On a class of pseudo calibrated generalized complex structures related to Norden, para-norden and statistical manifolds

Transcription

1 On a class of pseudo calibrated generalized complex structures related to Norden, para-norden and statistical manifolds Antonella Nannicini, E0,1 Abstract. We consider pseudo calibrated generalized complex structures,, defined by a pseudo Riemannian metric g and a g symmetric operator H such that H 2 = µi, µ R, on a smooth manifold M. These structures include the case of complex Norden manifolds for µ =, studied in [20], the case of almost tangent structures for µ = 0, ImH = KerH, and the case of para Norden manifolds for µ = 1. The special case H = O is described in [19]. We study integrability conditions of, with respect to a linear connection, and we describe examples of geometric structures that naturally give rise to integrable pseudo calibrated generalized complex structures. We prove that for µ integrability implies that the ±i eigenbundles of, E1,0, are complex Lie algebroids. We define the concept of generalized -operator of M, H, g, and we study holomorphic sections. M.S.C. 2010: 53C15, 53D18, 53C80, 53C38. Key words: general geometric structures on manifolds; Norden manifolds; para Norden manifolds; Lie algebroids. generalized geometry; 1 Introduction Generalized complex structures were introduced by Hitchin in [9], and further investigated by Gualtieri in [10], in order to unify symplectic and complex geometry. In this paper we consider the concept of generalized complex structure introduced in [16], [17] and also studied in [5], [18], [19], [20]. Let M, g be a smooth pseudo Riemannian manifold, let T M be the tangent bundle, let T M be the cotangent bundle and let E = T M T M be the generalized tangent bundle of M. A pseudo calibrated generalized complex structure of M is a complex structure on E which is pseudo calibrated with respect to the canonical symplectic structure of E. A linear connection,, on M defines a bracket, [, ], Balkan Journal of Geometry and Its Applications, Vol.22, No.1, 2017, pp c Balkan Society of Geometers, Geometry Balkan Press 2017.

2 52 Antonella Nannicini on sections of E and we can define the concept of integrability for generalized complex structures. We consider pseudo calibrated generalized complex structures defined by a pseudo Riemannian metric g and a g symmetric operator H such that H 2 = µi, µ R, on M. These structures include the case of complex Norden manifolds for µ =, studied in [20], the case of almost tangent structures for µ = 0, ImH = KerH, and the case of para Norden manifolds for µ = 1. The special case H = O is described in [19]. We study the integrability conditions of, with respect to a linear connection with torsion T, and we describe examples of geometric structures that naturally give rise to integrable pseudo calibrated generalized complex structures. Then we prove that for µ integrability implies that the ±i eigenbundles of, E 0,1, are complex Lie algebroids. We define the concept of generalized operator it follows 2 = 0 and, as is the, we get that C,,,, [, ] is a differential Gerstenhaber algebra, where denotes the Schouten bracket, [13], [28]. Finally we study certain holomorphic sections. The paper is organized as in the following. In section 2 we introduce preliminary material of the generalized tangent bundle and of generalized complex structures; in section 3 we compute integrability conditions and in section 4 we give examples of integrable structures; section 5 is devoted to the study of complex Lie algebroids naturally associated to integrable pseudo calibrated generalized complex structures; in section 6 we define the concept of generalized operator on M and in section 7 we study some generalized holomorphic sections, in particular, in this context, Hessian manifolds occur as interesting examples. This paper is a generalization of our previous papers [19], [20] and allows us to unify complex Norden and para Norden manifolds through almost tangent structures and statistical manifolds. The theory reveals that the case of complex Norden manifolds is special. of M, H, g, ; from the Jacobi identity on exterior derivative of the Lie algebroid, 2 Preliminaries Let M be a smooth manifold of real dimension n and let E = T M T M be the generalized tangent bundle of M. Smooth sections of E are elements X + ξ C E where X C T M is a vector field and ξ C T M is a 1 form. E is equipped with a natural symplectic structure,,, defined by: 2.1 X + ξ, Y + η = 1 ξy ηx 2 and a natural indefinite metric,,, defined by: 2.2 X + ξ, Y + η = 1 ξy + ηx. 2, is non degenerate and of signature n, n.

3 On a class of pseudo-calibrated generalized complex structures 53 A linear connection on M,, defines, in a canonical way, a bracket [, ] on C E, as follows: 2.3 [X + ξ, Y + η] = [X, Y ] + X η Y ξ. Like in [16], a direct computation gives the following: Lemma 2.1. For all X, Y C T M, for all ξ, η C T M f C M we have: 1. [X + ξ, Y + η] = [Y + η, X + ξ], 2. [fx + ξ, Y + η] = f [X + ξ, Y + η] Y f X + ξ, 3. Jacobi s identity holds for [, ] if and only if has zero curvature. We consider the following concept of generalized complex structure: and for all Definition 2.2. A generalized complex structure on M is an endomorphism : E E such that 2 = I. Definition 2.3. A generalized complex structure is called pseudo calibrated if it is, invariant and if the bilinear symmetric form defined by, on T M is non degenerate, moreover is called calibrated if it is pseudo calibrated and, is positive definite. From the definition we get that a pseudo calibrated complex structure can be written in the following block matrix form: H I + H 2 g 2.4 = g H where g : T M T M is identified to the bemolle musical isomorphism of the pseudo Riemannian metric g on M, H : T M T M is a g-symmetric operator and H : T M T M is the dual operator of H defined by: H ξx = ξhx. We have: 2.5 gx Y = gx, Y = 2 X, Y for all X, Y T M. In the following we will consider g-symmetric operators H : T M T M such that H 2 = µi where µ R and I denotes identity. In this case we have: H λg 2.6 = where λ = µ. g H We remark that for µ = M, H, g is a Norden manifold, [20], for µ = 0 and ImH = KerH, M, H is an almost tangent manifold, [2], and for µ = 1 M, H, g is a para Norden manifold. The special case H = O is described in [19].

4 54 Antonella Nannicini 3 Integrability Let be a linear connection on M and let [, ] be the bracket on C E defined by, the following holds: Lemma 3.1. [17] Let : E E be a generalized complex structure on M and let 3.1 N : C E C E C E defined by: ] 3.2 N [σ, σ, τ = τ ] [σ, τ ] [σ, τ for all σ, τ C E; N is a skew symmetric tensor. [σ, τ], Definition 3.2. N is called the Nijenhuis tensor of with respect to. Definition 3.3. Let : E E be a generalized complex structure on M, is called integrable if N = 0. Let T be the torsion of : 3.3 T X, Y = X Y Y X [X, Y ] and let d be the exterior differential associated to : 3.4 d g X, Y = X g Y Y g X + g T X, Y for all X, Y C T M. We have the following: H λg Proposition 3.4. Let = be the pseudo calibrated generalized complex structure on M defined by a pseudo Riemannian metric g and a g symmetric g H operator H of T M such that H 2 = λi. Let N be the generalized Nijenhuis tensor of defined by 3.2, then for all X, Y C T M we have: 3.5 N X, Y = NHX, Y λg d gx, Y + +d ghx, Y + d gx, HY + g Y H X X H Y 3.6 N X, gy = λ XH Y Y H X + λg HX g Y X g HY + +λg T HX, Y HT X, Y + +λd gx, Y g HX H Y H X H Y

5 On a class of pseudo-calibrated generalized complex structures N gx, gy = λ2 g d gx, Y + +λg Y HX X H Y where NH is the Nijenhuis tensor of H defined by: 3.8 NHX, Y = [HX, HY ] H [HX, Y ] H [X, HY ] + H 2 [X, Y ]. Proof. Direct computations give: N X, Y = [HX + gx, HY + gy ] + [HX + gx, Y ] [X, HY + gy ] [X, Y ] = [HX, HY ] + HX gy HY gx+ [HX, Y ] Y gx + [X, HY ] + X gy [X, Y ] = [HX, HY ] H [HX, Y ] H [X, HY ] + λg Y gx X gy + [X, Y ] + HX gy HY gx g [HX, Y ] + H Y gx g [X, HY ] + H X gy = NHX, Y + λ [X, Y ] + λg Y gx g X Y + + HX g Y HY g X + g Y HX X HY + H Y g X + H X g Y + g T HX, Y + T X, HY = NHX, Y + λg Y gx X gy λt X, Y + + HX g Y HY g X + g Y HX X HY + Y g HX + X g HY + g T HX, Y + T X, HY = NHX, Y λg d gx, Y + +d ghx, Y + d gx, HY + g Y HX X H Y N X, gy = [HX + gx, λy ghy ] [HX + gx, gy ] + [X, λy ghy ] XgY = λ [HX, Y ] HX ghy λ Y gx λ [X, Y ] + + XgHY HXgY X gy = λ [HX, Y ] HX ghy λ Y gx λ [X, Y ] + λh [X, Y ] λg [X, Y ] + λg X ghy λg HX gy Y gx +H HX gy H X ghy X gy = λ X HY Y H X g HX gy X ghy + +gt HX, Y HT X, Y + λd gx, Y g HX HY X H Y N gx, gy = [λx ghx, λy ghy ] +

6 56 Antonella Nannicini [λx ghx, Y ] [gx, λy ghy ] = λ 2 [X, Y ] λ { λg X gy Y gx } + λ { H X gy + H Y gx} = λ 2 T X, Y g X g Y Y g X + λ g Y HX X H Y = λ 2 g d gx, Y + λg Y HX X H Y. In particular we get: Theorem 3.5. For λλ = H the following conditions hold: 3.9 For λ = 0 = H hold: 3.10 g NH = 0 O H g d g = 0 NH = 0 H = 0. λg H is integrable if and only if is integrable if and only if the following conditions HX H = H X H d g HX, Y + d gx, HY g X H Y Y H X = 0. For λ = = H hold: 3.11 g g H is integrable if and only if the following conditions d g = 0 NH = 0 X H Y = Y H X HX H = H X H. Proof. If λ 0 then from 3.7 and 3.5 we get immediately the first and second condition in 3.9 and 3.11 and the third in Moreover: HX g Y X g HY + gt HX, Y HT X, Y = d g HX, Y + Y g HX X g HY ght X, Y = d g HX, Y + H Y g X X g Y gt X, Y = d g HX, Y H d g X, Y ;

7 On a class of pseudo-calibrated generalized complex structures 57 then we get In order to obtain 3.9 remark that if λ, we have X H Y = H 1 HX H Y = λ H HXH Y = 1 λ H Y H HX = 1 λ H Y H 2 X H Y HX = 1 λ H2 H Y X Y HX = Y H X = X H Y ; thus the third condition in 3.9 is obtained. Finally, 3.10 and 3.12 immediately follow. On the other hand if 3.9, respectively 3.10, 3.11 hold, then N = 0, and the proof is complete. O g Corollary 3.6. If H = O = is integrable if and only if g O 3.12 d g = 0. 4 Examples Examples of integrable structures with H = 0 can be found in the context of quasi statistical manifolds. Definition 4.1. [1], [21] Let M, g, be a pseudo Riemannian manifold with a torsion free linear connection, if g is symmetric then M, g, is called a statistical manifold. The concept of statistical manifold can be generalized to statistical manifolds admitting torsion or quasi statistical manifolds [12]: Definition 4.2. Let M, g be a pseudo Riemannian manifold and let be a linear connection on M with torsion T then M, g, is called a quasi statistical manifold or statistical manifold admitting torsion if, for all X, Y C T M, the following formula holds: 4.1 X gy Y g X + gt X, Y = 0. As a direct consequence of 3.4 and 3.12 we get the following: Corollary 4.3. Let M, g be a pseudo Riemannian manifold and let be a linear connection on M with torsion T, let O g 4.2 = g O

8 58 Antonella Nannicini be the generalized complex structure on M defined by g, is integrable if and only if M, g, is a quasi statistical manifold. Examples of integrable structures with H 2 = I can be found in the context of Norden manifolds. Norden manifolds were introduced by A. P. Norden in [22] and then studied also under the names of almost complex manifolds with B-metric and anti Kählerian manifolds, [3], [11]. They have applications in mathematics and in theoretical physics. Definition 4.4. Let M, H be an almost complex manifold of real dimension 2n and let g be a pseudo Riemannian metric on M, if H is a g-symmetric operator then g is called Norden metric and M, H, g is called Norden manifold. If M, H, g is a Norden manifold with H integrable then it is called complex Norden manifold. Let M, H, g be a complex Norden manifold, the following holds: Theorem 4.5. [11] On a complex manifold with Norden metric exists a unique linear connection D with torsion T such that: M, H, g there 4.3 D X g Y, Z = T HX, Y = T X, HY 4.5 gt X, Y, Z + gt Y, Z, X + gt Z, X, Y = 0 for all vector fields X, Y, Z on M. D is called the natural canonical connection of the Norden manifold or B connection and it is defined by: 4.6 D X Y = X Y 1 2 H XHY where is the Levi-Civita connection of g. In particular, if D is the natural canonical connection of the complex Norden manifold M, H, g, then 4.7 DH = 0. Corollary 4.6. Let M, H, g be a complex Norden manifold and let D be the natural canonical connection, let H O 4.8 = g H be the generalized complex structure defined by H and g, is D integrable. Definition 4.7. Let M, H, g be a Norden manifold and let be the Levi-Civita connection of g, if H = 0 then M, H, g is called Kähler Norden manifold.

9 On a class of pseudo-calibrated generalized complex structures 59 We remark that for a Kähler Norden manifold M, H, g the structure H is integrable and the natural canonical connection is the Levi Civita connection. Examples of integrable structures with H 2 = I are given by para Norden manifolds, [4], [25]. Definition 4.8. An almost product structure on a differentiable manifold M is a 1, 1 tensor field H on M such that H 2 = I. The pair M, H is called an almost product manifold. Definition 4.9. An almost paracomplex manifold is an almost product manifold M, H such that the two eigenbundles, T + M, T M, associated to the two eigenvalues, +1 and of H respectively, have the same rank. Definition An almost paracomplex Norden manifold M, H, g is a real smooth manifold of dimension 2n with an almost paracomplex structure H and a pseudo Riemannian metric g such that H is a g symmetric operator. Definition A paraholomorphic Norden manifold, or para Kähler Norden manifold, is an almost paracomplex Norden manifold M, H, g such that H = 0, where is the Levi Civita connection of g. We remark that for an almost paracomplex structure H the vanishing of the Nijenhuis tensor NH is equivalent to the existence of a torsion free linear connection such that H = 0, [25]. In particular from 3.9 we get immediately the following: Corollary Let M, H, g be a paraholomorphic Norden manifold and let be the Levi Civita connection of g, let H 2g 4.9 = g H be the generalized complex structure on M defined by H and g, is integrable. 5 Complex Lie algebroids Lie algebroids were introduced by J. Pradines in [23]; we recall here the definition and the main properties. Definition 5.1. A complex Lie algebroid is a complex vector bundle L over a smooth real manifold M such that: a Lie bracket [, ] is defined on C L, a smooth bundle map ρ : L T M C, called anchor, is defined and, for all σ, τ C L, for all f C M, the following conditions hold: 1. ρ [σ, τ] = [ρ σ, ρ τ] 2. [fσ, τ] = f [σ, τ] ρ τ f σ.

10 60 Antonella Nannicini Let L and its dual vector bundle L be Lie algebroids; on sections of L, respectively L, the Schouten bracket is defined by: 5.1 [, ] L : C p L C q L C p+q L 5.2 = p i=1 j=1 [X 1... X p, Y 1... Y q ] L = q i+j [X i, Y j ] L X 1..î.. X p Y 1..ĵ.. Y q and, for f C M, X C L 5.3 [X, f] L = [f, X] L = ρxf; respectively, by: 5.4 [, ] L : C p L C q L C p+q L 5.5 = p i=1 j=1 [ ] X 1... Xp, Y1... Yq q [ i+j Xi, Y ] j L = X L 1..î.. Xp Y1..ĵ.. Yq and, for f C M, X C L 5.6 [X, f] L = [f, X] L = ρxf. Moreover the exterior derivatives d and d associated with the Lie algebroid structure of L and L are defined respectively by: 5.7 d : C p L C p+1 L 5.8 dα σ 0,..., σ p = = p i ρ σ i α σ 0,..î.., σ p + i j i+j α [σ i, σ j ] L, σ 0,..î..ĵ.., σ p i=0 for α C p L, σ 0,..., σ p C L, and: 5.9 d : C p L C p+1 L 5.10 d α σ 0,..., σ p = = p i ρ σ i α σ 0,..î.., σ p + i j i+j α [σ i, σ j ] L, σ 0,..î..ĵ.., σ p i=0 for α C p L, σ 0,..., σ p C L.

11 On a class of pseudo-calibrated generalized complex structures 61 H λg Let M be a smooth manifold and let = g H be the generalized pseudo calibrated complex structure on M defined by a pseudo Riemannian metric g and a g symmetric operator H of T M such that H 2 = λi. Let 5.11 E C = T M T M C be the complexified generalized tangent bundle. The splitting in ±i eigenspaces of is denoted by: 5.12 E C = with 5.13 E 0,1 A direct computation gives: E 0,1 = = {Z ihz + gw + ihw iz + i λw Z, W C T M C}, equivalently is generated by elements of the following type: 5.15 Z ihz igz with Z C T M, 5.16 λiw + gw + ihw with W C T M. Analogously we have: 5.17 E 0,1 = {Z + ihz + gw ihw + iz i λw Z, W C T M C} and E 0,1 is generated by elements of the following type: 5.18 Z + ihz + igz with Z C T M, 5.19 λiw + gw ihw with W C T M. Lemma 5.2. defined by: For λ 0 the map ψ : T M C T M C 5.20 ψz = Z + ihz is an isomorphism and 5.21 ψz ihz igψz = λz igz + ihz = i λiz + gz + ihz.

12 62 Antonella Nannicini Proof. We have that ψ injective if and only if I + ih is invertible, or i is not an eigenvalue of H. Moreover a direct computation, by using the condition H 2 = λi, gives Corollary 5.3. If λ 0 then: 5.22 = { λz igz + ihz Z C T M C}, 5.23 E 0,1 = { λz + igz ihz Z C T M C}. Moreover, for any linear connection, the following holds: Lemma 5.4. and E 0,1 Proof. Let P + : E C are [, ] involutive if and only if N = 0. and P : E C E 0, P ± = 1 I i, 2 be the projection operators: for all σ, τ C E C we have: [ ] 1 P [P ± σ, P ± τ] = P 2 σ iσ, 1 2 τ iτ 5.25 = 1 8 N σ, τ ± in σ, τ = 1 4 P N σ, τ, and the proof is complete. H λg Theorem 5.4. Let = be the pseudo calibrated generalized complex g H structure on M defined by a pseudo Riemannian metric g and a g symmetric operator H of T M such that H 2 = λi with λ 0, let be a linear connection on M, if is integrable then E1,0 and E 0,1 are complex Lie algebroids. Proof. The anchor 5.26 ρ : is defined by T M C 5.27 ρ λz igz + ihz = λz. Conditions 1. and 2. of the definition are trivially satisfied then we need to prove the the Jacobi identity holds. Let σ, τ, υ, we denote Jacobiator with Jac, that is: 5.28 Jac [[σ, τ], υ] = [[σ, τ], υ] + [[τ, υ], σ] + [[υ, σ], τ].

13 On a class of pseudo-calibrated generalized complex structures 63 We may assume, without loss of generality, that: 5.29 σ = λz igz + ihz τ = λw igw + ihw υ = λu igu + ihu, with Z, W, U C T M C and use integrability conditions. We compute Jac [[ λz igz + ihz, λw igw + ihw ], λu igu + ihu]. As well, we have and then or, [ λz igz + ihz, λw igw + ihw ] = λ 2 [Z, W ] + iλ Z gw + ihw iλ W gz + ihz = iλ { iλ [Z, W ] Z g W + ihw + W g Z + ihz} + +iλ {g Z W + ihw g W Z + ihz} = iλ { iλ [Z, W ] g [Z, W ] ih [Z, W ]} + d g Z, W ih d gz, W = iλ { iλ [Z, W ] + g [Z, W ] + ih [Z, W ]}, Jac [[ λz igz + ihz, λw igw + ihw ], λu igu + ihu] = iλ { iλjac [[Z, W ], U] + g Jac [[Z, W ], U] + ihjac [[Z, W ], U]} = O, A similar computation for E 0,1 Jac [[σ, τ], υ] = O. gives the statement, and the proof is complete. We remark that in the case λ = 0, integrability of is not sufficient to have the Jacobi identity on and E 0,1. Namely, in this case, we get the following: Proposition 5.5. [20] Let M, H, g be a complex Norden manifold, the Jacobi identity holds on and E 0,1 if and only if the following conditions are satisfied: 5.30 R D HY, HZ HR D HY, Z HR D Y, HZ R D Y, Z = O 5.31 R D HX, Y + R D X, HY Z + R D HZ, X + R D Z, HX Y + + R D Y, HZ + R D HY, Z X = O for all X, Y, Z C T M, where R D denotes the curvature operator of the natural canonical connection D on M, 5.32 R D X, Y = D X D Y D Y D X D [X,Y ]. In particular we have the following, [20]: Theorem 5.6. Let M, H, g be a Kähler Norden manifold then complex Lie algebroids. and E 0,1 are

14 64 Antonella Nannicini 6 The generalized -operator The following holds: H λg Proposition 6.1. Let M be a smooth manifold and let = g H be the generalized pseudo calibrated complex structure on M defined by a pseudo Riemannian metric g and a g symmetric operator H of T M such that H 2 = λi. The natural symplectic structure on E defines a canonical isomorphism between E 0,1 and the dual bundle of,. Proof. We define 6.1 φ : E 0,1 by: 6.2 φz + ihz + gw ihw + iz + iλw X ihx + gy + ihy ix iλy = = Z + ihz + gw ihw + iz + iλw, X ihx + gy + ihy ix iλy, for all X, Y, Z, W C T M C and we extend by linearity. A direct computation gives: 6.3 φz + ihz + gw ihw + iz + iλw X ihx + gy + ihy ix iλy = = gy, Z gw, X + i gw, HX + gy, HZ gx, Z. We have immediately that φ is injective and furthermore φ is an isomorphism. The canonical isomorphism φ between E 0,1 and the dual bundle allows us to define the operator associated to the complex structure, and to the fixed linear connection on M, as in the following: let f C M and let df C T M C T M T M, we pose 6.4 f = 2 df 0,1 = df + idf, or, 6.5 f = df ij df = df i df J; moreover, we define: 6.6 : C E 0,1 C 2 E 0,1

15 On a class of pseudo-calibrated generalized complex structures 65 via the natural isomorphism 6.7 E 0,1 as: 6.8 : C 6.9 for α C In general: φ, C 2 α σ, τ = ρ σ α τ ρ τ α σ α [σ, τ],, σ, τ C 6.10 : C p is defined by: 6.11 = p i=0 for α C p i ρ σ i α σ 0,..î.., σ p., σ 0,..., σ p C C p+1 α σ 0,..., σ p = + i j i+j α [σ i, σ j ], σ 0,..î..ĵ.., σ p, Definition 6.2. is called generalized operator of M, J, g, or generalized operator.. We get the following: Proposition 6.3. If 2 = 0. and E 0,1 are complex Lie algebroids then 2 = 0 and Proof. It follows from the fact that Jacobi identity holds on and. It turns out that operator is the exterior derivative, d L, of the Lie algebroid L =. Moreover the exterior derivative d L of L = is given by the operator defined by: 6.12 : C p C p+1 σ α0,..., αp = 6.13 = p i ρ αi α σ 0,..î.., αp + i j [α ] i+j σ i, αj, α 0,..î..ĵ.., αp i=0 for σ C p In particular C, α 0,..., α p C.,,,, [, ] is a differential Gerstenhaber algebra, where denotes the Schouten bracket, [13], [28].

16 66 Antonella Nannicini 7 Generalized holomorphic sections Definition 7.1. α C p 7.1 α = 0. is called generalized holomorphic section if We remark that for all f C M we have f = 0 if and only if df = 0, so the generalized holomorphic condition for functions gives only constant functions on connected components of M. The following holds: H λg Proposition 7.2. Let = be the pseudo calibrated generalized complex structure on M defined by a pseudo Riemannian metric g and a g symmetric g H operator H of T M, assume that H 2 = λi with λ 0. Let be a linear connection on M such that is integrable, let W C T M and let σ = λw + igw ihw E 0,1, then σ = 0 if and only if gw is a Lagrangian submanifold of T M with respect to the standard symplectic structure. Proof. Let X, Y C T M, direct computations give: σ λx igx + ihx, λy igy + ihy = 7.2 = λx 2iλg W, Y + λy 2iλgX, W + σ[λx, λy ] + igx + ihy, λy igλx, Y + ihy = 2iλ 2 {XgW, Y Y gw, X g[x, Y ], W }. In particular we have σ = 0 if and only if: 7.3 dgw X, Y = 0, and then, by using a classical result in symplectic geometry, [15], we have that σ = λw + igw ihw is a generalized holomorphic section of E 0,1 if and only if gw is a Lagrangian submanifold of T M with respect to the standard symplectic structure. Examples of generalized holomorphic sections can be obtained naturally in the field of Hessian geometry, [24], [26]. The concept of Hessian manifold was inspired by the Bergmann metric on bounded domains in C n and now is a very interesting topic, related to many other fields in mathematics and theoretical physics: Kähler and symplectic geometry, affine differential geometry, special manifolds, string theory and mirror symmetry, [6], [7], [8], [14], [26], [27]. Definition 7.3. Let M, g be a pseudo Riemannian manifold, g is called of Hessian type if there exists u C M such that g = Hessu = 2 u, where is the Levi Civita connection of g. M, g is called Hessian pseudo Riemannian manifold if g is of Hessian type.

17 On a class of pseudo-calibrated generalized complex structures 67 Proposition 7.4. Let M, H, g be a Hessian pseudo Riemannian manifold such that H is a g symmetric operator and H 2 = λi, λ 0. Let be the Levi Civita H λg connection of g, assume = is integrable. Let {x 1,..., x n } be { g H } local coordinates on M, let,..., be local frames for T M, if the curvature x 1 x n tensor of, R, vanishes, then for all k = 1,..., n the local section σ k = λ + ig ih C E 0,1 x k x k x k is closed. Proof. Let g = 2 u, then: g jk = 2 u x j x k l=1 Γ l jk u, x l { } where Γ l jk are Christoffel s symbols of g. In particular g is d closed if and x k only if for all i, j, k = 1,..., n : = = 0 3 u x j x k x i l=1 3 u + x i x k x j l=1 Γ l ik Γ l jk l=1 x i 2 u x l x j Γ l ik x j u x l r=1 u x l + Γ r lj l=1 Γ l jk Γ l ik 2 u + x l x i 2 u x l x j l=1 u Γ l 2 u jk x r x l x i r=1 Γ r li u x r or: = l=1 l=1 Γ l ik Γl jk + x j x i R i ijk u = 0 x l r=1 Γ r ik Γ l rj Γ r jkγ l ri u x l and thus, by using Proposition 7.2., we have the statement. The holomorphic sections in the case λ = 0 have been studied in [20]. Acknowledgement. The author acknowledges the support of PRIN Proprietà geometriche delle varietà geometriche reali e complesse of MIUR 2010NNBZ78 and of GNSAGA of INDAM.

18 68 Antonella Nannicini References [1] S. Amari, Differential-Geometrical Methods in Statistics, Lectures Notes in Statistics [2] A.M. Blaga, M. Crasmareanu, A class of almost tangent structures in generalized geometry, Balkan J. Geom. Appl , no. 2, [3] A. Borowiec, M. Francaviglia, I. Volovich, Anti-Kählerian manifolds, Diff. Geom. Appl , [4] V. Cruceanu, P. Fortuny, P. M. Gadea, A survey on paracomplex geometry, Rocky Mountain J. Math , [5] L. David, On cotangent manifolds, complex structures and generalized geometry, Annals-Institut Fourier , [6] J. Diustermaat, On Hessian Riemannian structures, Asian J. Math , [7] D. S. Freed, Special Kähler manifolds, Comm. in Math. Physics , [8] N. Hitchin, The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci , [9] N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford 54, 2003, [10] M. Gualtieri, Generalized Complex Geometry, Ann. of Math , [11] G. Ganchev, V. Mihova, Canonical connection and the canonical conformal group on an almost complex manifold with B-metric, Ann. Univ. Sofia Fac. Math. et Inf. 81, 1987, [12] T. Kurose, Statistical manifolds admitting torsion, Geometry and Somethong, Fukuoka University, [13] Y. Kosmann-Schwarzbach, Exact Gerstenhaber Algebras and Lie Bialgebroids, Acta Applicandae Mathematicae , [14] J. Loftin, S.T. Yau, E. Zaslow, Affine manifolds, SYZ Geometry and the Y vertex, Journal of Differential Geometry , [15] D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford Science Publications [16] A. Nannicini, Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold, Journal of Geometry and Physics , [17] A. Nannicini, Almost complex structures on cotangent bundles and generalized geometry, Journal of Geometry and Physics , [18] A. Nannicini, Special Kähler manifolds generalized geometry, Differential Geometry and its Applications , [19] A. Nannicini, Generalized geometry of pseudo-riemannian manifolds and generalized operator, Adv. Geom , [20] A. Nannicini, Generalized geometry of Norden manifolds, Journal of Geometry and Physics , [21] M. Nogushi, Geometry of statistical manifolds, Differential Geometry and its Applications ,

19 On a class of pseudo-calibrated generalized complex structures 69 [22] A. P. Norden, On a class of four-dimensional A-spaces, Russian Math. Izv VUZ , [23] J. Pradines, Théorie de Lie pour les groupoides différentiables. Calcul différentiel dans la catégorie des groupoides infinitésimaux, C.R.Acad. Sc. Paris, t. 264, Série A, 1967, [24] A. A. Salimov, A. Gezer, Norden structures of Hessian type, Turk. J. Math , [25] A. A. Salimov, M. Iscan, F. Etayo, Paraholomorphic B-manifolds and its properties, Topology and its Applications , [26] H. Shima, K.Yagi, Geometry of Hessian manifolds, Differential geometry and its applications , [27] B. Totaro, The curvature of a Hessian manifold, Int. J. Math , [28] P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys , Author s address: Antonella Nannicini Università di Firenze, Italy, Dipartimento di Matematica e Informatica U. Dini Viale Morgagni 67/a Firenze, Italy. antonella.nannicini@unifi.it