On Nearly Kähler Finsler Spaces

Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. An International Journal ISSN: 1932-9466 (AAM)

Vol. 14, Issue 2 (December 2019), pp. 1243 – 1268

1Z. Didehkhani, 2∗B. Najaﬁ and 3N. Heidari

1Faculty of Science, Department of Mathematics Shahed University Tehran. Iran [email protected]

2Department of Mathematics and Computer Sciences Amirkabir University Tehran. Iran behzad.najaﬁ@aut.ac.ir

3Nikrooz Heidari Department of Mathematics University of Mohaghegh Ardabili (UMA) Ardabil.Iran [email protected]

∗Corresponding Author

Received: April 17, 2019; Accepted: September 19, 2019

Abstract

Ichijyo¯ introduced (a, b, J)-manifolds as a special class of generalized Randers manifolds. We introduce generalized (a, b, J)-manifolds. A partial negative answer to Ichijyo’s¯ open problem on nearly Kähler Finsler manifolds is given. The condition under which generalized (a, b, J)- manifolds are Berwaldian is obtained. Finally, we prove that under a mild assumption a nearly Kähler Finsler manifold is Landsbergian.

Keywords: Rizza manifold,(a, b, J)-metric; Nearly Kählerian Finsler manifold; Projective equivalent; Berwald metric; Landsberg metric

MSC 2010 No.: 53B40, 53B35 1243 1244 Z. Didehkhani et al.

1. Introduction

Riemannian geometry has many important subcategories, such as Hermitian geometry, including Kähler geometry. The first discussions and analyses of Riemannian and semi-Riemannian nearly Kähler manifolds emerged during the 1970s with Alfred Gray’s works. Gray and Hervella classi- fied almost Hermitian manifolds in 16 classes in 1976 (Gray and Hervella (1980)), one of these classes, known as nearly Kähler manifolds, attracted attention. In his later papers (Gray (1970); Gray (1972); Gray (1976)), Gray studied these manifolds in great detail and surveyed some of the topological characteristics of these manifolds. In (Gray (1972)) he explored the relationship between the nearly Kähler manifolds and the 3-symmetric spaces and proposed a basic classifi- cation theorem and provided a decomposition for the nearly Kähler manifolds to the Kähler and strictly nearly Kähler manifolds, which had nice characteristics, such as Einsteinism and the vanishing of the first Chern class (Gray (1976)) (cf. Nagy’s papers (Nagy)). It was Grunewald who proved that in dimension 6, nearly Kähler manifolds are related to the existence of a Killing spinor (Grunewald). Relation and position of nearly Kähler manifolds in string theory were revealed by Friedrich and Ivanove’s papers (Friedrich and Ivanov (2002)). In 2002, they expressed that nearly Kähler manifolds admit a Hermitian connection with totally skew-symmetric torsion.

The study of homogeneous nearly Kähler manifolds began with Buteruille (Butruille (2010)). He classiﬁed six-dimensional complete homogeneous nearly Kähler manifolds and responded positively to the Gray-Wolf conjecture that every homogeneous nearly Kähler manifold is a Rieman- nian 3-symmetric space. According to the Butruille, only the 6-dimensional, complete, homogeneous and simply connected strictly nearly Kähler manifolds examples are S3 × S3, the complex projective space CP 3, the ﬂag manifold F3, and the sphere S6, all of which are 3-symmetric.

Finsler geometry, a natural generalization of Riemannian geometry, has been considered noticeable in recent years particularly for its applications in physics, biology, etc. Randers metrics F = α+β, where α is a Riemannian metric and β is a 1-form, are the ﬁrst and easiest examples that come to the β mind from the Finslerian manifolds, which are a special case of (α, β)- metrics, i.e., F = αϕ( α ). Most of the theorems in Finsler geometry were ﬁrst investigated for Randers metrics, and possibly later they were probed for a more general case. Recently, researchers have shown an increased interest in Finsler spaces equipped with complex structures. Several attempts have been made to illuminate nature of these spaces (Fukui; Heil; Prakashi). In complex Finsler spaces quite a few efforts had been done to introduce Hermitian manifolds and extend this concept to Finslerian space, but after encountering a few failures, it was understood that this is not a light task. The correct generalization of Hermitian manifolds in Finslerian setting was done by Rizza (Rizza (1962); Rizza (1963)) and we call an almost Hermitian Finsler manifold, a Rizza manifold.

Considering Finsler metric’s role in physics and biology and on the other hand the role of nearly Kähler Riemannian manifolds in theoretical physics, it seems that study of nearly Kähler manifolds in Finslerian case may lead to a new approach and understanding of both mathematics and physics phenomenon. The systematic study of Kählerian Finsler manifold was done by Ichijyo.¯ He introduced a special class of Rizza manifolds and named them (a, b, J)- manifolds, which are special generalized Randers metrics, and by contributing of these manifolds, he demonstrated examples AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1245

of Kählerian Finsler manifold (Ichijyo¯ (1988)-Ichijyo¯ and Hashiguchi (1995b)). Preliminary work on nearly Kähler Finsler manifolds was undertaken by Ichijyo.¯ By presenting concepts of normal and nearly normal manifolds, Ichijyo¯ in (Ichijyo¯ and Hashiguchi (1995b)) made an example of nearly Kähler Finsler manifolds. H-S Park and H-T Lee studied nearly Kähler Finsler structures as well and extended some theorems which was obtained by Ichijyo¯ for Kählerian Finsler manifolds to nearly Kähler Finsler manifolds under some additional conditions (Park (1993); Park and Lee (1993)).

In the purpose of achieving more instances of nearly Kähler Finsler spaces, Ichijyo¯ questioned that with having a nearly Kähler Riemannian manifold (M, α, J), can we ﬁnd a non-zero 1-form b on M such that the corresponding (a, b, J)-manifold becomes a nearly Kählerian Finsler manifold? This paper comes out with a negative answer, in nearly normal case, to Ichijyo¯’s question (see Proposition 3.3).

Two Finsler metrics are said to be projectively equivalent if they have the same geodesics as point sets. Hilbert’s fourth problem deals with projectively ﬂat metrics. Here, a situation under which an (a, b, J)-metric and generalized (a, b, J)-metric are projectively equivalent to their Riemannian parts is attained at Proposition 4.11 and Proposition 4.12, respectively.

Non-Riemannian curvatures such as Berwald and Landsberg curvatures have a major position in Finsler geometry. It is known that Berwlad metrics are nearest Finsler metrics to Riemannian ones. We characterize Berwaldian generalized (a, b, J)-manifolds (see Theorem 4.8). In (Ichijyo¯ (1994)), Ichijyo¯ showed that a Kählerian Finsler manifold is Landsbergian. We also acquire the condition under which the same thing holds in nearly Kähler case (see Theorem 4.14).

2. Preliminaries

∞ Let M be an n-dimensional C manifold. Denote by TxM the tangent space at x ∈ M, and by TM = ∪x∈M TxM the tangent bundle of M. Let TMg be TM − {0}, i.e., the slit tangent bundle of M and let ρ : TMg → M be the natural projection. A Finsler metric on M is a function F : TM → [0, ∞) which has the following properties: (i) F is C∞ on slit tangent bundle, (ii) F is positively 1-homogeneous on the fibers of tangent bundle TM, and (iii) for each y ∈ TxM, the following quadratic form gy on TxM is positive definite, 1 ∂2 g (u, v) := F 2(y + su + tv) | , u, v ∈ T M. y 2 ∂s∂t s,t=0 x The geodesic curves of a Finsler metric F on a smooth manifold M are determined by the system of 2ed order differential equations cï + 2Gi(˙c) = 0, where the local functions Gi = Gi(x, y) are called the spray coefficients and defined by 1 ∂2F 2 ∂F 2 Gi = gil − , (1) 4 ∂xk∂yl ∂xl 1 ∂2F 2 ij −1 i where gij := 2 ∂yi∂yj and (g ) = (gij) . F is called a Berwald metric, if G are quadratic in i 1 i j k i y ∈ TxM for any x ∈ M, and called a Douglas metric if G = 2 Γjk(x)y y + P (x, y)y (Bácsó 1246 Z. Didehkhani et al.

1 ∂F i i ∂3Gi and Matsumoto (1997); Najafi et al. (2007)). Let Ljkl := − 2 F ∂yi Bjkl, where Bjkl := ∂yj ∂yk∂yl is the Berwald curvature of F . The Landsberg curvature Ly : TxM ⊗ TxM ⊗ TxM → R is defined i j k by Ly(u, v, w) := Lijk(y)u v w . A Finsler metric is called a Landsberg metric if L = 0. It is easy to see that every Berwald metric is a Landsberg metric, but the converse is still an open problem in Finsler geometry. On a Berwald manifold (M,F ), all tangent spaces TxM with the induced Minkowski norm Fx are linearly isometric. Moreover, F is affinely equivalent to a Riemannian metric g, namely, F and g have the same spray. Using this fact, S. Zabó determined the local structure of Berwald metrics (Chern and Shen (2004)). Let x ∈ M and Fx := F |TxM . Putting i ∂Gi i i i Gj = ∂yj , the Cartan connection of the Finsler metric F is denoted by CF = (Fjk,Gj,Cjk) and defined by the following, 1 1 ∂g ∂g ∂g F i = gir(δ g + δ g − δ g ),Ci = gir jr + rk − kj , (2) jk 2 k jr j rk r kj jk 2 ∂yk ∂yj ∂yr

∂ i ∂ i ir 1 ∂gij where δk = ∂xk − Gk ∂yi . Indeed, Cjk = g Crjk, where Cijk = 2 ∂yk is the Cartan tensor of F , which measures the non-Euclidean feature of F . It is well known that C = 0 if and only if i F is Riemannian. For y ∈ TxM0, deﬁne the mean Cartan torsion Iy by Iy(u) := Ii(y)u , where jk Ii := g Cijk. By Deicke Theorem, F is Riemannian if and only if I = 0.

i For any Finsler tensor Sj(x, y), the h-covariant and v-covariant derivatives with respect to CΓ are deﬁned as follows, respectively δSi Si = j + SmΓi − Si Γm , (3) j|k δxk j mk m jk

i i ∂Sj m i i m S = + Sj Cmk − SmCjk. (4) j k ∂yk

i i ∂Gj i i Putting Gjk = ∂yk , it is well known that Fjk and Gjk are positively homogeneous functions of i i k degree 0 with respect to y and the relation Gj = Fjky holds. Moreover, we have this important i i i i it identity Fjk = Gjk − Ljk, where Ljk = g Ltjk. Unlike Riemannian geometry, there are more than one Finsler connection associated to any Finsler metric. The Berwald connection of the Finsler i i i metric F is deﬁned by BΓ = (Gjk,Gj, 0). For any Finsler tensor Sj(x, y), the h-covariant derivative with respect to BΓ is denoted and deﬁned as follows (Matsumoto (1986)) δSi Si = j + SmGi − Si Gm . (5) j;k δxk j mk m jk

Deﬁnition 2.1. Two Finsler metrics F and F¯ on a manifold M are said to be projectively equivalent if they have the same geodesics as point sets. More precisely, for any geodesic σ¯(t) of F¯, after an appropriate oriented reparametrization, t¯ = t¯(t), the new map σ(t) :=σ ¯(t¯(t)) is a geodesic of F, and vice versa.

Deﬁnition 2.2.

A Finsler metric F = F (x, y) on an open subset U ∈ Rn is said to be projectively ﬂat if all geodesics are straight in U, i.e., a(t) = f(t)a + b for some constant vectors a, b ∈ Rn. A Finsler AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1247

metric F on a manifold M is said to be locally projectively ﬂat if at any point, there is a local coordinate system (xi) in which F is projectively ﬂat.

Let us recall Rapcsak’s characterization of two projectively related Finsler metrics.

Theorem 2.3. (Rapcsak’s Theorem (Chern and Shen (2004))) Let F and F¯ be Finsler metrics on a manifold M. F is projectively equivalent to F¯ if and only if F satisﬁes the following system, k F;k.ly − F;l = 0, (6) k i ¯i i F;k in which case, their spray coefﬁcients are related by G = G + P y where P = y 2F . Here ¯ F;k denote the horizontal covariant of F with respect to the Berwald connection of F and F;k.l = (F;k)yl .

i j ∂ Let M be a 2n-dimensional manifold. A (1, 1)-tensor field J = J jdx ⊗ ∂xi on M is called an almost complex structure on M if J ◦ J = −ITM , where ITM is the identity map of TM. A Riemannian metric h is called compatible with the almost complex structure J, if at each point x ∈ M the endomorphism Jx preserves the inner product hx induced from h on TxM. It means that for all tangent vectors u, v ∈ TxM hx(Jx(u),Jx(v)) = hx(u, v). In this case, the triple (M, J, h) is called an almost Hermitian manifold. The Nijenhuis tensor corresponding to J is defined as follows: NJ (X,Y ) = [JX,JY ] − J [X,JY ] + [JX,Y ] − [X,Y ], (7) where X and Y are any two arbitrary vector fields on M. The almost complex structure is called integrable if NJ vanishes. In this case, manifold is complex and the almost Hermitian manifold (M, J, h) is Hermitian manifold.

Finsler geometry is a natural extension of Riemannian geometry. Hence, it is natural to extend the compatibility of almost complex structure from Riemannian setting to Finslerian one.

Let (M,F ) be a Finsler manifold. At each point x ∈ M, the Finsler metric F induces a Minkowski ∗ norm on TxM, instead of inner product. It is known that F makes the pull-back bundle p TM to a ∗ Riemmanian bundle over TMg . The fiber of p TM at a point (x, y) ∈ TMg is just {(x, y)} × TxM. Hence, every vector field X on M is naturally lifted to a section of the pull-back bundle p∗TM by setting X˜(x, y) = (x, y, X(x)). Similarly, any almost complex structure J on M is lifted to ∗ ˜ an endomorphism of p TM as follows J(x,y)(x, y, u) := (x, y, Jx(u)). Noticing these facts, N. Prakash tried to extend the compatibility of an almost complex structure J on a manifold M with a Finsler metric F on M as follows, g(J˜X,˜ J˜Y˜ ) = g(X,˜ Y˜ ), (8) where g is the Riemannian metric on the pull-back bundle p∗TM which is constructed from F , and X,Y are any two vector fields on M. Later on, E. Heil and Y. Ichijyo¯ showed that (8) implies that F is Riemannian (Fukui (1989)). In the next step, generalized Finsler metrics on a manifold M were considered. Every Riemannian structure on the pull-back bundle p∗TM is called a generalized 1248 Z. Didehkhani et al.

Finsler metric on M. It is obvious that every Finsler metric F induces a generalized Finsler metric, i j ∂ but the converse is not true. Let J = J jdx ⊗ ∂xi and F be an almost complex structure and 1 ∂2F 2 Finsler metric on a manifold M, respectively. Suppose that gij = 2 ∂yi∂yj is the fundamental tensor associated to F . Then, the Moor metric associated to J and F is a generalized Finsler metric on M and deﬁned by the following, 1 G˜ = (g + g J pJ q). (9) ij 2 ij pq i j ˜ ˜ p q It is seen that Gij = GpqJi Jj . This means that the Moor metric associated to J and F is compatible with the complex structure J, while F is not necessary a Riemannian metric.

i j ∂ Furthermore, another stream of thoughts appeared in Rizza’s mind. Once more, let J = J jdx ⊗ ∂xi and F be an almost complex structure and Finsler metric on a manifold M, respectively. He asked himself: “How one can make any Minkowski space (TxM,Fx) to a complex Banach space?" Then he proposed the following compatibility between J and F ,

F (x, y cos θ + Jx(y) sin θ) = F (x, y), ∀θ ∈ R, ∀y ∈ TxM. (10) Ichijyo¯ baptized such Finsler manifolds as Rizza manifolds (Ichijyo¯ (1988)). If F is Riemannian, then (M,F,J) is a Rizza manifold if and only if it is an almost Hermitian manifold. Thus, Rizza manifolds are natural extension of almost Hermitian manifolds. Some equivalent conditions to (10) were obtained as follows

i k j (1) gijJky y = 0, m m m r (2) gimJj + gjmJi + 2CijmJr y = 0.

Y. Ichijyo¯ and M. Hashiguchi gave an important class of non-Riemannian Rizza manifolds, namely (a, b, J)-manifolds (Ichijyo¯ and Hashiguchi (1995a)). Let (M, α, J) be a 2n-dimensional almost Hermitian manifold. For a non-vanishing 1-form bi(x) on M, we have a symmetric quadratic form i j 1 β(x, y) = (bij(x)y y ) 2 , (11)

r where bij = bibj + JiJj,Ji = brJi . Indeed, Ji are the local component of the 1-form b ◦ J. Now, it is easy to see that the Finsler metric F = α + β is a typical example of Rizza manifolds (Ichijyo¯ (1988)). In this case, (M,F,J) is called an (a, b, J)-manifold (Ichijyo¯ and Hashiguchi (1995a)). An (a, b, J)-manifold is called normal if it satisﬁes

i ∇kbi = 0, ∇kJj = 0. (12)

Two 1-forms bi and Ji given on a Riemannian manifold (M, α) are called cross-recurrent if there exist a 1-form λk satisfying

∇kbi = λkJi, ∇kJi = −λkbi, (13) where ∇ is the Levi-Civita connection of α (for more details see (Ichijyo¯ and Hashiguchi (1995a))). An (a, b, J)-manifold is called nearly normal if bi and Ji are cross-recurrent and the following holds i i ∇kJj + ∇jJk = 0. (14) AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1249

We deﬁne the kernel of the symmetric quadratic form β at x ∈ M as follows,

i j Ker(β) = {y ∈ TxM| bij(x)y u = 0 ∀u ∈ TxM}. Note that an (a, b, J)-manifold F = α + β is singular at every tangent vector y which is in the kernel of β. In Lee ((2003)), N. Lee has determined the maximal domain D of TMg , on which an (a, b, J)-manifold is regular. Then, N. Lee computed the fundamental tensor and Cartan tensor of F = α + β, F F F F L g = a + b b + J J + F F − α α − β β , ij α ij β i j β i j i j α i j β i j

∂α ∂β where αi = ∂yi , βi = ∂yi and Fi = αi + βi,

1 2 1 1 αij βij C := (F ) i j k = (g ) k = σ {( − )(αβ − βα )}. ijk 4 y y y 2 ij y 2 (i,j,k) α β k k

The notation σ(i,j,k) denotes the summation of the cyclic permutation of indices i, j, k.

In Ichijyo¯ and Hashiguchi (1995a), Ichijyo¯ studied geometric properties of (a, b, J)-metrics and found a condition under which an (a, b, J)-metric is Berwaldian.

Theorem 2.4.

An (a, b, J)-manifold (M, α + β) is a Berwald space if and only if ∇kbij = 0, where ∇ is the Levi-Civita connection of α.

A Rizza manifold (M,F,J) is called a Kählerian Finsler manifold if it satisﬁes the following

i J j|k = 0, where “|” denotes the h-covariant differentiation with respect to the Cartan connection of F . As an important class of Kählerian Finsler manifolds, we have the class of normal (a, b, J)-manifolds (Ichijyo¯ and Hashiguchi (1995a)).

Let us consider an almost Hermitian manifold (M, h, J) with the Levi-Civita connection ∇. As an object of particular importance for nearly Kähler (M, h, J), we have the canonical Hermitian connection deﬁned by 1 ∇¯ Y = ∇ Y + (∇ J)JY. X X 2 X We know that ∇¯ is the unique Hermitian connection with totally skew-symmetric torsion given by

T (X,Y ) = (∇X J)JY. (15) It is known that T vanishes if and only if (M, h, J) is a Kähler manifold (Nagy (2002b)).

A Rizza manifold (M, L, J) is called a nearly Kählerian Finsler manifold if the following holds

i i J j|k + J k|j = 0 1250 Z. Didehkhani et al.

If a nearly Kählerian Finsler manifold is Riemannian, then it is also called Tachibana manifold or a K-space (Ichijyo¯ and Hashiguchi (1995b); Park and Lee (1993)). It was proved that every nearly normal (a, b, J)-manifold is a nearly Kählerian Finsler manifold (Ichijyo¯ and Hashiguchi (1995b)).

3. Non-existence theorem on nearly normal (a, b, J)-manifold

i Let us investigate some severe consequences of (13). We deﬁne the norm of 1-form b = bi(√x)dx i j i on a manifold M with respect to a Riemannain metric α = aij(x)dx ⊗ dx by ||b||α := b bi, i ir i where b = a br. On a nearly normal (a, b, J)-manifold, contracting ∇kbi = λkJi with b and i i using b Ji = 0, one gets b ∇kbi = 0. Thus the norm of b with respect to α is a constant function, 2 i since ∇k||b||α = 2b ∇kbi.

Now, suppose that b and b ◦ J are cross-recurrent, i.e., for any vector ﬁeld X and Y on M we have

(∇X b)Y = ∇X (bY ) − b(∇X Y ) = λ(X)b ◦ J(Y ), (16)

(∇X b ◦ J)Y = ∇X (b ◦ J)(Y ) − b ◦ J(∇X Y ) = −λ(X)b(Y ). (17) Putting J(Y ) in palce of Y in (17) and using J ◦ J = −Id, we get

−(∇X b ◦ J)J(Y ) = ∇X (bY ) + b ◦ J(∇X J(Y )) = λ(X)b(J(Y )). (18) Comparing (16) and (18), we obtain b T (X,Y ) = b (∇X J)JY = −b J(∇X J(Y ) + ∇X Y = 0, (19) which says b ◦ T = 0. Summarizing up, we get the following.

Proposition 3.1. If (M, α + β) is a nearly normal (a, b, J)-manifold, then the norm of the 1-form b with respect to α is a constant function on M. Moreover, b annihilates the torsion T , i.e., b(T (X,Y )) = 0 for any two vector ﬁelds X and Y on M, where T is given by (15).

In Ichijyo¯ and Hashiguchi (1995a), Ichijyo¯ proposed a problem on nearly Kähler Finsler manifolds:

“Let (M, α, J) be a nearly Kählerian Riemannian manifold. Can we ﬁnd a non-zero 1-form b on M such that the corresponding (a, b, J)-manifold (M, α + β) becomes a nearly Kählerian Finsler manifold?"

We know that every nearly normal (a, b, J)-manifold is a nearly Kählerian Finsler manifold. So, it is natural to propose the following question: AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1251

“Let (M, α, J) be a nearly Kählerian Riemannian manifold. Can we ﬁnd a non-zero 1-form b on M such that the corresponding (a, b, J)-manifold (M, α + β) becomes a nearly normal (a, b, J)- manifold?"

Let us consider the standard unit 6-dimensional sphere S6 with its almost complex structure. We consider S6 as a totally umbilical submanifold of the 7-dimensional Euclidean space R7. It is well known that the algebraic structure of octonions induces a so-called cross product “×00 on R7 (Gray (1966)). Choosing a unit normal vector ﬁeld N on S6, we deﬁne the canonical almost complex structure on S6 by J(X) := N × X. The following relation is a useful one,

(∇X J)(Y ) = ∇X J(Y ) − J(∇X Y ) = ∇X (N × Y ) − N × ∇X Y

= ∇X N × Y + N × ∇X Y − N × ∇X Y

= ∇X N × Y = X × Y, where we have used totally umbilicness of S6 in R7. As a consequence, we get (14). Therefore S6 is a nearly Kähler manifold. Now, we seek a desired 1-form b such that b and b ◦ J are cross- recurrent. Proposition 3.1 infers that b must be of constant length. On the other hand, every vector ﬁeld on S6 is zero at some point of S6. Thus, we conclude that b = 0. Therefore, the only 1-form b that makes S6 as a nearly normal (a, b, J)-manifold is null 1-form. Hence, we give a negative answer to Ichijyo¯’s question, in the case of nearly normal.

Indeed, we can generalize above observation to any orientable compact manifold M with non-zero Euler characteristic. Hopf’s well-known theorem states that in this case, every vector ﬁeld on such a manifold vanishes at some point of the manifold. Due to the constancy of the length of b, the desired 1-form b must be zero, which is a contradiction.

The only four complete, homogeneous and simply connected examples of strictly nearly Kähler manifolds in dimension 6 are S3 ×S3, the complex projective space CP 3, the ﬂag manifold F 3 and the sphere S6 (Butruille(2010)). Thus, all of theme are compact with (except S3 × S3) non-zero Euler characteristics.

For the case S3 × S3 we use the nearly Kähler structure which has been used and described in (Bolton et al. (2015)). As they considered the 3-sphere in R4 as the set of all unit quaternions. The vector ﬁelds X1,X2 and X3 given by

X1(p) = pi = −x2 + x1i + x4j − x3k,

X2(p) = pj = −x3 − x4i + x1j + x2k,

X3(p) = pk = x4 − x3i + x2j − x1k, at the point p = x1 + x2i + x3j + x4k form a basis of tangent vector ﬁelds. Thus a tangent 3 vector in TpS can be expressed as pα where α is an imaginary quaternion. Using the quaternion relations ij = k, jk = i and ki = j one shows that the Lie brackets are given by [Xi,Xj] = −2εijkXk. Based on their notations εijk is the Levi-Civita symbol. Using the nat- 3 3 ∼ 3 3 ural identiﬁcation T (p, q)(S × S ) = TpS ⊕ TqS , they wrote a tangent vector at (p, q) as 1252 Z. Didehkhani et al.

Z(p, q) = U(p, q),V (p, q) or simply Z = (U, V ). They deﬁned the vector ﬁelds

E1(p, q) = (pi, 0),F1(p, q) = (0, qi),

E2(p, q) = (pj, 0),F2(p, q) = (0, qj),

E3(p, q) = −(pk, 0),F3(p, q) = −(0, qk).

These vector fields are mutually orthogonal with respect to the usual product metric on S3 × S3. The Lie brackets are [Ei,Ej] = −2εijkEk, [Fi,Fj] = −2εijkFk and [Ei,Fj] = 0. The almost complex structure J on S3 × S3 is defined as JZ(p, q) = √1 2pq−1V U, 2qp−1U + V for Z ∈ 3 T (p, q)(S3 × S3) (see (Butruille (2010))). We have 1 JEi = −√ (Ei + 2Fi), 3 1 JFi = √ (2Ei + Fi). 3 Furthermore, they defined another metric g on S3 × S3 by 1 g(Z,Z0) = (hZ,Z0i + hJZ,JZ0i) 2 4 2 = (hU, U 0i + hV,V 0i) − (hp−1U, q−1V 0i + hp−1U 0, q−1V i), 3 3 where Z = (U, V ),Z0 = (U 0,V 0) and h., .i is the product metric on S3 × S3. By definition the almost complex structure is compatible with the metric g. Therefore one should obtain g(Ei,Ej) = 4 2 4 3 δij, g(Ei,Fj) = − 3 δij and g(Fi,Fj) = 3 δij. Note that this metric differs up to a constant factor from the one introduced in (Butruille (2010)). They set everything up so that it equals the Hermitian metric associated with the usual metric. In (Butruille (2010)), the factor was chosen in such a way that the standard basis E1,E2,E3,F1,F2,F3 has volume 1. Using the Koszul formula, they express the following lemma.

Lemma 3.2. (Bolton et al. (2015)) The Levi-Civita connection ∇ on S3 × S3 with respect to the metric g ε ∇ E = −ε E , ∇ F = ijk (E − F ), Ei j ijk k Ei j 3 k k ε ∇ E = ijk (F − E ), ∇ F = −ε F . Fi j 3 k k Fi j ijk k

Then, the following has been obtained 2 2 (∇E J)Ej = − √ εijk(Ek + 2Fk), (∇E J)Fj = − √ εijk(Ek − Fk), i 3 3 i 3 3 2 2 (∇F J)Ej = − √ εijk(Ek − Fk), (∇F J)Fj = − √ εijk(2Ek + Fk). i 3 3 i 3 3 AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1253

As mentioned the torsion of Hermitian connection given by T (X,Y ) = (∇X J)JY and according to relation (19) we have boT = 0. We compute torsion for different vector ﬁelds as follows. 2 T (E ,E ) = − ε E , i j 3 ijk k 2 T (F ,F ) = ε (F − 4E ), i j 9 ijk k k 2 T (F ,E ) = − ε (5E + F ), i j 9 ijk k 3 2 T (E ,F ) = − ε (E + F ). (20) i j 3 ijk k k

i i 1 2 3 Suppose that {E ,F } are dual of {Ei,Fi}. Let us consider a 1-form b = b1E + b2E + b3E + 1 2 3 3 3 b4F + b5F + b6F making S × S to a nearly normal (a, b, J)-manifold. Thus, we have 2 2 2 bT (E ,E ) = − b , bT (E ,E ) = b , bT (E ,E ) = b , 2 1 3 3 2 3 3 1 3 1 3 2 10 2 10 2 10 2 bT (F ,E ) = b + b , bT (F ,E ) = b + b , bT (F ,E ) = − − b . (21) 2 3 9 1 9 4 3 1 9 2 9 5 2 1 9 9 6

Therefore bi’s should be zero, which one concludes that b = 0. Hence, the only 1-form b that makes S3 × S3 as a nearly normal (a, b, J)-manifold is null 1-form. Finally, we express the following proposition.

Proposition 3.3. Any complete homogeneous strictly 6-dimensional nearly Kähler manifold does not admit nearly normal (a, b, J)-structure.

It may be reasonable to consider non-comapct (nearly) Kähler manifolds. In the sequel, we are going to show that on every even dimensional Euclidean space R2n with its canonical almost complex structure J and Euclidean metric α, one can always ﬁnd the desired 1-form b making R2n as a Kählerian Finsler (a, b, J)-manifold. Let us denote the Cartesian coordinates of R2n by (x1, y1...., xn, yn). Then J is given by ∂ ∂ ∂ ∂ J = ,J = − , i = 1, ..., n. (22) ∂xi ∂yi ∂yi ∂xi

It is easy to see that (R2n, α, J) is a Kählerian manifold. For simplicity, let us denote the 1- form b by its components in Cartesian coordinates b = (b1, b2, ..., b2n−1, b2n). Thus, the components of the 1-form b ◦ J are given by b ◦ J = (−b2, b1, ..., −b2n, b2n−1). Suppose that 2n λ = (λ1, λ2, ..., λ2n−1, λ2n) is a constant vector in R whose components are positive. If we 1 2n 1 1 n n put u = (u , ..., u ) = (x , y ...., x , y ), then ∇kbi = λkJi, ∇kJi = −λkbi if and only if bi satisﬁes the following second order linear PDEs system, ∂2b i + λ λ b = 0, i, j, k = 1, 2, ..., 2n. (23) ∂uj∂uk j k i 1254 Z. Didehkhani et al.

2n One can see that for every point P = (A1,B1...., An,Bn) ∈ R the following 1-form b is a solution of (23),

b2k−1 = A2k−1cos(< λ, u >) + B2k−1sin(< λ, u >),

b2k = A2kcos(< λ, u >) − B2ksin(< λ, u >), k = 1, ..., n, (24)

where <, > is the Euclidean inner product on R2n. Moreover, the norm of b with respect to α is the Euclidean norm of the ﬁxed point P .

4. Generalized (a, b, J)-metrics

p i j i Let α = aij(x)y y be a Riemannian metric and β = bi(x)dx be a 1-form on a manifold M with ||β||α < 1. Then F = α + β is called a Randers metric. Replacing the 1-form β with a symmetric i j quadratic form β = bijdx ⊗dx of rank 0 ≤ r < n, we get a generalized Randers metric F = α+β. Every (a, b, J)-metric is a generalized Randers metric. It is well known that the class of Randers metrics is an special case of a general class of Finsler metrics so-called (α, β)-metrics. A Finsler β metric F is called an (α, β)-metric if it can be expressed as F = αϕ( α ), where ϕ :(−b0, b0) → R is a positive smooth function satisfying some regularity conditions. Similarly, in order to extend the class of Rizza manifolds introduced by Ichijyo¯, we deﬁne generalized (a, b, J)-metrics as follows.

Deﬁnition 4.1.

Consider an (a, b, J)-metric F = α + β. Let Ψ:(−b0, b0) → R be a positive smooth function. β Then, a Finsler metric in the form F = αΨ( α ) is called a generalized (a, b, J)-metric.

β One can compute the fundamental tensor of generalized (a, b, J)-metric F = αΨ(s), where s := α as follows 1 g =Ψ2 − sΨΨ0 a + ΨΨ0b − s (−s(ΨΨ00 + Ψ0Ψ0) + ΨΨ0) α α ij ij s ij i j 00 0 0 0 +(−s(ΨΨ + Ψ Ψ ) + ΨΨ )(αiβj + αjβi) 1 + − ΨΨ0 + ΨΨ00 + Ψ0Ψ0 β β (25) s i j

To ﬁnd the inverse of (gij), we need the following lemma.

Lemma 4.2. (Lee (2003))

ij Let (Pij) be a real symmetric non-singular matrix with the inverse (P ). And let (Qij) = (Pij ± 2 2 ij cicj) with 1 ± c 6= 0 and c := ciP cj. Then the matrix (Qij) is non-singular and its inverse is ij ij 1 i j i ij 2 (Q ) = (P ∓ 1±c2 c c ) where c = P cj and det(Qij) = (1 ± c )det(Pij) .

Proposition 4.3. β For the fundamental tensor (gij) of a generalized (a, b, J)-metric F = αΨ( α ), the determinant of AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1255

(gij) is given by

b2 det(g ) = Ψn+1 (Ψ − sΨ0)n−3 (Ψ − sΨ0) + (b2 − s2)Ψ00 Ψ + ( − s)Ψ0 det(a ). ij s ij

Proof: First, we set

ρ ρ ρ ρ g = ρ a + 0 b + 1 α α + 2 (α β + α β ) + 3 β β , ij ij ρ ij ρ i j ρ i j j i ρ i j where

ρ : = Ψ(Ψ − sΨ0), 1 ρ : = ΨΨ0, 0 s 00 0 0 0 ρ1 : = −s(−s(ΨΨ + Ψ Ψ ) + ΨΨ ), 00 0 0 0 ρ2 : = −s(ΨΨ + Ψ Ψ ) + ΨΨ , 1 ρ : = − ΨΨ0 + Ψ0Ψ0 + ΨΨ00. 3 s

We want to simplify the formula and use the Lemma to calculate the determinants. We have the following equalities,

ρ2 2 ρ2 2 ρ3 − ( ) ρ1 ( ) ρ1 ρ1 ρ1 ρ2 ρ2 ρ3 ρ1 βiβj + (αi + βi)(αj + βj) = βiβj − βiβj ρ ρ ρ1 ρ1 ρ ρ ρ1 ρ1 ρ2 2 ρ2 ρ2 + αiαj + ( ) βiβj + αiβj + αjβi. ρ ρ ρ1 ρ ρ gij can be written as follows,

gij = ρ {aij + ηbij + δβiβj + µYiYj} , (26) where Yi = αi + βi and

ρ − 2ρ δ := 3 1 = 0, ρ ρ 1 := 2 = − , ρ1 s ρ s {s(ΨΨ00 + Ψ0Ψ0) − ΨΨ0} µ := 1 = , ρ Ψ(Ψ − sΨ0) ρ Ψ0 η := 0 = . ρ s(Ψ − sΨ0) 1256 Z. Didehkhani et al.

Now we check whether Equation (26) lead us to Equation (25). ( Ψ0 g = Ψ(Ψ − sΨ0) a + b + 0β β ij ij s(Ψ − sΨ0) ij i j ) s{s(ΨΨ00 + Ψ0Ψ0) − ΨΨ0} 1 1 + (α + (− β ))(α + (− β )) Ψ(Ψ − sΨ0) i s i j s j 1 1 1 1 = Ψ(Ψ − sΨ0)a + ΨΨ0b + s{s(ΨΨ00 + Ψ0Ψ0) − ΨΨ0}(α α + β β − α β − α β ) ij s ij i j s2 i j s i j s j i 1 = Ψ(Ψ − sΨ0)a + ΨΨ0b + (s2(ΨΨ00 + Ψ0Ψ0 − sΨΨ0))α α ij s ij i j 1 + ((ΨΨ00 + Ψ0Ψ0) − ΨΨ0)β β − {s(ΨΨ00 + Ψ0Ψ0) − ΨΨ0}α β s i j i j 00 0 0 0 − {s(ΨΨ + Ψ Ψ ) − ΨΨ }αjβi, and we see that it is exactly Equation (25).

∂α ∂β First, we express some relationships that are used in the calculations: αi = ∂yi , βi = ∂yi . We set i ir i ir i ij i ij α = a αr, β = a βr, b = a bj,J = a Jj. Hence, y yi αi = air r = , α α airb ys bi ys βi = rs = s , β β y yi α2 α αi = i = = 1, i α α α2 yi b yj b yiyj β2 β αiβ = ij = ij = = , i α β αβ αβ α y airb ys b yrys β2 β α βi = i rs = rs = = . i α β αβ αβ α Also, i ij 4 i ir i i Jr = biJr, b bij = 2b , bj = a brj, b Ji = biJ = 0, j 2 ij jr i s rs 2 i i i J Jj = b , b = a br, b bs = a brbs = b , bj = b bj + J Jj, i i j 2 ij 2 is 2 i ij i j i j yiy = aijy y = α , a bij = 2b , b bjs = b bj, b = b b + J J ,

y y bi y yr bi a ysyr b yrys β2 bijα α = ajrbi α α = ajrbi i j = r i = r is = rs = , i j r i j r α α α2 α2 α2 α2 y b ys ajrbi b y ys bi bry ys b2bi a ypys b2b ypys b2β2 b2β bijα β = bij i js = r js i = r s i = s pi = sp = = , i i α β αβ αβ αβ αβ αβ α

b yj airb ys b bi yjys b2b yjys b2β2 β βi = ij rs = ij s = js = = b2, i β β β2 β2 β2 AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1257

b yp (airb ys = brys) bijβ β = ajrbi β β = bi β βr = bi ip is s i j r i j r i r β β b2bi b ysyp b2(b2b )ysyp b4β2 = s ip = sp = = b4. β2 β2 ββ2 To calculate the determinant we use the lemma. Attaining the formula of the determinant is done in three stages.

˜ ˜ −1 ˜ij Let’s ﬁrst assume Aij := aij + ηbibj. The inverse of Aij = A is given as follows, √ i √ i 2 ij √ ij√ i 2 c˜i = ηbi, c˜ = ηb , c˜ =c ˜ia c˜j = ηbia ηbj = ηbib = ηb ,

A˜ij = aij − τbibj, where η Ψ0 τ := = , 1 + ηb2 sΨ + (b2 − s2)Ψ0 ˜ 2 2 and det(Aij) = (1 +c ˜ ) det aij = (1 + ηb ) det(aij). ˜ Now we set Aij = Aij + ηJiJj and again use Lemma 4.2, √ i √ i 2 ˜ij ij i j i i j 2 c˜i = ηJi, c˜ = ηJ , c˜ = ηJi(A )Jj = ηJi(a − τb b )Jj = ηJi(J − τb b Jj ) = ηb , |{z} 0

˜ci˜cj J iJ j Aij = A˜ij − = aij − τbibj − η = aij − τbibj − τJ iJ j = aij − τbij. 1 +˜c2 1 + ηb2 Therefore, Aij = aij − τbij and 2 ˜ 2 2 det(Aij) = (1 +˜c ) det(Aij) = (1 + ηb ) det(aij).

Next, let’s put Qij = Aij + δβiβj and utilize Lemma 4.2, √ ˜˜˜ci = δβi, √ √ √ i ij ij ij i ij 2 i ˜˜˜c = A ˜˜˜cj = δ(a − τb )βj = δ(β − τ b βj) = δ(1 − τb )β , |{z} b2βi 2 i 2 i 2 2 ˜˜˜c =˜˜˜ci˜˜˜c = δ(1 − τb )βiβ = δb (1 − τb ),

˜˜˜ci˜˜˜cj δ(1 − τb2)2βiβj Qij = Aij − = (aij − τbij) − . 1 +˜˜˜c2 1 + δb2(1 − τb2) δ(1−τb2)2 We put σ := 1+δ(1−τb2)b2 . Therefore, Qij = aij − τbij − σβiβj,

2 2 2 2 det Qij = (1 + δb (1 − τb ))(1 + ηb ) det aij.

For the third step, we set Pij = Qij + µYiYj and using Lemma 4.2 we get √ i √ i cˆi = µYi, cˆ = µY , 1258 Z. Didehkhani et al.

i √ ij √ ij ij i j cˆ = µQ Yj = µ(a − τb − σβ β )(αj + βj) √ i ij i j i ij i j = µ(α − τ b αj −σβ β αj +β − τ b βj −σβ β βj) |{z} |{z} |{z} |{z} β βi β b2βi b2 √ α α = µ(αi + βi(−τs − σs + − τb2 − σb2)).

We place σ0 := −τs − σs + − τb2 − σb2. Then, √ cˆi = µ(αi + σ0βi),

we obtain

2 i 0 i 0 i i 0 i cˆ = µ(αi + βi)(α + σ β ) = µ(1 + σ αiβ + α βi + σ βiβ ) β β = µ(1 + σ0 + + σ0b2) = µ(1 + σ0s + s + σ0b2) α α 1 b2 = µ(σ0s − b2σ0) = µ((s − )σ0), s s

cˆicˆj µ(αi + σ0βi)(αj + σ0βj) P ij = Qij − = aij − τbij − σβiβj − . 1 +c ˆ2 b2 0 1 + µ(s − s )σ

00 µ We set σ := 2 . Hence, by computing the following product, 1+µ((s− b )σ0) s

(αi + σ0βi)(αj + σ0βj) = αiαj + σ02βiβj + σ0αiβj + σ0αjβi,

we gain

P ij = aij − τbij − (σ + σ00σ02)βiβj − σ00αiαj − σ00σ0(αjβi + αiβj),

and according to Lemma 4.2 we have

b2 det P = (1 +c ˆ2) det Q = (1 + µ((s − )σ0))(1 + δb2(1 − τb2))(1 + ηb2)2 det a . ij ij s ij Therefore, inverse of fundamental tensor of generalized (a, b, J)-metric is as follows,

1 gij = P ij = ρ−1{aij − τbij − σβiβj − σ00Y iY j}, (27) ρ

n gij = ρPij, det gij = ρ det Pij,

b2 det g = ρn(1 + µ((s − )σ0))(1 + δb2(1 − τb2))(1 + ηb2)2 det a . (28) ij s ij AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1259

Now we compute τ, δ, σ0, µ and η in the above relation base on Ψ and then replace them in it, 1 = − , s s{s(ΨΨ00 + Ψ0Ψ0) − ΨΨ0} µ = , Ψ(Ψ − sΨ0) Ψ0 η = , s(Ψ − sΨ0) sΨ + (b2 − s2)Ψ0 1 + ηb2 = , s(Ψ − sΨ0) Ψ0 τ = , sΨ + (b2 − s2)Ψ0 δ = 0, σ = 0, −Ψ σ0 = , sΨ + (b2 − s2)Ψ0 (sΨΨ00 + sΨ0Ψ0 − ΨΨ0)(sΨ + (b2 − s2)Ψ) σ00 = , Ψ2((Ψ − sΨ0) + (b2 − s2)Ψ00)

−b2 + s2 s(s(ΨΨ00 + Ψ0Ψ0) − ΨΨ0) Ψ b2 − s2 1 + µ( )σ0 = 1 + s Ψ(Ψ − sΨ0) sΨ + (b2 − s2)Ψ0 s sΨ(Ψ − sΨ0 + (b2 − s2)Ψ00) = . (Ψ − sΨ0)(sΨ + (b2 − s2)Ψ0) Now replace these in (28), sΨ(Ψ − sΨ0 + (b2 − s2)Ψ00) (sΨ + (b2 − s2)Ψ0)2 det g = Ψn(Ψ − sΨ0)n . ij (Ψ − sΨ0)(sΨ + (b2 − s2)Ψ0) s2(Ψ − sΨ0)2 Finally, by simpliﬁcation, we arrive at the following formula, 2 n+1 0 n−3 0 2 2 00 b 0 det(gij) = Ψ (Ψ − sΨ ) (Ψ − sΨ ) + (b − s )Ψ Ψ + ( − s)Ψ det(aij). s

One can see that for Ψ = 1 + s, Ψ0 = 1, Ψ00 = 0 the result is the same as the formula which has been gotten in (Proposition 3.1, Lee (2003)).

Lemma 4.4. β F = αΨ( α ) is a Finsler metric, for any Riemannian metric α and symmetric quadratic form β if and only if Ψ = Ψ(s) satisﬁes the following conditions, b2 Ψ(s) > 0, (Ψ − sΨ0) + (b2 − s2)Ψ00 > 0, Ψ + ( − s)Ψ0 > 0, (29) s 0 dΨ where Ψ = ds . 1260 Z. Didehkhani et al.

Proof: Assume that (29) is satisﬁed. Then by taking b = s in (29), we see that the following inequality holds, Ψ − sΨ0 > 0. Consider the following family of functions,

Ψ(s) := 1 − + Ψ(s). β 1 2 Let F := αΨ( α ) and gij := 2 [F ]yiyj (y), 0 ≤ ≤ 1. We have 0 0 Ψ − sΨ = 1 − + (Ψ − sΨ ) > 0,

0 2 2 00 0 2 2 00 (Ψ − sΨ ) + (b − s )Ψ = 1 − + (Ψ − sΨ ) + (b − s )Ψ > 0,

b2 b2 Ψ + ( − s)Ψ 0 = 1 − + (Ψ + ( − s)Ψ0) > 0. s s For = 0, F = α since det(gij ) is continuous for , all the eigenvalues of (gij ) are positive by intermediate value theorem. Therefore, all the eigenvalues of (gij) are positive.

β Conversely, suppose that F = αΨ( α ) is a regular Finsler metric. we can always ﬁnd a vector y ∈ D such that β(x0, y) = sα(x0, y). By assumption, F (x0, y) = αΨ(s) > 0, we conclude that Ψ(s) > 0. By another assumption, det(gij(x0, y)) > 0, we conclude from (29) that Ψ(s) − sΨ0(s) 6= 0. provided that n > 3. Since Ψ(0) > 0, and s arbitrary we must have Ψ(s) − sΨ0(s) > 0. Now by (29), we conclude that b2 Ψ(s) − sΨ0(s) + (b2 − s2)Ψ00(s) > 0, Ψ + ( − s)Ψ0 > 0. s

If n = 3, we still get the above inequality from (29). This proves the lemma.

Here, we state that every generalized (a, b, J)-metric is a Rizza manifold.

Proposition 4.5. β A generalized (a, b, J)-metric F = αΨ( α ) is a Rizza manifold.

Lemma 4.6. β Mean Cartan torsion of a generalized (a, b, J)-metric F = αΨ( α ) is given by ( 0 00 2 2 000 00 b2 00 b2 0 ) 1 Ψ sΨ (b − s )Ψ − 3sΨ ( − s)Ψ − 2 Ψ I = (n + 1) − (n − 3) + + s s h , i 2α Ψ Ψ − sΨ0 (Ψ − sΨ0) + (b2 − s2)Ψ00 b2 0 i Ψ + ( s − s)Ψ

k bkiy yi yi where hi = β − α s = βi − α s. AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1261

Proof: jk ∂ p ∂s hi We know that Ii = g Cijk = ∂yi Ln det(gjk) . A direct computation shows that ∂yi = α k bkiy yi yi where hi := β − α s = βi − α s. Thus ∂ n + 1 n − 3 1 I = LnΨ + Ln(Ψ − sΨ0) + Ln (Ψ − sΨ0) + (b2 − s2)Ψ00 i ∂yi 2 2 2 1 b2 1 + Ln Ψ + ( − s)Ψ0 + Ln(det(a )) 2 s 2 ij

Finally, the desired result is obtained.

2 2 n+1 b α yi β Therefore, mean Cartan torsion of an (a, b, J)-metric is Ii = ( 2(α+β) − β(β+b2α) )(βi − α α ).

To state our result, we shall ﬁrst introduce some notations. Let ∇jbi and ∇jJi denote the horizontal i i p i j covariant derivative of bidx and Jidx with respect to α = aijy y , respectively. Let 1 1 r := (∇ b + ∇ b ) , s := (∇ b − ∇ b ) , ij 2 j i i j ij 2 j i i j 1 1 r˜ := (∇ J + ∇ J ) , s˜ := (∇ J − ∇ J ) , ij 2 j i i j ij 2 j i i j

i j i j i j i j r00 := rijy y = ∇ibjy y , r˜00 :=r ˜ijy y = ∇iJjy y ,

i i j i i i j i 2 ij s0 = sjy , s0 := siy , s˜0 =s ˜jy , s˜0 :=s ˜iy , b := a bibj. It is easy to see that i j i j sij + sji = 0, s00 := sijy y = 0, s˜ij +s ˜ji = 0, s˜00 :=s ˜ijy y = 0. Let Gi = Gi(x, y) and G¯i = G¯i(x, y) denote the spray coefﬁcients of F and α, respectively, in the i i same coordinate system. Put b0 = biy and J0 = Jiy . According to (1) and considering (27) spray coefﬁcients Gi are given by ∇ F yk F Gi = Gi + k yi + gil{(∇ F ) yk − ∇ F }, (30) α 2F 2 k .l l where ∇iF denote the covariant derivatives of F with respect to α and (∇iF ).j = [∇iF ]yj .

Lemma 4.7. β Spray coefﬁcients of a generalized (a, b, J)-metric F = αΨ( α ) are given by

Gi = G¯i + P yi + Qi, where

l l J0 P = Ξs0 + (Θ1 + Q)(b0r00 + J0r˜00) + Ξ(sl0J +s ˜l0b0) , b0 1262 Z. Didehkhani et al.

i i i i i i i Q = Θ2b + Θ3J + Θ4b0 + Θ5 r00 + (Θ6 + Θ8)b0 + Θ7J + Θ9 r˜00 + Θ10b0 + Θ11,

0 0 0 0 0 b0Ψ (sΨΨ +sΨ Ψ −ΨΨ ) where Ξ = β2Ψ(Ψ−sΨ0)((Ψ−sΨ0)−(b2−s2)Ψ00) and Θis are given in the Appendix.

β In (Ichijyo¯ and Hashiguchi (1994)), Ichijyo¯ says irrationality of λ = α in terms of y yields that a generalized Randers space is a Berwald space if and only if ∇kbij = 0. Here, we extend Ichijyo¯’s result to generalized (a, b, J)-metrics.

Theorem 4.8. β s(Ψ−sΨ0) Let F = αΨ( α ) be a generalized (a, b, J)-metric. Suppose the λ = Ψ0 is an irrational function of y. Then, (M,F ) is a Berwald space if and only if two 1-forms bi and Ji are cross-recurrent.

Proof: Suppose F is a Berwald metric. We denote the h-covariant differentiation with respect to the Berwald connection of F with “;”. We have β α − βα F = α Ψ(s) + Ψ0(s)( ;k ;k ). (31) ;k ;k α On the other hand, it is well known that F;k = 0. Multiplying (31) with α, we obtain 0 0 αΨα;k + Ψ αβ;k − βΨ α;k = 0. (32)

Substituting α;k and β;k in (32), we rewrite as following i j i j λaij;ky y + bij;ky y = 0. (33) i (M,F ) is a Berwald space. Thus, the horizontal Christoffel coefficients G jk of the Berwald con- i j i j nection BΓ are functions of position alone. Consequently, aij;ky y and bij;ky y become polyno- i j i j mials of y. Now, the irrationality of λ and (33) infer that aij;ky y = 0 and bij;ky y = 0, that is, aij;k = 0 and bij;k = 0. The former implies that the Levi-Civita connection of α and the Berwald connection of F coincide. Therefore, we have ∇kbij = 0, which is equivalent to bi and Ji are cross- 1 0 recurrent. The converse is also true. In fact, if ∇kbij = 0 is satisfied, then ∇kF = α ∇kβΨ = 0 it follows from Theorem 4.1.3 in (Chern and Shen (2004)) that F and α are affinely equivalent.

Hence, F is Berwaldian. Remark 4.9. If in Theorem 4.8 we put Ψ(s) = 1 + s, then we get Ichijyo’s¯ assumption on generalized Randers metrics.

β We call a generalized (a, b, J)-metric F = αΨ( α ) normal if (12) holds. It follows from (12) that β is parallel with respect to α. Thus, Theorem 4.8 implies that every normal generalized (a, b, J)- metric is Berwaldian.

β We call also a generalized (a, b, J)-metric F = αΨ( α ) nearly normal if bi and Ji are cross- recurrent and the following holds , i i ∇kJj + ∇jJk = 0. (34) AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1263

Thus every nearly normal generalized (a, b, J)-metric is a Berwald space. Thus, the horizontal covariant differentiation with respect to Cartan and Berwald connections coincide with the one i i with respect to Levi-Civita connection of Riemannian metric α. Therefore, ∇kJj = Jj|k. Hence, we get the following proposition.

Proposition 4.10. Every normal (nearly normal) generalized (a, b, J)-metric is a Kählerian (nearly Kählerian) Finsler metric.

In Theorem 4.8, we characterize those (a, b, J)-metrics F = α + β whose Riemannian parts α is afﬁnely equivalent to F . Now, we deal with projectively equivalency of F and α.

Proposition 4.11. Let F = α + β be an (a, b, J)-metric. Then F is projectively equivariant to α if and only if for l = 1, ..., n

i σ(ijkm) − (∇kbij)bml + (∇kblj)bimy + (∇kbil)bjm − (∇kblj)bim = 0, (35)

where σ(ijkm) denotes the summation over all permutations of indices i, j, k and m.

Proof:

k By (6), it sufﬁces to prove that (∇kF ).ly − ∇lF = 0 is equivalent to (35). One can see

k k (∇kF ).ly − ∇lF = (∇kβ).ly − ∇lβ 1 −b ym = (∇ b )( ml )yiyjyk + (∇ b )yjyk + (∇ b )yiyk − (∇ b )yiyj . 2β k ij β2 k lj k il l ij (36) Suppose that F is projectively related to α. Thus, (36) implies that −b ym (∇ b )( ml )yiyjyk + (∇ b )yjyk + (∇ b )yiyk − (∇ b )yiyj = 0. (37) k ij β2 k lj k il l ij Multiplying (37) by β2, we get a polynomial equation of degree four in terms of y as follows i j k m Sijkmly y y y = 0, (38) in which Sijkml := −bml∇kbij + bim∇kblj + bjm∇kbil − bkm∇lbij are functions independent of direction. We get (35) from (37). This completes the proof.

It is natural to study the projectively equivalency problem for generalized (a, b, J)-metrics.

Theorem 4.12.

β Ψ00 Let F = αΨ( α ) be a generalized (a, b, J)-metric. Suppose that sΨ0 is an irrational function of y. Then F is projectively equivalent to α if and only if (35) and σijk∇kbij = 0 are satisﬁed. 1264 Z. Didehkhani et al.

Proof: To use for the case F = αΨ(β/α), We need the following β α − βα (∇ F ) yk − ∇ F = (∇ β) Ψ0(s)yk + ∇ β .l .l Ψ00(s)yk − ∇ βΨ0(s). k .l l k .l k α2 l k Suppose that F is projectively related to α. Then Rapcsak’s Theorem implies that (∇kF ).ly − ∇lF = 0. Therefore, we have 1 Ψ00 (∇ β) yk − ∇ β + ∇ β(β − sα )yk = 0. (39) k .l l α k .l .l Ψ0 Multiplying (39) with α4β2 we get

4 m i j k j k 2 i k 2 i j 2 α −(∇kbij)bmly y y y + (∇kblj)y y β + (∇kbil)y y β − (∇lbij)y y β Ψ00 + β2(∇ b )ykyiyj(α2b − β2y ) = 0. (40) k ij l0 l sΨ0 Ψ00 All terms appeared in (40) except sΨ0 , are rational functions of y. Therefore, (40) is equivalent to following

m i j k j k 2 i k 2 i j 2 −(∇kbij)bmly y y y + (∇kblj)y y β + (∇kbil)y y β − (∇lbij)y y β = 0, (41)

k i j 2 2 (∇kbij)y y y (α bl0 − β yl) = 0. (42)

2 2 As we saw in the previous theorem, (41) is equivalent to (35). We claim that α bl0 − β yl is non- zero. Otherwise, we have

i t s (bliats − btsali)y y y = 0. (43) Therefore,

(bliats − btsali) + (bltais − bisalt) + (blsati − btials) = 0. (44)

ts ts If we denote the inverse of (ats) by (a ) and contract (44) with a , we obtain

nbli = µali, (45)

ts where µ = a bts and we have used the symmetry of β. One can see (45) contradicts β being of k i j rank 2. Hence, ∇kbijy y y = 0 which is equivalent to ∇kbij + ∇jbik + ∇ibkj = 0.

Ichijyo¯ in (Ichijyo¯ (1994)) proved that a Kählerian Finsler manifold is a Landsberg manifold. We are going to generalize this fact to nearly Kählerian Finsler manifolds. First, we prove that the Berwlad curvature of a nearly Kähler Finsler manifold and its almost complex structure have a delicate relation. Let us recall two important identities,

i i i (a) gij;k = −2Lijk, (b) Fjk = Gjk − Ljk, (46) where “;" stands for the h-covariant derivative with respect to the Berwald connection BΓ = i i i (Gjk,Gj, 0) and Fjk are given by (2). AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1265

Proposition 4.13. Let (M,F,J) be a nearly Kähler Finsler manifold. Then ∂Li (a)J r Li + ykJ r rj = 0, (b)J rLi − 2J iLr + J r Li + ykJ rBi = 0. (47) m rj k ∂ym j rm r mj m rj k rjm

Proof:

i i Using (3), we rewrite Jj|k + Jk|j = 0 as follows, i r i i r i i r ∂kJj + Jj Frk + ∂jJk + Jk Frj − 2JrFjk = 0. (48) We multiply (48) by yk and obtain k i r i k i i r k r i y ∂kJj + Jj Gr + y ∂jJk − 2JrGj + y Jk Frj = 0, (49) k r r m where we have used y Fkj = Gj . Differentiating (49) with respect to y , we get ∂F i ∂ J i + ∂ J i + J rGi + J r Gi − 2J iGr − J r Li + ykJ r rj = 0. (50) m j j m j rm m rj r jm m rj k ∂ym Using (5) , (50) and (46b), we reformulate (50) as follows, ∂Li J i + J i = J r Li + ykJ r rj − ykJ rBi . (51) j;m m;j m rj k ∂ym k rjm Transvecting (49) with yj yields j k i k r i i r y y ∂kJj + y Jk Gr − 2JrG = 0, (52) j r r j m where we have used y Gj = 2G . Differentiating (55) with respect to y and y , respectively, leads us to i i r i i r r i k r i ∂mJj + ∂jJm + Jj Grm − 2JrGmj + JmGrj + y Jk Brjm = 0, (53) j i where we have used y Bjkl = 0. One can rewrite (53) as follows, i i k r i Jj;m + Jm;j = −y Jk Brjm. (54)

Comparing (51) with (54), we get (47a).

Now we multiply (49) with yj and get the following, j k i k r i j i r j r i 2y y ∂kJj + y Jk Gr − 2y JrGj + y Jj Gr = 0. (55) First differentiating (55) with respect to ys and then with respect to yt , by using (46) we achieve r i i r r i k r i Jj Lrm − 2JrLmj + JmLrj + y Jk Brjm = 0. (56)

Theorem 4.14. Let (M,F,J) be a nearly Kähler Finsler manifold satisfying (57). Then F is a Landsberg metric p r r p Jr Lik = Ji Lrk. (57) 1266 Z. Didehkhani et al.

Proof: Using (57), we have p p p r r p p Ji|m = Ji;m + Jr Lim − Ji Lrm = Ji;m. (58) i i k r i Thus, by (54) and Jj|m + Jm|j = 0, we have y Jk Brjm = 0 and concequently (47b) reduces to the following, r i r i i r Jj Lrm + JmLrj = 2JrLjm. (59) k By contracting (59) with Ji we get k r i k r i k Ji Jj Lrm + Ji JmLrj = −2Ljm. (60) j k j i j Transvecting (60) with y and using Ljmy = Lrjy = 0, we get j k r i y Ji Jj Lrm = 0. (61)

Taking vertical differentiating with respect to yl from (61) and taking into account (47a) implies that k r i Ji Jl Lrs = 0. (62)

Substituting (62) into (60) completes the proof.

5. Conclusion

In this paper, we introduced generalized (a, b, J) -manifolds. A partial negative answer to Ichijyo’s¯ problem on nearly Kähler Finsler manifolds was given. The condition under which generalized (a, b, J) - manifolds are Berwaldian was obtained. Finally, we proved that under a mild assumption a nearly Kähler Finsler manifold is Landsbergian.

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APPENDIX

 2 00  0 00 0 0 0 00 2 2 0 0 b s 1 Ψ 2 s l  Ψ (sΨΨ + sΨ Ψ − ΨΨ )[Ψ s(sΨ + (b − s )Ψ ) + αΨ ( β (− β + α )) − α (b − αβ b0yl)]  Θ1 = + 0 0 2 2 00  2αβΨ 2αβΨ(Ψ − sΨ )((Ψ − sΨ ) + (b − s )Ψ )  1 b2 1 Ψ0 b2b2 b2 b2Ψ00 b2Ψ0 Θ = αΨ0 (− 0 + ) − (− 0 + ) + 0 1 − 2 2(Ψ − sΨ0) β3 β sΨ + (b2 − s2)Ψ0 β3 β β2 sΨ + (b2 − s2)Ψ0 b J αΨ0 Ψ0b2 Θ = 0 0 ( − Ψ00) −1 + 3 2(Ψ − sΨ0)β2 β sΨ + (b2 − s2)Ψ0 1 Θ = × 4 2(Ψ − sΨ0) " # α(sΨΨ00 + sΨ0Ψ0 − ΨΨ0) b b s Ψ00(sΨΨ00 + sΨ0Ψ0 − ΨΨ0) b s b b2 ( 0 − 0 ) − (− 0 + 0 ) 0 2 2 00 2 2 2 0 0 2 2 00 3 l 2 Ψ((Ψ − sΨ ) + (b − s )Ψ ) αβ β3 (sΨ + (b − s Ψ ))((Ψ − sΨ ) + (b − s )Ψ αβ b0yl β Ψ00 sb Ψ0 b (sΨΨ00 + sΨ0Ψ0 − ΨΨ0)(sΨ + (b2 − s2)Ψ0 Θ = 0 bily − 0 5 2α(Ψ − sΨ0) sΨ + (b2 − s2)Ψ0 l αΨ2((Ψ − sΨ0) + (b2 − s2)Ψ00) 1 b J Ψ0 b J b2 b2 b J b2Ψ0 Θ = αΨ0 − 0 0 − (− 0 0 + ) − Ψ00 0 0 1 + 6 2(Ψ − sΨ0) β3 sΨ + (b2 − s2)Ψ0 β3 β β2 sΨ + (b2 − s2)Ψ0 1 J2 1 Ψ0 J2b2 J2 b2Ψ0 Θ = αΨ0 (− 0 ) + + 0 − Ψ00 0 1 + 7 2(Ψ − sΨ0) β3 β sΨ + (b2 − s2)Ψ0 β3 β2 sΨ + (b2 − s2)Ψ0 1 Θ = × 8 2(Ψ − sΨ0) α(sΨΨ00 + sΨ0Ψ0 − ΨΨ0) J J s Ψ00(sΨΨ00 + sΨ0Ψ0 − ΨΨ0) J s J b2 ( 0 − 0 ) − (− 0 bl y + 0 ) Ψ((Ψ − sΨ0) + (b2 − s2)Ψ00 αβ2 β3 (sΨ + (b2 − s2)Ψ0)((Ψ − sΨ0) + (b2 − s2)Ψ00) αβ3 0 l β2 Ψ00 sJ Ψ0 J (sΨΨ00 + sΨ0Ψ0 − ΨΨ0)(sΨ + (b2 − s2)Ψ0 Θ = 0 bily − 0 9 2α(Ψ − sΨ0) sΨ + (b2 − s2)Ψ0 l αΨ2((Ψ − sΨ0) + (b2 − s2)Ψ00) −αΨ0(sΨΨ00 + sΨ0Ψ0 − ΨΨ0) h i Θ = b2s + s JlJ b +s ˜ bl J bi 10 (Ψ − sΨ0)β3(sΨ + (b2 − s2)Ψ0)((Ψ − sΨ0) + (b2 − s2)Ψ00) 0 0 l0 0 0 l0 0 0 0 αΨ0 αΨ0Ψ0 Θ = (b si + J s˜i ) − (b s )bi + (b s Jl)Ji +s ˜ J bil 11 (Ψ − sΨ0β) 0 0 0 0 β(Ψ − sΨ0)(sΨ + (b2s2)Ψ0) 0 0 0 l0 l0 0