GENERALIZED QUATERNIONIC STRUCTURES ON THE TOTAL SPACE OF A COMPLEX FINSLER SPACE

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1 Bulletin of the Transilvania University of Braşov Vol 554, No Series III: Mathematics, Informatics, Physics, GENERALIZED QUATERNIONIC STRUCTURES ON THE TOTAL SPACE OF A COMPLEX FINSLER SPACE Annamária SZÁSZ 1 Communicated to the conference: Finsler Extensions of Relativity Theory, Braşov, Romania, Aug Sept. 4, 2011 Abstract In this note our goal is to introduce a generalized quaternionic structures, on the total space of a complex Finsler space. Some important properties of this structures are emphasized. A special approach is devoted to the commutative almost quaternionic connections Mathematics Subject Classification: 53B40, 53C26. Key words: commutative quaternion structure; generalized Sasaki metric; almost hyper-hermitian structures; almost hyper-kähler structures; metric compatible connections. 1 Introduction Let M, F be a complex Finsler manifold, i.e., M is a smooth manifold and F is a Finsler metric on M. In this paper, we introduce the following metric G on T M cf. Section 3: Gz, η = g i jz, ηdz i d z j alg i jz, ηδη i δ η j, 1.1 where a : ImL R R, and L := F 2. We define an almost hyper-complex structure G, J 1, J 2, on the complexified holomorphic tangent bundle T M of a complex manifold M, where J 1 is the natural complex structure and J 2 is an almost complex structure defined by the help of a = al. We demonstrate that T M, J 1, J 2, J 3 is a commutative quaternion structure, [Mu2], where J 3 = J 1 J 2. In the rest of the 4 we are concerned with the integrability conditions of the structures. How J 1 is the natural complex structure, his Nijenhuis tensor field is vanishing, but the case of J 2 is more complicated. Theorem 3.3. is the main result of this section, and tells when the J 1, J 2 structure is integrable. 1 Transilvania University of Braşov, Faculty of Mathematics and Informatics Iuliu Maniu 50, Braşov , Romania, am.szasz@yahoo.com

2 86 Annamária Szász In Section 4 we evaluate under what conditions T M, G, J 1, J 2 an almost hyper- Hermitian is almost hyper-kählerian or at least hyper-kählerian structure. This conditions are in the Proposition 4.1., Theorem 4.2. and Theorem 4.3. The last part ot this paper is about the construction of a metric compatible linear connection with the commutative quaternion structure J 1, J 2. 2 Preliminaries Let M be a complex manifold, dim C M = n and z k local complex coordinates in a chart U, ϕ. The holomorphic tangent bundle T M has a natural structure of complex manifold, dim C T M = 2n, and the induced coordinates in a local chart in u T M are u = z k, η k. The changes of local coordinates in u are given by: z k = zh z k η k = zh z k η h. z h 2 z h z j z k ; 2.2 η h Consider the sections of the complexified tangent bundle of T M. Let V T M T T M be the vertical bundle, locally spanned by { }, and V T M its conjugate. The idea of η k complex nonlinear connection, briefly c.n.c., is an instrument in linearization of the geometry of T M manifold. A c.n.c. is a supplementary complex subbundle to V T M in T T M, i.e. T T M = HT M V T M. The horizontal distribution H u T M is locally spanned by δ δz k = z k N j k η j, 2.3 where N j k z, η are the coefficients of the c.n.c.. The pair {δ k :=, δz k k := } will be η k called the adapted frame of the c.n.c. which obey to the change rules δ k = zj δ z k j and k = zj z k j. By conjugation everywhere we obtain an adapted frame {δ k, k} on T u T M. The dual adapted bases are {dz k, δη k } and {d z k, δ η k }. The action of natural complex structure on T C T M is wich in view of 2.3 yields J k = i k ; J k = i k ; J k = i k; J k = i k 2.4 Jδ k = iδ k ; J k = i k ; Jδ k = iδ k; J k = i k 2.5 and hence HT M and HT M are J invariant. The base manifold of a complex Finsler space is T M and the main objects of this geometry operate on the section of the complexified tangent bundle T C T M, which itself is decomposed into horizontal, vertical and their conjugates subbundles by a complex nonlinear connection N, uniquely determined by the complex Finsler function, [3], [5]. δ

3 Generalized quaternionic structures on the total space of a complex Finsler space 87 Definition 2.1. A complex Finsler metric on M is a continuous function F : T M R satisfying: a L := F 2 is smooth on T M := T M \ {0}. b F z, η 0, the equality holds if and only if η = 0; c F z, λη = λ F z, η for λ C; d the Hermitian matrix is positive-definite on T M. g i jz, η = 2 L η i η j Definition 2.2. The pair M, F is called a complex Finsler space. The assertion c says that L is positively homogeneous with respect to the complex norm, i.e. Lz, λη = λ λlz, η for any λ C. The assertion d allows us to define a Hermitian metric structure on T M, because g i j is a d-tensor complex nondegenerate, called in [5] as the fundamental metric tensor of the complex Finsler space M, F, with the inverse g ji, and g ji g i k = δ j k. The homogenity condition of the complex Finsler metric allows us to enumerate some important results. Applying the Euler s Theorem for L = F 2, we have: Proposition 2.1. The complex Finsler metric satisfies the conditions a L η k η k = L; L η k η k = L; b g i jη i = L η j ; g i j η j = L η i ; L = g i jη i η j ; c g i j η k η k = 0; g i j η k η k = 0; g i j η k η i = 0; d g ij η i = 0; g i j η k η j = g ik, where g ij = 2 L η i η j. A fundamental problem in a complex Finsler space remains that of determinating the c.n.c function only on complex Finsler metric F. A well-known solution is provided by the complex Chern-Finsler connection, [3]. Determined from the technique of good vertical connection, it is proved that the Chern-Finsler connection is a unique N c.l.c of 1, 0- CF CF type. With the notations in [5], the Chern-Finsler connection is: DΓ = Nj i CF, L i CF jk, Cjk i, where and CF Nj i = g mi g l m z j ηl = g mi 2 L z j η m ; CF CF Lī jk = Cī jk = 0. CF L i jk = g mi δg j m δz k ; With a straightforward computation we obtained that CF C i jk = g mi g j m η k, 2.6 CF L i jk = j CF N i k.

4 88 Annamária Szász Observation 2.1. As a direct consequance of 2.6 it results L δ k L = δ kl = δ k η j = Observation 2.2. We will use the notation for the Chern-Finsler connection whithout the indexes CF. Locally, in adapted frame fields of the c.n.c Chern-Finsler N, the components of the Lie brackets are: A simple computation get: [δ j, δ k ] = δ k N i j δ j N i k i = 0; 2.8 [δ j, δ k] = δ kn j i i δ j N [δ j, ] k = k Nj i i ; [ ] δ j, k = kn j i i ; [ j, ] [ k] k = 0; j, = 0; ī k ī; kn i jg i m = m N i jg i k. 2.9 Now we can add that the nonzero torsion of the complex Chern-Finsler connection are only: T l jk = Ll jk Ll kj = j N l k k N l j; Q l jk = Cl jk ; Θl j k = δ k N l j; ρ l j k = kn l j 2.10 In the terminology of Abate and Patrizio, [3], the complex Finsler space M, F is strongly Kähler iff T i jk = 0, Kähler iff T i jk ηk = 0, and weakly Kähler iff g i lt i jk ηk η l = 0. In [4] is proved that the strongly Kähler and the Kähler notions actually coincide. 3 Integrabilty of J 1, J 2, J 3 structure Consider a generalized Sasaki metric G on T M given by Gz, η = g i jz, ηdz i d z j alg i jz, ηδη i δ η j, 3.11 where a : ImL R R. Let J 1 the natural complex structure on T M, and J 2 an other almost complex structure on T M defined by: J 1 δ k = iδ k ; J 2 δ k = 1 a k 3.12 J 1 δ k = iδ k ; J 2 δ k = 1 a k J 1 k = i k ; J 2 k = aδ k J 1 k = i k ; J 2 k = aδ k.

5 Generalized quaternionic structures on the total space of a complex Finsler space 89 We will denote whith J 3 := J 1 J 2. The following relations are true: J 2 1 = J 2 2 = I, J 2 3 = I 3.13 J 1 J 2 = J 2 J 1 = J 3 J 1 J 3 = J 3 J 1 = J 2 J 2 J 3 = J 3 J 2 = J 1 Using 3.12 and 3.13, according to [6], we obtain the next theorem: Theorem 3.1. T M, J 1, J 2, J 3 is a commutative quaternion structure. Now we shall study the integrability problem for the obtained almost commutative quaternion structure. The integrability conditions for such a structure are expressed with the help of various Nijenhuis tensor fields obtained from the tensor fields J 1, J 2,J 3 = J 1 J 2. For a tensor field K of type 1,1 on a given manifold, we can consider its Nijenhuis tensor field N K defined by N K X, Y = [KX, KY ] K [X, KY ] K [KX, Y ] K 2 [X, Y ], where X, Y are vector fields on the given manifold. For two tensor fields K, L of type 1,1 on the given manifold, we can consider the corresponding Nijenhuis tensor field N K,L defined by N K,L X, Y = [KX, LY ] [LX, KY ] K[X, LY ] [LX, Y ] L[KX, Y ] [X, KY ] KL LK [X, Y ]. The almost commutative quaternion structure defined by J 1, J 2, J 3 is integrable if N 1 = 0, N 2 = 0, where N 1, N 2 are the Nijenhuis tensor fields of J 1, J 2. Equaivalently, the structure is integrable if N 1 N 2 N 3 = 0, or if N 12 = 0, where N 3 is the Hijenhuis tensor field of J 3 = J 1 J 2 and N 12 is the NIjenhuis tensor field of J 1, J 2. Since J 1 is the natural complex structure, then it is integrable, i.e. N 1 = 0. Remains the study of N 2. With the help of the Lie brackets 2.8 we have obtained: N 2 [δ j, δ k ] = a L 2a 2 η j δl k L η k δl j l L i kj Li jk δ l N 2 δ j, δ k = a L 2a 2 η k j L η j k δ kn j l l δ j N l k l j N l kδ l kn jδ l l N 2 δ j, k = a L 2a η k δ j L η j δ k j Nk l l k Nj l l N 2 δ j, k = a L 2a η k δ j L η j δ k a δ j N l kδ l δ kn jδ l l j N l k j kn j l l N 2 j, k = a L 2a η j k L η k j a j Nk l δ l k Njδ l l N 2 j, k = a L 2a η j k L η k j a δ kn j l l δ j N l k l j N l kδ l kn jδ l l

6 90 Annamária Szász We have replaced the expressions for the torsions 2.10 in the above relations, and is obtain: N 2 δ j, δ k = a L 2a 2 η j δl k L η k δl j l Tjk l δ l N 2 δ j, δ k = a L 2a 2 η k j L η j k ρ l jk δ l ρl j k δ l Θ l j k l Θ l kj l N 2 δ j, k = a 2a N 2 δ j, k = a 2a N 2 j, k = a 2a N 2 j, k = a 2a L η k δl j L η j δl k L η k δ j L η j δ k L η j δl k L η k δl k L η j k L η k j δ l Tjk l l a Θ l kj δ l Θ l j k δ l ρ l kj l ρ l j k l l atjk l δ l a Θ l j k l Θ l kj l ρ l kj δ l ρ l j k δ l From the linear independence of the base fields it results that T M, G, J 2 is complex if and only if: and their conjugates. L a η j δl k L η k δl j = 0 and T l jk = 0 Θ l j k = 0 and ρ l j k = 0, 3.14 Theorem 3.2. The manifold T M, G, J 2 is complex if and only if M, F is Kähler, the torsions Θ l j k and ρl are zero and j k L a η j δl k L η k δl j = Corollary 3.1. T M, G, J 2 is a complex manifold if and only if M, F is a generalized complex Berwald space and Θ i j k = 0. Observation 3.1. The notion of generalized complex Berwald space is described in [AM]. We have seen, that J 1 is integrable, J 2 is integrable when the conditions in the Theorem 3.2. are fullfield, then the integrability condition for the J 1, J 2, J 3 quaternion structure are in the next theorem: Theorem 3.3. The commutative quaternion structure J 1, J 2, J 3 is intergable if and only if M, F is a generalized complex Berwald space and Θ i j k = 0.

7 Generalized quaternionic structures on the total space of a complex Finsler space 91 4 Hyper-Kähler Structures on T M The structure defined in 3.13 is one hypercomplex four dimensional. Moreover, T M, G, J 1, J 2 has an almost hyper-kählerian structure if the following conditions are satisfied: a T M, G, J 1, J 2 is an almost hyper-hermitian manifold; b The fundamental 4-form Ω is closed. For the point a we shall be interested in the conditions under wich the metric G is almost Hermitian with respect to the almost complex structures J 1, J 2, considered in 3.12, i.e. GJ 1 X, J 1 Y = GX, Y GJ 2 X, J 2 Y = GX, Y, X, Y T C T M. We have Proposition 4.1. T M, G, J 1 T M, G, J 2 and T M, G, J 3 are almost Hermitian manifolds, i.e. GJX, JY = GX, Y X, Y. Proof. For J 1 the condition GJ 1 X, J 1 Y = GX, Y is { verified imediatly, } and for J 2 it s enough to verify for the elements of the adapted frame δ k, k, δ k, k the above relations. The nonzero values of GJ 2 X, J 2 Y are GJ 2 δ j, J 2 δ k = G 1 1 j, a a k = 1 a G j, k = = 1 a ag j k = g j k = Gδ j, δ k GJ 2 j, J 2 k = G aδ j, aδ k = agδ j, δ k = = a g j k = G j, k, For the almost hyper-hermitian manifold T M, G, J 1, J 2 the fundamental 2 forms φ 1, φ 2 are defined by φ 1 X, Y = GX, J 1 Y, φ 2 X, Y = GX, J 2 Y, where X, Y are vector fields on sections of T C T M. Since we have a third almost complex structure J 3 = J 1 J 2 which is almost Hermitian with respect to G, we can consider a third 2 form φ 3 defined by φ 3 X, Y = GX, J 3 Y, next we have the fundamental 4-form Ω, defined by Ω = φ 1 φ 1 φ 2 φ 2 φ 3 φ 3. The almost hyper-hermitian manifold T M, G, J 1, J 2 is almost hyper-kählerian if the fundamental 4-form Ω is closed, i.e. dω = 0. The condition for Ω to be closed is equivalent

8 92 Annamária Szász to the conditions for φ 1, φ 2 and hence for φ 3 too to be closed, i.e. dφ 1 = 0, dφ 2 = 0. In our case, it is more convenient to study the conditions under which the 2-forms φ 1, φ 2 are closed. The expressions of φ 1, φ 2 in adapted local frames are φ 1 z, η = ig j kdz i d z j ialg j kδη j δ η k φ 2 z, η = alg j kdz j δ η k alg j kδη j d z k With a straightforward computation using properties of the Chern-Finsler c.n.c. results { dφ 1 = i δ i g j kdz i dz j d z k δīg j kd z i dz j d z k i ag j kδη j δ η k δη i īag j kδη j δ η k δ η i [ i g j kδη i dz j d z k ag j kδ h N k l δηj d z l dz h] [ īg j kδ η i dz j d z k ag j kδ hn j l δ ηk dz l d z h] [ δ i ag j kdz i δη j δ η k ag j k h N j l dzl δη h δ η k] [ δīag j kd z i δη j δ η k ag hn k j k l δηj d z l δ η h] ag hn j j k l dzl δ η h δ η k ag j k h Nl kδηj d z l δη h} = { 1 = i 2 δ ig j k δ j g i kdz i dz j d z k 1 2 δ īg j k δ jg i kd z i dz j d z k a L η i g j k δηj δ η k δη i a L η i g j k δηj δ η k δ η i [ ] [ ] i g j k ag i lδ j Nk l dz j d z k δη i īg j k ag l hδ kn h l dz j d z k δ η i} So we have deduced Theorem 4.1. The manifold T M, G, J 1 is Kähler if and only if: a L = 0 al = c R δ i g j k = δ j g i k 4.18 g li i g j k = aδ j N l k and their conjugates.

9 Generalized quaternionic structures on the total space of a complex Finsler space 93 Analogous for dφ 2 we have: dφ 2 = 2 a δ ilg j k aδ i g j k dz j δ η k dz i a δ ilg j k aδ i g j k δη j d z k dz i 2 a δ īlg j k aδīg j k dz j δ η k d z i 2 a δ īlg j k aδīg j k δη j d z k d z i 2 a i Lg j k a i g j k dz j δ η k δη i 2 a i Lg j k a i g j k δη j d z k δη i 2 a īlg j k a īg j k dz j δ η k δ η i 2 a īlg j k a īg j k δη j d z k δ η i ag j kδ h N k l dzj d z l dz h ag j k hn k l dzj d z l δ η h ag j k h N k l dzj d z l δη h ag j kδ hn j l dzl d z h d z k ag j k h N j l dzl δη h d z k ag j k hn j l dzl δ η h d z k. Theorem 4.2. The almost complex manifold T M, G, J 2 is almost Kähler if and only if one of the next condition sets are fullfield: a = 0 and i g j k = 0; 4.20 or and their conjugates. δ i g j k = δ j g i k, Θ k lh = 0, L l ki g l j = k N l j g i l, Using the integrability conditions for J 2 in Theorem 3.2., we obtain: a 2a i Lg j k = i g j k, 4.21 Theorem 4.3. The manifold T M, G, J 2 is Kähler if and only if, one of the next condition sets are fullfield: and their conjugates. a = 0 and G is purely Hermitian; 4.22 or L l ki g l j = 0, a 2a i Lg j k = i g j k, 4.23

10 94 Annamária Szász Corollary 4.1. The structure T M, G, J 1, J 2, J 3 is Hyper-Kählerian if and only if M, F is a complex Berwald manifold with Θ i j k = 0, a = 0, G is purely Hermitian, and or a = 0 or L l ki g l j = 0. 5 Metric compatible linear connection with the commutative quaternion structure Further we will deal with linear connections compatible with a commutative quaternion metric structure. Definition 5.1. A linear connection D on T M is called metric connection commutative quaterion if: DJ i = 0, i = 1, 2, 3; and DG = The general family of the linear connections D compatible with the metric G, according to [6], is where Ď is an arbitrary linear connection. Let us consider the connection transformations: D X Y = ĎXY 1 2 g 1 ĎXg Y, 5.25 Ď X Y T 1 DXY 1 = ĎXY 1 2 J 1ĎXJ 1 Y 5.26 Ď X Y T 2 DXY 2 = ĎXY 1 2 J 2ĎXJ 2 Y 5.27 Ď X Y T 3 DXY 3 = ĎXY 1 2 J 3ĎXJ 3 Y 5.28 Ď X Y T 4 DXY 4 = ĎXY 1 2 ĎXg Y 5.29 where ĎXg Y is a 1-form defined as follows ĎXg Y Z = ĎXgY, Z. Obviously D i X J i = 0, i = 1, 2, 3, X T C T M. Then, according to [6], we consider the commutative quaternion connection: D X Y = 1 4 {ĎX Y J 1 ĎXJ 1 Y J 2 ĎXJ 2 Y J 3 ĎXJ 3 Y } 5.30 where Ď is an arbitrary linear connection. Proposition 5.1. The following relation is true: DD 4 = D 4 D, where DD 4 respectively D 4 D is a connection obtained from D respectively D 4 by replacing Ď with D4 respectively D.

11 Generalized quaternionic structures on the total space of a complex Finsler space 95 Proof. DD 4 X Y = 1 { D 4 4 X Y J 1 DXJ 4 1 Y J 2 DXJ 4 2 Y J 3 DXJ 4 3 Y } = 5.31 = 1 { Ď X Y G 1 ĎXG Y J 1 ĎXJ 1 Y 1 2 G 1 ĎXG J1 Y J 2 ĎXJ 2 Y 1 2 G 1 ĎXG J2 Y J 3 ĎXJ 3 Y 1 } 2 G 1 ĎXG J3 Y = On the other hand: = 1 4 {ĎX Y J 1 ĎXJ 1 Y J 2 ĎXJ 2 Y J 3 ĎXJ 3 Y } 1 { G 1 ĎXG Y J 1 G 1 ĎXG 8 J1 Y J 2 G 1 ĎXG J2 Y J 3 G 1 } ĎXG J3 Y D 4 DX Y = D X Y 1 2 G 1 D X G Y = 5.32 = 1 4 {ĎX Y J 1 ĎXJ 1 Y J 2 ĎXJ 2 Y J 3 ĎXJ 3 Y } 1 8 G 1 { ĎXG Y J 1 ĎXG J1 Y J 2 ĎXG J2 Y J 3 ĎXG J3 Y } = = 1 4 {ĎX Y J 1 ĎXJ 1 Y J 2 ĎXJ 2 Y J 3 ĎXJ 3 Y } 1 8 { G 1 ĎXG Y G 1 J 1 ĎXG J1 Y G 1 J 2 ĎXG J2 Y G 1 J 3 ĎXG J3 Y }, where D X G Y Z = D X GY, Z. Therefore DD 4 = D 4 D. Theorem 5.1. The following linear connection: D X Y = DD 4 X Y, X, Y T C T M or equivalently D X Y = 1 4 {ĎX Y J 1 ĎXJ 1 Y J 2 ĎXJ 2 Y J 3 ĎXJ 3 Y } { G 1 ĎXG Y G 1 J 1 ĎXG J1 Y G 1 J 2 ĎXG J2 Y G 1 } J 3 ĎXG J3 Y 8 is a metric commutative quaternion connection, where on T M. Ď an arbitrary linear connection Proof. D X J i = 0, i = 1, 2, 3, because D is obtained from D that is commutative quaternion by replacing the arbitrary connection with D 4. Similary, D X G = 0 because, based on Proposition 5.1., D = D 4 D, i.e. D is obtained from the metric connection D 4 by replacing the arbitrary connection with D.

12 96 Annamária Szász Theorem 5.2. If is the Levi-Civita connection defined by the metric G, then the connection ˆD X Y = 1 4 { XY J 1 X J 1 Y J 2 X J 2 Y J 3 X J 3 Y } 5.34 has properties a ˆD X G = 0, ˆD X J i = 0, i = 1, 2, 3, X T C T M; b ˆD is uniquily determined by the metric commutative quaternion structure. Proof. Both results from 5.33 considering X G = 0. The local expression of the Levi-Civita connection defined by the metric G will be studied in a forthcoming paper. References [1] Aldea, N., The Holomorphic Flag Curvature of the Kählr Model of a Complex Lagrange Space, Bull. of the Transilvanian Univ. Braşov 944, 2002, [2] Aldea, N., Munteanu G., On complex Landsberg and Berwald spaces, Journal of Geometry and Physics , [3] Abate, M., Patrizio, G., Finsler Metrics - A Global Approach, Lecture Notes in Math., 1591, Springer-Verlag, [4] Chen, B., Shen, Y., Kähler Finsler metrics are actually strong Kähler, Chin. Ann. Math. Ser. B , [5] Munteanu, Gh., Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Acad. Publ., [6] Munteanu, Gh., Metric Almost Semiquaternion Structures, Bull. Math. Soc. Sci. Roumanie, , no.4, [7] Oproiu, V., Hyper-Kähler structures on the tangent bundle of a Kähler manifold, Balkan J. of Geom. and its Appl., 15, no.1, 2010, [8] Peyghana, E., Tayebi, A., Finslerian complex and Kählerian structures, Nonlinear Analysis: Real World Applications ,