = = 0 and rJ = 0 by rst of (4.1). Otherwise > 0, by Corollary 3.3 the existence of a compatible almost complex structure J implies that (M 4n ; g) is homothetic to the Grassmannian G2 (C m ) equipped with the symmetric metric. By a strong result which was proved in [BGMR, theorem 1.7], a complex structure which is Hermitian with respect to a symmetric metric is necessarily parallel. Therefore J is a compatible parallel complex structure on G2(C m ). But this is impossible by theorem 2.4. Remark 4.4. If < 0 but M is non-compact the conclusion is not true: it is sucient to think of the hyperbolic quaternionic space H Hn [P2]. M The following two Propositions hold on any (not necessarily complete) quaternionKahler manifold. Proposition 4.5. Let (M 4n ; g; Q) be a quaternion-Kahler manifold with a compatible complex structure J and Lee form . Then the following identities hold for any vector elds X; Y on M 4n : (rX )(Y ) + (rJ2Y )(J2 X ) = 12 (JX )(JY ) + 12 (J3 X )(J3 Y ) ? 2g(X; Y ) (4:4) (rX )(Y ) + (rJ3Y )(J3 X ) = 21 (JX )(JY ) + 12 (J2 X )(J2 Y ) ? 2g(X; Y ) (4:5) (rX )(Y ) + 21 (X )(Y ) = (rJX )(JY ) + 12 (JX )(JY ) (4:6) 16 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO Proof. From lemma 4.1 we have = 2!2 J2 ; !2 = ? 12 J2 ; !3 = ? 21 J3. For = 2 the identity (2.2) gives (rX !2 )(Y ) ? (rY !2)(X ) + !3 (X )!1 (Y ) ? !3(Y )!1 (X ) = g(X; J2Y ) that is ? 21 (rX )(J2 Y ) ? 21 ((rX J2)Y ) + 12 (rY )(J2 X ) + 21 ((rY J2)X ) ? 21 (J3 X )!1 (Y ) + 12 (J3 Y )!1 (X ) = g(X; J2Y ) that is ? 21 (rX )(J2 Y ) ? 12 (!1 (X )J3 Y ? !3(X )JY ) + 21 (rY )(J2 X ) + 21 (!1 (Y )J3X ? !3 (Y )JX ) ? 21 (J3 X )!1 (Y ) + 12 (J3 Y )!1 (X ) = g(X; J2Y ) that is, after suitable cancellations, ? 12 (rX )(J2 Y ) + 12 (rY )(J2 X ) + 12 !3 (X )(JY ) ? 12 !3 (Y )(JX ) = g(X; J2Y ) By changing Y with J2Y one nds 1 (r )(Y ) + 1 (r )(J X ) + 1 ! (X )(J Y ) ? 1 ! (J Y )(JX ) = ?g(X; Y ) 3 2 X 2 J2 Y 2 2 3 2 3 2 and (4.4) follows by expressing !3 by . For = 3 we get (rX !3 )(Y ) ? (rY !3)(X ) + !1 (X )!2 (Y ) ? !1(Y )!2 (X ) = g(X; J3Y ) that is equivalent to ? (rX )(J3 Y ) ? (!2 (X )JY ? !1 (X )J2 Y ) + (rY )(J3 X ) ? (!2 (Y )JX ? !1(Y )J2X ) ? !1 (X )(J2 Y ) + !1(Y )(J2 X ) = 2g(X; J3Y ) and, after cancellations, 2g(X; J3Y ) = ?(rX )(J3 Y ) + (rY )(J3 X ) + 21 (J2 X )(JY ) ? 21 (J2 Y )(JY ) After substitution of Y with J3Y we get (4.5). The (4.6) follows by substraction of (4.5) to (4.4) and substitution of X; Y with J2X; J2Y . For any point p 2 M 4n let be the orthogonal projection from the space of bilinear forms dened on TpM 4n onto the subspace of Q-Hermitian bilinear forms, that is for any ! 2 Bilp one has X ! = 41 [! + !( ; )] ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 17 Proposition 4.6. For a compatible complex structure J with associated Lee form on a quaternion-Kahler manifold (M n ; g; Q) one has: 1) The bilinear form r + is J -Hermitian. 2) The Q-Hermitian part r = r and the skew-Q-Hermitian part r? = (1 ? )r of the covariant derivative r are given by (4:7) r = 21 ( ) ? g + 21 d 4 + 1 2 + r? = (r)s ? 21 ( ) + g where (r)s is the symmetric part of r. (4:8) 3) The 2-form d is Q-Hermitian and harmonic. Proof. Statement 1) is equivalent to the identity (4.6). To prove identity (4.7) , we calculate X 4r+(X; Y ) 4r(X; Y ) = (rX )(Y ) + (rJX )(J Y ) using identities (4.4), (4.5), (4.6) to epress (rJ X )(J Y ), = 1; 2; 3. The (4.8) and the rst statement of 3) follow from (4.7). To prove the last statement in 3) we can argue as in [MS] (where Q-Hermitian 2-forms are called self-dual and it was proved that such a 2-form is closed if and only if it is harmonic): for a Q-Hermitian 2-form ! one has ?! = c2! ^ n?1 where c2 = ?1=(2n ? 1)! (see [GP]). Hence d = ? ? d ? d = ?c2 ? d(d ^ n?1) = 0 and 4d = 0. Remark 4.7. (1) When J is a compatible complex structure more can be said about the 2-form J considered in the previous section x2: J = ? 1 ( J2) ^ ( J3) ? F (4.9) 4 and (4.10) J (; J ) = ? 41 [( J2) ( J2) + ( J3) ( J3)] ? g Hence, if > 0 then ? J tames J , that is g0 := 41 [( J2) ( J2) + ( J3) ( J3)] + g (4.11) is a Riemannian metric which is almost-Kahler with respect to J and in fact, by a classical result, g0 is Kahler and J is parallel with respect to rg . (2) On a non-compact complete quaternion-Kahler manifold (M 4n ; g; Q) , a complex structure compatible with Q may exist. For example, any quaternionKahler manifold (M 4n ; g; Q) which admits a simply transitive solvable group G of isometries and dierent from quaternion hyperbolic space has a Ginvariant compatible complex structure, see [A]. 0 M 18 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO Conformally balanced complex structures, Dirac and Twistor operators. Referring to the fundamental work of Salamon [S1], we would now like to establish a relation between compatible almost complex structures on a quaternion-Kahler manifold (M 4n ; g; Q) and two rst order dierential operators acting on sections of Q which we will call Dirac and Twistor opertors by analogy with the four-dimensional case; as usual they are dened by rst taking covariant derivative with respect to the Levi-Civita connection and then projecting onto the two irreducible Sp(n) Sp(1)submodules of the target space. Our notations are taken from the four-dimensional case and are dierent from those used by Salamon who denotes the complexied decomposition (4.12) by S 2H E H = (E H ) (E S 3H ); what we called Twistor operator is denoted by Salamon by D and our Dirac operator by . Salamon also introduced complex analytic methods in quaternionic geometry by dening the twistor bration t : Z ! M . The total space Z is a (2n +1)-dimensional complex manifold equipped with a holomorphic contact structure and the bers t?1 (p) are holomorphically imbedded C P1's The general principle of Penrose is that holomorphic properties of Z reect geometric properties of (M 4n ; g; Q) . With respect to this, Salamon shows that the kernel of the twistor operator is isomorphic to the real part of the space of holomorphic sections of the contact line bundle O(2) over Z [S1, Lemma 6.4]. He also proves that in the compact non-Ricci-at case the Dirac operator maps the kernel of the twistor operator isomorphically onto the space of Killing vector elds of (M; g). In what follows we will let V denote the tangent bundle TM . Then using notations of [AM2] we have: where (V Q)0 V Q = V(1) = fA = X( J ) J ; 2 V g V(1) =V (4.12) and (V Q)0 are the trace-free tensors. P More precisely, given a tensor T 2 V Q we can write T = J and set X X = ? 31 ( J) and T 0 = ( J) J 2 V(1) Then the above direct sum decomposition is T = T 0 + (T ? T 0 ). Finally if p1 and p2 denote the projections onto the irreducible components we will call ) D : ?(Q) ! ?(V(1) the Dirac operator given by covariant dierentiation r followed by p1 and D : ?(Q) ! ?((V Q)0 ) ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 19 the Twistor operator p2 r. A non zero section of Q , restricted to the open submanifold M^ = f 6= 0g can always be written as = fJ for some smooth function f and compatible almost complex structure J on M . To compute the covariant derivative of we choose a local admissible basis (J) of Q such that J = J1. We have r = rfJ = f (!3 J2 ? !2 J3) + df J and 3D = [f (!3 J3 + !2 J2) + df ] J1 + [f (!3 ? !2 J1) ? df J3] J2 + [f (?!3 J1 ? !2 ) + df J2] J3 X = ? [f + df ] J J J Which can also be rewritten as 3D = ? X (fF ) J J = ? X J J (4.13) (4:14) so that D can be identied with the codierential : ?(Q) ?! T M . We now compute the twistor operator: = f!3 J2 ? f!2 J3 + df J ? D D (4.15) We will now use the above facts to prove the following result which holds whether M is compact or not, and generalizes [Theorem 2.1, P1] to higher dimensions. Denition 4.8. A Hermitian metric (g; J ) is said to be balanced if (J is integrable and) the Lee form = 0; conformally balanced if is exact. For the sake of simplicity, when g is xed we also say that J is balanced or, respectively, conformally balanced. Of course when n = 1 one has = 0 if and only if F is symplectic. For n > 1, by rst of (4.1) it follows immediately that a compatible complex structure J on the quaternion-Kahler manifold (M 4n ; g; Q) is balanced if and only if it is parallel. Theorem 4.9. Let (M 4n ; g; Q) be quaternion-Kahler if n 2 or anti-self-dual if n = 1 with twistor space Z . Suppose J is a compatible almost complex structure and let X denote the image of the sections J : M ! Z in the twistor space. Then, X is associated to the holomorphic contact line bundle O(2) if and only if (g; J ) is conformally balanced. Proof. The smooth submanifold X given by the (disjoint) union of J (M ) and ?J (M ) is a divisor of O(2) if and only if there is a holomorphic section s 2 H 0(Z; O(2)) vanishing exactly on X . By Salamon correspondence [S1, Lemma 6.4] this is equivalent 20 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO (s) = 0 2 to having a smooth section (s) 2 ?(Q) satisfying the Twistor equation D ?(V Q)0 . Now, Salamon correspondence tells us that (s) = fJ where J is the given almost complex structure and f is a smooth real function on M which never vanishes because X contains no twistor bers, for example. By equations (4.13) and = 0 is equivalent to !2 J2 = !3 J3 - i.e. J is an (4.15) we easily see that DfJ integrable complex structure - and furthermore d log f 2 = !2 J2 + !3 J3 - where = !2 J2 + !3 J3 is the Lee form of (g; J ). It is now easy to check that this is equivalent to the fact that the Hermitian metric (jf jg; J ) is conformally balanced. Remark 4.10. We can now give a more self-contained proof of the statement that there is no compatible complex structure on a complete quaternion-Kahler manifold (M 4n ; g; Q) of positive scalar curvature; see theorem 4.3. Recall that M is necessarily compact and simply connected [S1]. By contradiction, let be the Lee form of a compatible complex structure J then d is harmonic by proposition 4.6 3), and since M is compact d = 0. In fact, since M is simply connected we conclude that the Lee form is exact: = dh for some smooth function h which is globally dened on M . In this situation (4:2) becomes kdhk2 + 2h = 8n showing that the function h cannot have a minimum; this is a contradiction because M is compact. M Proposition 4.11. Assume that the compatible almost complex structure J on M is globally conformally balanced - i.e. J is integrable and = dlogf 2 is exact. Then df J 12 f J 12 fF is a Killing 1-form (that is the vector eld dual to df J is a Killing vector eld). Furthermore, if = 0 then the scalar curvature = 0. Proof. We must show that under the present hypothesis the covariant derivative of 1-form df J is skew-symmetric. For any vectors X; Y , by taking into account that df J 12 f J , one has rX (df J )(Y ) + rY (df J )(X ) = 21 df (X )(JY ) + 21 f (rX )(JY ) + df (!3 (X )J2 Y ? !2(X )J3 Y ) + 21 df (Y )(JX ) + 12 f (rY )(JX ) + df (!3 (Y )J2X ? !2 (Y )J3X ) = 21 f [ 12 (X )(JY ) + 21 (Y )(JX ) + (rX )(JY ) + (rY )(JX )] (Last equality being deduced from identities df = 12 f and !2 = ? 21 J2 , !3 = ? 21 J3). By (4.6) one gets rX (df J )(Y ) + rY (df J )(X ) = 0. Finally, = 0 if and only if = 0: in this case (g; J ) is Kahler by (3.5) and therefore the holonomy of g is in Sp(n) Sp(1) \ U (2n) Sp(n). ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 21 Conformally balanced hypercomplex structures. It is well known that the n quaternionic projective space H P is locally hypercomplex: local admissible hypercomplex bases H = (J) on H Pn are obtained by considering systems of quaternionic projective coordinates. But Theorem 4.3 excludes the existence of global hypercomplex bases on H Pn (In fact a classical result of W.S. Massey states that there does'nt exist any almost complex structure on H Pn). On the other hand, as recalled by Remark 4.4, the hyperbolic quaternionic space H Hn admits global hypercomplex bases. In fact we wonder if for < 0 this is essentially the only possibility of complete quaternion-Kahler manifolds having this property, up to a Riemannian covering. A rst result in this direction is the following one. Proposition 4.12. Let (M 4n ; g; Q) be a complete quaternion-Kahler manifold. Assume that there exists a global admissible basis H = (J) of Q, consisting of integrable complex structures. Then 1) 0 and < 0 only if (M 4n ; g) is non-compact. Moreover 2) if < 0 and one of the complex structures J ( = 1; 2; 3) is conformally balanced then there exists an (eventually singular) integrable distribution D on M 4n which is Q-invariant and whose regular orbit is a (non trivial) totally geodesic quaternionic submanifold with constant quaternionic curvature, that is locally isometric to H Hk , 1 k n, with a standard metric. Proof. Let H = (J) be a global admissible basis which is hypercomplex. Then by theorems 4.3 and 2.4 we exclude that 6= 0 if M 4n is compact. It remains to prove the second statement. Hence let assume < 0, M 4n non-compact and, say J1 conformally balanced. Then formula (2.9) and similar formulas for J2; J3 together with Lemma 4.1 imply that the three complex structures J have equal Lee forms : = 1 = 2 = 3 and, moreover, does not vanishes. By 1) of Proposition 4.6 the 2-tensor r + 21 is Q-Hermitian. In fact, by summing up the (4.6) for J = J, = 1; 2; 3 respectively, one gets r? = ? 21 + 21 ( ) and by adding to (4.7) it results, since d = 0, (4.16) r = ? 21 + ( ) ? g By assumption, = dh for some function h. We dene the 1-form := e 21 h . Then r = e 21 h [r + 12 ( )] and hence r is a symmetric Q-Hermitian 2-tensor on M , that is g?1 is a quaternionic non-isometric innitesimal transformation on M 4n (see [AM1]). By result of [AM3, Proposition 6] we can conclude. 22 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO 5. Laplacians of self-dual forms Let (M 4n ; g; Q) be a quaternion-Kahler manifold if n > 1 and a self-dual Einstein 4-manifold if n = 1. Then the following formulas hold. Proposition 5.1. Let H = (J) be an admissible basis of Q. Then 4F1 = (4 + k!2k2 + k!3k2)F1 (5:1) + (!3? < !1; !2 >)F2 ? (!2+ < !1; !3 >)F3 and r rF1 = (k!2 k2 + k!3k2)F1 (5:2) + (!3 ? < !1 ; !2 >)F2 ? (!2 + < !1 ; !3 >)F3 (Note that if > 0 then 4F1 6= 0 everywhere). Moreover, for F = F1 one has 4F ? r rF = 4F (5:3) Proof. Let recall the expression of F given by (2.12). Now we compute dF . We have dF (X; Y ) = [rX (!3 J2 ? !2 J3)](Y ) ? [rY (!3 J2 ? !2 J3)](X ) = (rX !3)(J2 Y ) ? (rY !3 )(J2X ) ? (rX !2 )(J3Y ) + (rY !2)(J3 X ) + !3[(!1 (X )J3 ? !3(X )J1 )(Y )] ? !2[(!2 (X )J1 ? !1 (X )J2 )(Y )] ? !3[(!1 (Y )J3 ? !3(Y )J1 )(X )] + !2[(!2 (Y )J1 ? !1(Y )J2)(X )] Hence dF (X; Y ) = (rX !3)(J2 Y ) ? (rY !3)(J2 X ) ? (rX !2)(J3 Y ) + (rY !2)(J3 X ) ? [!2(X )!2 (J1Y ) + !3(X )!3 (J1 Y ) ? !2(Y )!2(J1 X ) ? !3(Y )!3 (J1X )] + !1(X )!2 (J2 Y ) ? !1(Y )!2(J2 X ) + !1(X )!3 (J3Y ) ? !1 (Y )!3 (J3X ) Let recall the expression of dF given by (2.1.2). Now we compute dF . We have dF (X; Y ) = ? =? X i X i (rEi dF )(Ei ; X; Y ) [(rEi !3)(Ei )F2 (X; Y ) + !3(Ei )g((rEi J2)X; Y ) ? (rEi !2)(Ei )F3 (X; Y ) ? !2 (Ei)g((rEi J3)X; Y )] ? !3(X )!1 (J3 Y ) + !3(X )!3 (J1 Y ) ? (rJ2Y !3)(X ) + !2(X )!2 (J1 Y ) ? !2(X )!1 (J2 Y ) + (rJ3Y !3)(X ) X ? [g((rEi J2)Ei ; X )!3 (Y ) + F2 (Ei; X )(rEi !3 )(Y ) i ? g((rEi J3)Ei ; X )!2 (Y ) ? F3 (Ei; X )(rEi !2 )(Y )] ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 23 Hence dF (X; Y ) = ? X i [(rEi !3 )(Ei )F2 (X; Y ) ? (rEi !2 )(Ei )F3(X; Y ) + !3 (Ei)g(!1 (Ei )J3X ? !3 (Ei)J1 X; Y ) ? !2 (Ei)g(!2 (Ei )J1X ? !1 (Ei)J2 X; Y ) + !3 (Y )g(!1 (Ei)J3 Ei ? !3(Ei )J1Ei ; X ) ? !2 (Y )g(!2 (Ei)J1 Ei ? !1(Ei )J2Ei ; X )] + (rJ2 X !3)(Y ) ? (rJ3X !2)(Y ) ? !3 (X )!1 (J3Y ) + !3(X )!3 (J1 Y ) ? (rJ2Y !3)(X ) + !2 (X )!2 (J1Y ) ? !2(X )!1 (J2 Y ) + (rJ3Y !3)(X ) That is dF (X; Y ) = !3F2 (X; Y ) ? !2F3 (X; Y ) ? < !3; !1 > F3(X; Y ) + k!3 k2F1(X; Y ) + k!2 k2F1(X; Y )? < !1; !2 > F2(X; Y ) + !1 (J3X )!3 (Y ) ? !3(J1X )!3 (Y ) ? !2(J1 X )!2(Y ) + !1(J2 X )!2 (Y ) + (rJ2 X !3)(Y ) ? (rJ3X !2 )(Y ) ? !3 (X )!1 (J3Y ) + !3(X )!3 (J1 Y ) ? (rJ2Y !3)(X ) + !2 (X )!2 (J1Y ) ? !2(X )!1 (J2 Y ) + (rJ3Y !3)(X ) By previous formulas one has 4F1(X; Y ) = (d + d)F1 (X; Y ) = (rX !3)(J2 Y ) ? (rY !3 )(J2X ) ? (rX !2)(J3 Y ) + (rY !2 )(J3X ) + (rJ2 X !3)(Y ) ? (rJ3X !2 )(Y ) ? (rJ2Y !3 )(X ) + (rJ2Y !3)(X ) + !1 (J3X )!3 (Y ) ? !3(J1X )!3 (Y ) ? !2(J1X )!2 (Y ) + !1(J2 X )!2 (Y ) ? !3 (X )!1 (J3Y ) + !3(X )!3 (J1 Y ) + !2(X )!2 (J1 Y ) ? !2(X )!1 (J2Y ) ? !2 (X )!2 (J1Y ) ? !3(X )!3 (J1 Y ) + !2(Y )!2(J1 X ) + !3(Y )!3(J1 X ) + !1 (X )!2 (J2Y ) ? !1(Y )!2 (J2X ) + !1(X )!3 (J3 Y ) ? !1(Y )!3(J3 X ) + (k!2 k2 + k!3k2)F1 (X; Y ) + (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1 ; !3 >)F3 (X; Y ) 24 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO that is 4F1(X; Y ) = d!3(X; J2 Y ) + d!3(J2 X; Y ) ? d!2(J3X; Y ) ? d!2(X; J3 Y ) + !1 ^ !2(J2X; Y ) + !1 ^ !2(X; J2 Y ) ? !3 ^ !1(J3X; Y ) ? !3 ^ !1(X; J3 Y ) + (k!2 k2 + k!3k2)F1 (X; Y ) + (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1; !3 >)F3(X; Y ) = g(J2X; J3 Y ) + g(X; J3J2Y ) ? g(J3X; J2 Y ) ? g(X; J2J3Y ) + (k!2 k2 + k!3k2)F1 (X; Y ) + (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1; !3 >)F3(X; Y ) = 4g(J1X; Y ) + (k!2 k2 + k!3k2)F1 (X; Y ) + (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1; !3 >)F3(X; Y ) and (5.1) follows immediately. Now we compute rrF1. By using (2.1.1) we have (r rF1)(X; Y ) = ? X i [(rEi !3)(Ei ))F2 (X; Y ) + !3(Ei )(rEi F2)(X; Y ) ? (rEi !2 )(Ei ))F3 (X; Y ) ? !2(Ei )(rEi F3)(X; Y )] = !3F2 (X; Y ) ? !2F3 (X; Y ) ? !3 (Ei)g(!1 (Ei )J3X ? !3(Ei )J1X; Y ) + !2(Ei )g(!2 (Ei)J1 X ? !1(Ei )J2X; Y ) and (5.2) follows. By (5.1), (5.2) it is straightforward to deduce (5.3). Now we want to compute the Laplacian on self-dual 2-forms. Let be a section of 2 + g Q, that is is a self-dual 2-form on M. Let write = fF where F F1 is the Kahler form of a compatible almost complex structure J J1 and H = (J1; J2; J3) is a local basis of Q. Now we calculate the Laplacian 4. Let (Ei)i=1;::;n be an (locally dened) orthonormal frame and X a vector eld on M . We have X X (fF )(X ) = ? df (Ei )F (Ei ; X ) ? f (rEi F )(Ei ; X ) Xi Xi = df (Ei )g(Ei ; JX ) ? f (rEi F )(Ei ; X ) i Hence i (fF ) = (df J ) + fF Remark 5.2. Note that (fF ) = 0 () J = d(?logf ). M (5.4) ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 25 Moreover, for any two vector elds X; Y we have d(fF )(X; Y ) = rX (df J )(Y ) ?rY (df J )(X )+ df ^ F (X; Y )+ fdF (X; Y ) (5.5) that is d(fF ) = d(df J ) + df ^ F + fdF (5.6) On the other hand, we have d d(fF ) = df ^ F + fdF and hence X d(fF )(X; Y ) = ? rEi (df ^ F )(Ei ; X; Y ) Xi ? df (Ei )dF (Ei ; X; Y ) + fF (X; Y ) i that is X d(fF )(X; Y ) = ? rEi (df ^ F )(Ei ; X; Y ) Xi ? df (Ei )(!3 ^ F2 ? !2 ^ F3)(Ei ; X; Y ) + fF (X; Y ) i Now we have X i (df ^ F )(Ei ; X; Y ) = X i [df (Ei )F (X; Y ) + df (X )F (Y; Ei ) + df (Y )F (Ei ; X )] and ? X i rEi (df ^ F )(Ei ; X; Y ) = (df )F (X; Y ) ? X i df (Ei )(rEi F )(X; Y ) ? (rEi df )(X )F (Y; Ei ) ? df (X )rEi F (Y; Ei) ? (rEi df )(Y )F (Ei ; X ) ? df (Y )(rEi F )(Ei ; X ) 26 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO hence ? X i rEi (df ^ F )(Ei ; X; Y ) X df (Ei ) !3 (Ei)F2 (X; Y ) ? !2 (Ei)F3 (X; Y ) X !3(Ei )F2 (Y; Ei) ? !2(Ei )(F3 (Y; Ei) ? (rJY df )(X ) ? df (X ) i X !3(Ei )F2 (Ei ; X ) ? !2 (Ei)F3 (Ei ; X ) + (rJX df )(Y ) ? df (Y ) = (df )F (X; Y ) ? i i = (df )F (X; Y ) ? hdf; !3iF2 (X; Y ) + hdf; !2 iF3(X; Y ) + (rJX df )(Y ) ? (rJY df )(X ) ? df (X )!3 (J2Y ) ? df (X )!2 (J3 Y ) + df (Y )!3 (J2X ) ? df (Y )!2(J3 X ) = (df )F (X; Y ) ? hdf; !3iF2 (X; Y ) + hdf; !2 iF3(X; Y ) + (rJX df )(Y ) ? (rJY df )(X ) ? df (X )(JY ) + df (Y )(JX ) Moreover we have ? X i X i df (Ei )(!3 ^ F2 ? !2 ^ F3)(Ei ; X; Y ) [?df (Ei)!3 (Ei )F2 (X; Y ) ? df (Ei )!3 (X )F2 (Y; Ei) ? df (Ei )!3 (Y )F2 (Ei; X ) + df (Ei )!2 (Ei )F3 (X; Y ) + df (Ei )!2 (X )F3 (Y; Ei ) + df (Ei )!2 (Y )F3 (Ei; X )] = ?hdf; !3iF2 (X; Y ) ? (df J2)(Y )!3(X ) + (df J2)(X )!3 (Y ) + hdf; !2 iF3 (X; Y ) + (df J3)(Y )!2 (X ) ? (df J3)(X )!2 (Y ) Hence d(fF )(X; Y ) = (df )F (X; Y ) ? hdf; !3iF2 (X; Y ) + hdf; !2 iF3 (X; Y ) + (rJX df )(Y ) ? (rJY df )(X ) ? df (X )( J )(Y ) + df (Y )( J )(X ) ? hdf; !3 iF2 (X; Y ) ? (df J2)(Y )!3 (X ) + (df J2)(X )!3 (Y ) + hdf; !2 iF3 (X; Y ) + (df J3)(Y )!2 (X ) ? (df J3)(X )!2 (Y ) + fdF (X; Y ) (5.7) ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 27 >From which we deduce that 4(X; Y ) 4(fF )(X; Y ) = (rX df )(JY ) ? rY (df )(JX ) + (rJX df )(Y ) ? (rJY df )(X ) + df !3 (X )J2 Y ? !2 (X )J3 Y ? df !3 (Y )J2X ? !2 (Y )J3 X ? df (J2 Y )!3 (X ) + df (J2 X )!3 (Y ) + df (J3Y )!2 (X ) ? df (J3 X )!2 (Y ) ? hdf; !3 iF2(X; Y ) + hdf; !2iF3 (X; Y ) + 4fF (X; Y ) + f 4 F (X; Y ) that is 4(fF )(X; Y ) = 4fF (X; Y ) + f 4 F (X; Y ) + (rX df )(JY ) ? (rY df )(JX ) + (rJX df )(Y ) ? (rJY df )(X ) ? hdf; !3iF2 (X; Y ) + hdf; !2iF3 (X; Y ) that is 4(fF ) = (4f )F + f 4 F ? hdf; !3 iF2 + hdf; !2iF3 (5.8) 4(fF ) = [f (4 + k!2k2 + k!3 k2) + 4f ]F [f (!3 ? h!1 ; !2i) ? hdf; !3i]F2 + [?f (!2 + h!1; !3 i) + hdf; !2i]F3 (5.9) and hence, by combining with formula (5.1), we have the following Proposition 5.3. For any self-dual 2-form = fF where F is the locally dened Kahler form of a compatible J one has We conclude this paragraph by showing that for Laplacian of self-dual 2-forms associated to integrable almost complex structures, on a not necessarily compact M , one has the following Proposition 5.4. Let assume that the compatible almost complex structure J on M is integrable. Then one has 4F = (4 + 21 kk2 )F (5.10) and rrF = 21 kk2 F (5.11) Proof. It is an immediate consequence of previous identities (5.1) and (5.2) and of the following Lemma. 28 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO Lemma 5.5. For J integrable the following identities hold: 1 Proof. We know that: !2+ < !1; !3 >= 0 !3? < !1; !2 >= 0 (5.12) (5.13) (d! + ! ^ ! )(; J ) = ?g ( = 1; 2; 3) Hence, in particular, the 2-forms at left member are Q-Hermitian and for = 3 one has the identities (rX !3 )(J3Y ) ? (rJ3Y !3 )(X ) + !1(X )!2 (J3Y ) ? !1(J3 Y )!2 (X ) = (rJX !3 )(J3JY ) ? (rJ3 JY !3)(J X ) ( = 1; 2; 3) + !1 (JX )!2 (J3JY ) ? !1(J3 JY )!2(JX ) and, by taking into account the identity rX !3 = ?(rX !2 ) J1 + !2(X )(!2 J3) + !2 (J1X )(!2 J2) (5.14) which is obtained by dierentiating the identity !2 J2 = !3 J3, it results (rX !2)(J2 Y ) ? !2 (X )!2 (Y ) + !2(J1X )!2 (J1 Y ) + (rJ3 Y !2)(J1 X ) ? !2(J3Y )!2 (J3X ) + !2(J2Y )!2 (J2X ) + !1 (X )!2 (J3Y ) ? !1 (J3Y )!2 (X ) = (rJX !2)(J2 JY ) ? !2 (JX )!2 (JY ) + !2(J1 JX )!2 (J1 JY ) + (rJ3 JY !2)(J1 JX ) ? !2(J3 JY )!2 (J3JX ) + !2(J2JY )!2 (J2JX ) + !1 (JX )!2 (J3JY ) ? !1(J3 JY )!2 (JX ) ( = 1; 2; 3) By choosing = 1 and by substituting J2Y to Y we get the identity ? (rX !2 )(Y ) ? (rJ1Y !2)(J1 X ) ? (rJ1X !2 )(J1Y ) ? (rY !2)(X ) = !2 (X )!2 (J2Y ) ? !2 (J1X )!2 (J3Y ) + !1(X )!2 (J1 Y ) ? !1(J1Y )!2 (X ) ? !2(J1Y )!2 (J3X ) + !2(Y )!2(J2 X ) ? !2(J1X )!2 (J3 Y ) + !2(X )!2 (J2 Y ) + !2(Y )!2(J2 X ) ? !2(J1Y )!2 (J3X ) ? !1(J1 X )!2(Y ) + !1(Y )!2(J1 X ) By contraction with respect to g one obtains 4!2 = ? < !1 ; !3 > ? < !1; !3 > ? < !1 ; !3 > ? < !1; !3 > that is !2+ < !1; !3 >= 0 By interchanging J3 and J2 one gets also !3? < !1; !2 >= 0 (It is sucient to repeat the computations for the admissible basis H 0 = (J0 ) where (J1 0 = J1; J20 = J3; J30 = ?J2) and to take into account that !1 0 = !1 , !20 = !3 , !3 0 = ?!2. ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS 29 6. Action of the real and quaternionic Weyl tensors on self-dual forms Let (M 4n ; g; Q), n > 1, be a quaternion-Kahler manifold with curvature tensor R. We have the formulas (see for example [AM2]) ( R = RH P + W Q ; = 4n(Kn+2) R = 4n(4Kn?1) RS(1) + W R (6.1) where RS(1) is the curvature tensor of the 4n-dimensional sphere of constant curvature 1, RH P is the curvature tensor of the metric of quaternionic curvature 1 for the ndimensional quaternionic projective space H Pn. We recall that for any vector elds X; Y; Z; T 2 M 4n RS(1)(X; Y; Z; T ) = g(X; T )g(Y; Z ) ? g(X; Z )g(Y; T ) and X RH P (X; Y; Z; T ) = 41 [g(X; T )g(Y; Z ) ? g(X; Z )g(Y; T ) + 2 g(X; J Y )g(JZ; T ) X X ? g(Y; JZ )g(JX; T ) + g(X; JZ )g(JY; T )] (6.2) W R is the Weyl curvature tensor, W Q is the quaternionic Weyl curvature tensor. Hence W R = 4n(nK+ 2) RH P + W Q ? 4n(4K (6.3) n ? 1) RS(1) By dening the action of curvature tensor on a 2-form F by X (6.4) R(F )hk = ? 21 R(Ei ; Ej ; Eh ; Ek )F (Ei ; Ej ) i;j one has the following results. Proposition 6.1. Let F be a self-dual 2-form on the quaternion-Kahler manifold (M; g; Q), n > 1. Then one has W Q (F ) = 0 ; RS(1)(F ) = F ; RH P (F ) = nF (6.5) W R (F ) = cF (6.6) ? 1)(2n + 1) ]g c = f 2Kn [ ((nn + 2)(4n ? 1) (6.7) and where c is constant, 30 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO Proof. Without any loss of generality we can assume that F is the Kahler form of a (local) compatible almost complex structure J on (M; g; Q). We have ?2W Q (F )hk = = X < W Q (Ei ; Ej )Eh ; Ek > g(IEi; Ej ) i;j X < W Q (Eh ; Ek )Ei ; Ej > g(IEi; Ej ) i;j X < W Q (Eh ; Ek )Ei ; JEi > i X = ? < JW Q(Eh ; Ek )Ei; Ei > = i = Trace[JW Q (Eh ; Ek )] = 0 as it was proved in [AM2]. We have ?2RS(1)(F )hk = = X [g(Ei ; Ek )g(Ej ; Eh ) ? g(Ei; Eh )g(Ej ; Ek )]g(JEi; Ej ) i;j X i [g(Ei ; Ek )g(JEi; Eh ) ? g(Ei; Eh )g(JEi; Ek )] = 2g(Eh ; JEk ) = ?2Fhk We have ?2RH P (F )hk = 41 [?2RS(1)(F )hk ] + 41 [2 X g(Ei; JEj )g(JEh ; Ek ) ? X g(Ej ; JEh )g(JEi ; Ek ) i;j ; X i;j; + g(Ei; JEh )g(JEj ; Ek )]g(JEi; Ej ) i;j ; X = 41 [?2F (Eh; Ek ) ? 2 g(JEj ; JEj )g(J Eh; Ek ) j ; X X ? g(JEi; JEh )g(JEi; Ek ) ? g(JEj ; J Eh)g(JEj ; Ek )] i; = 41 [?2F (Eh; Ek ) ? 8ng(JEh; Ek ) ? X g(JEk ; JJEh ) ? X j ; g(JEk ; JJEh )] = 41 [?2F (Eh; Ek ) ? 8nF (Eh; Ek ) + 2F (Eh ; Ek )] = ?2nF (Eh; Ek ) ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS Hence, 31 W R(F ) = [ 4n(nK+ 2) (n) + 0 ? 4n(4K n ? 1) 1]F ? 1)(2n + 1) ]F = 2Kn [ ((nn + 2)(4n ? 1) Remark 6.2. From (6.6) it is possible to show how the formulas (2.11), (4.2) and (5.3) could be obtained respectively from formulas (12), (26) and (7) of [G1]. Our formulas agree with Gauduchon formulas, by taking into account that for the scalar product between 2-forms (; ) used in [G1] one has M [AGS] [A] [ABM] [AM1] [AM2] [AM3] [AMP] [Be] [B] [BGMR] [G1] [G2] [G3] [GBNV] (F; F ) = 21 kF k2 References E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifold, Preprint 1995. D.V. Alekseevsky, Classication of quaternionic spaces with a transitive solvable group of motions, Math. USSR Izv. 39 (1975), 315-362. D.V. Alekseevsky, E. Bonan, S. Marchiafava, On some structure equations for almost quaternionic Hermitian manifolds, Proceedings of \Second International workshop on complex structures and vector elds, Pravetz, Bulgaria, 94" (S. Dimiev, K. Sekigawa, eds.), World Scientic, 1995. D.V. Alekseevsky, S. Marchiafava,, Hypercomplex structures on quaternionic manifolds, New developments in dierential geometry (L. Tamassy, J. Szenthe, eds.), Kluwer, 1996. , Quaternionic structures on a manifold and subordinated structures, Annali di Mat. Pura e Appl. 171 (1996), 205-273. , Quaternionic transformations of a non positive quaternionic Kahler manifold, Preprint MPI/95-126 (1996). D.V. Alekseevsky, S. Marchiafava, M. Pontecorvo, Compatible almost complex structures on almost quaternionic manifolds, Preprint ESI n. 404 (1996). A. Besse, Einstein manifolds, Ergebnisse der Math., vol. 3, Springer-Verlag, 1987. C. Boyer, A note on hyperhermitian four-manifolds, Proc. Am. Math. Soc. 102 (1988), 157-164. F. E. Burstall, G. Grantcharov, O. Muskarov, J. H. Rawnsley, Hermitian structures on Hermitian symmetric spaces, J. Geom. Phys. 10 (1993), 245-249. P. Gauduchon, Complex structures on compact conformal manifolds of negative type, Complex analysis and Geometry (V. Ancona, E. Ballico, S. Silva, eds.), Proceedings of the conference at Trento, Marcel Dekker, New York, Basel, Hong Kong, 1996, pp. 201-212. , Canonical connections for almost-hypercomplex structures, Preprint 1996. P. Gauduchon, Alcuni spunti di geometria quasi hermitiana e hermitiana (1983), Dip. Mat. Roma "La Sapienza". A. Gray, M. Barros, A.M. Naveira, L. Vanhecke, The Chern numbers of holomorphic vector bundles and formally holomorphic connections of complex vector bundles over almost complex manifolds, J. Reine Angew. Math. 314 (1980), 84-98. 32 ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO [GP] [Gr] [H] [K] [LS] [MS] [M] [P1] [P2] [P3] [S1] [S2] [S3] [T1] [T2] [W] K. Galicki, Y.S. Poon, Duality and Yang-Mills elds on quaternionic Kahler manifolds, J. Math. phys. 32 (1997), 1263-1268. A. Gray, L.M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123, 35-58. N. Hitchin, Linear eld equations on self-dual spaces, Proc. R. Lond. Soc. A 370 (1980), 173-191. P. Kobak, Explicit doubly-hermitian metrics, ESI preprint (1995). C. LeBrun, S. Salamon, Strong rigidity of positive quaternion-Kahler manifolds, Invent. Math. 118 (1994), 109-132. M. Mamone Capria, S.M. Salamon, Yang-Mills elds on quaternionic spaces, Non linearity 1 (1988), 517-530. S. Marchiafava, Sulla geometria locale delle varieta' kahleriane quaternionali, Bollettino U.M.I. (7) 5-B (1991), 417-447. M. Pontecorvo, On twistor spaces of anti-self-dual hermitian surfaces, Trans. Am. Math. Soc. 331 (1992), 653-661. , Complex structures on quaternionic manifolds, Di. Geometry and its Applications 4 (1992), 163-177. , Complex structures on riemannian 4-manifolds, Math. Ann. (1997) (to appear). S. Salamon, Quaternionic Kahler manifolds, Invent. Math. 67 (1982), 143-171. , Special structures on four-manifolds, Riv. Mat. Univ. Parma (4) 17 (1991), 109123. , Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics, vol. 201, Longman Scientic, 1989. F. Tricerri, Sulle varieta dotate di due strutture quasi complesse linearmente indipendenti, Riv. Mat. Univ. Parma 3 (1974), 349-358. , Connessioni lineari e metriche hermitiane sopra varieta dotate di due strutture quasi complesse, Riv. Mat. Univ. Parma 4 (1975), 177-186. J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. of Math. and Mech. 14 (1965), 65-70. Gen. Antonova 2 - 99, 117279 Moscow, Russian Federation E-mail address : [email protected] Dipartimento di Matematica, Universita di Roma \La Sapienza", P.le A. Moro 2, 00185 Roma, Italy E-mail address : [email protected] Dipartimento di Matematica, Universita di Roma Tre, Via C. Segre 2, 00146 Roma, Italy E-mail address : [email protected]

© Copyright 2023 Paperzz