On Twisted Products Finsler Manifolds

On the product of two Finsler manifolds M1 M2, we consider the twisted metric F which is construct by using Finsler metrics F1 and F2 on the manifolds M1 and M2, respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study Creducible and semi-C-reducible properties of this manifold. Then we obtain the Riemannian curvature and some of non-Riemannian curvatures of the twisted product Finsler manifold such as Berwald curvature, mean Berwald curvature and we find the relations between these objects and their corresponding objects on M1 and M2. Finally, we study locally dually flat twisted product Finsler manifold.


Introduction.
Twisted and warped product structures are widely used in geometry to construct newexamples of semi-Riemannian manifolds with interesting curvature properties. (see [3][10] [13][18] [21]). Twisted product metric tensors, as a generalization of warped product metric tensors, have also been useful in the study of several aspects of submanifold theory, namely, in hypersurfaces of complex space forms [14], in Lagrangian submanifolds [7] and in decomposition of curvature netted hypersurfaces [12], etc.
The notion of twisted product of Riemannian manifolds was mentioned first by Chen in [5], and was generalized for the pseudo-Riemannian case by Ponge and Reckziegel [22]. Chen extended the study of twisted product for CR-submanifolds in Kähler manifolds [6].
On the other hand, Finsler geometry is a natural extension of Riemannian geometry without the quadratic restriction. Therefore, it is natural to extending the construction of twisted product manifolds for Finsler geometry. In [13], Kozma-Peter-Shimada extended the construction of twisted product for the Finsler geometry.
Let (M 1 , F 1 ) and (M 2 , F 2 ) are two Finsler manifolds with Finsler metrics F 1 and F 2 , respectively, and f : M 1 × M 2 −→ R + be a smooth function. On the product manifold M 1 × M 2 , we consider the metric The manifold M 1 × M 2 endowed with this metric, we call the twisted product of the manifolds M 1 and M 2 and denote it by M 1 × f M 2 . The function f will be called the twisted function. In particular, if f is constant on M 2 , then M 1 × f M 2 is called warped product manifold.
Let (M, F ) be a Finsler manifold. The second and third order derivatives of 1 2 F 2 x at y ∈ T x M 0 are the symmetric trilinear forms g y and C y on T x M , which called the fundamental tensor and Cartan torsion, respectively. A Finsler metric is called semi-C-reducible if its Cartan tensor is given by where p = p(x, y) and q = q(x, y) are scalar function on T M , h ij is the angular metric and C 2 = I i I i [17]. If q = 0, then F is called C-reducible Finsler metric and if p = 0, then F is called C2-like metric.
The geodesic curves of a Finsler metric F on a smooth manifold M , are determined by the system of second order differential equationsc i + 2G i (ċ) = 0, where the local functions G i = G i (x, y) are called the spray coefficients. F is called a Berwald metric, if G i are quadratic in y ∈ T x M for any x ∈ M . Taking a trace of Berwald curvature yields mean Berwald curvature E. Then F is said to be isotropic mean Berwald metric if E = n+1 2 cF −1 h, where h = h ij dx i ⊗ dx j is the angular metric and c = c(x) is a scalar function on M [19].
The second variation of geodesics gives rise to a family of linear maps R y = R i k dx k ⊗ ∂ ∂x i | x : T x M → T x M at any point y ∈ T x M . R y is called the Riemann curvature in the direction y. A Finsler metric F is said to be of scalar flag curvature, if for some scalar function K on T M 0 the Riemann curvature is in the form R i k = KF 2 h i j . If K = constant, then F is said to be of constant flag curvature.
In this paper, we introduce the horizontal and vertical distributions on tangent bundle of a doubly warped product Finsler manifold and construct the Finsler connection on this manifold. Then, we study some geometric properties of this product manifold such as C-reducible and semi-C-reducible. Then, we introduce the Riemmanian curvature of twisted product Finsler manifold (M 1 × f M 2 , F ) and find the relation between it and Riemmanian curvatures of its components (M 1 , F 1 ) and (M 2 , F 2 ). In the cases that (M 1 × f M 2 , F ) is flat or it has the scalar flag curvature, we obtain some results on its components. Then, we study twisted product Finsler metrics with vanishing Berwald curvature and isotropic mean Berwald curvature, respectively. Finally, we study locally dually flat twisted product Finsler manifold. We prove that there is not exist any locally dually flat proper twisted product Finsler manifold.

Preliminary
Let M be an n-dimensional C ∞ manifold. Denote by T x M the tangent space at x ∈ M , by T M = ∪ x∈M T x M the tangent bundle of M , and by T M • = T M \ {0} the slit tangent bundle on M [20]. A Finsler metric on M is a function F : T M → [0, ∞) which has the following properties: (ii) F is positively 1-homogeneous on the fibers of tangent bundle T M ; (iii) for each y ∈ T x M , the following quadratic form g y on T x M is positive definite, The family C := {C y } y∈T M • is called the Cartan torsion. It is well known that C = 0 if and only if F is Riemannian [23]. For y ∈ T x M • , define mean Cartan torsion I y by I y (u) : h ij := F F y i y j is the angular metric. In [16]), it is proved that a Finsler metric F on a manifold M of dimension n ≥ 3 is a Randers metric if and only if M y = 0, ∀y ∈ T M 0 . A Randers metric F = α + β on a manifold M is just a Riemannian metric α = a ij y i y j perturbated by a one form β = b i (x)y i on M such that β α < 1. A Finsler metric is called semi-C-reducible if its Cartan tensor is given by where p = p(x, y) and q = q(x, y) are scalar function on T M and C 2 = I i I i with p + q = 1. In [17], Matsumoto-Shibata proved that every (α, β)-metric on a manifold M of dimension n ≥ 3 is semi-C-reducible.
Given a Finsler manifold (M, F ), then a global vector field G is induced by F on T M • , which in a standard coordinate (x i , y i ) for T M • is given by G is called the spray associated to (M, F ). In local coordinates, a curve c(t) is a geodesic if and only if its coordinates (c i (t)) satisfyc i + 2G i (ċ) = 0 [2]. A Finsler metric F = F (x, y) on a manifold M is said to be locally dually flat if at any point there is a coordinate system (x i ) in which the spray coefficients are in the following form where H = H(x, y) is a C ∞ scalar function on T M • satisfying H(x, λy) = λ 3 H(x, y) for all λ > 0. Such a coordinate system is called an adapted coordinate system. In [24], Shen proved that the Finsler metric F on an open subset U ⊂ R n is dually flat if and only if it satisfies ( B and E are called the Berwald curvature and mean Berwald curvature, respectively. Then F is called a Berwald metric and weakly Berwald metric if B = 0 and E = 0, respectively [23]. It is proved that on a Berwald space, the parallel translation along any geodesic preserves the Minkowski functionals [11]. A Finsler metric F is said to be isotropic Berwald metric and isotropic mean Berwald metric if its Berwald curvature and mean Berwald curvature is in the following form, respectively B i jkl = c{F y j y k δ i l + F y k y l δ i j + F y l y j δ i k + F y j y k y l y i }, (2 .1) where c = c(x) is a scalar function on M [8] [25].
is a family of linear maps on tangent spaces, defined by The flag curvature in Finsler geometry is a natural extension of the sectional curvature in Riemannian geometry was first introduced by L. Berwald [4]. For a flag P = span{y, u} ⊂ T x M with flagpole y, the flag curvature K = K(P, y) is defined by K(P, y) := g y (u, R y (u)) g y (y, y)g y (u, u) − g y (y, u) 2 .
(2 .4) We say that a Finsler metric F is of scalar curvature if for any y ∈ T x M , the flag curvature K = K(x, y) is a scalar function on the slit tangent bundle T M • . If K = constant, then F is said to be of constant flag curvature.

Nonlinear Connection
Let (M 1 , F 1 ) and (M 2 , F 2 ) be two Finsler manifolds. Then the functions Then by using (3 .5) we conclude that . . , n 2 } and a, b, . . . ∈ {1, . . . , n 1 + n 2 }. Now we consider the the spray coefficients of F 1 , F 2 and F as Taking into account the homogeneity of both F 2 1 and F 2 2 , and using (3 .7) and (3 .8), we can conclude that G i and G α are positively homogeneous of degree two with respect to (y i ) and (v α ), respectively. Hence from Euler theorem for homogeneous functions, we infer that By setting a = i in (3 .9) we have

Direct calculations give us
Putting these equations together g ih = g ih in the above equation and using (3 .7) imply that Similarly, by setting a = α in (3 .9) and using (3 .8) we obtain where G a , G i and G α are given by (3 .9), (3 .10) and (3 .11), respectively. Now, we put Then we have the following.
Next, V T M • kernel of the differential of the projection map is spanned by the natural vector fields { ∂ ∂y 1 , . . . , ∂ ∂y n 1 , ∂ ∂v 1 , . . . , ∂ ∂v n 2 } and it is called the twisted vertical distribution on T M • . Then, using the functions given by (3 .14)-(3 .17), the nonholonomic vector fields are defined as following which make it possible to construct a complementary vector subbundle HT M • is called the twisted horizontal distribution on T M • . Thus the tangent bundle of T M • admits the decomposition In the following, we compute the non-linear connection of a twisted product Finsler manifold.
Definition 1. Using decomposition (3 .20), the twisted vertical morphism v t : For this projective morphism, the following hold From the above equations, we conclude that This mapping is called the twisted vertical projective. Definition 2. Using decomposition (3 .20), the doubly warped horizontal pro- For this projective morphism, the following hold Thus we result that Definition 3. Using decomposition (3 .20), the twisted almost tangent structure Thus we result that Here, we introduce some geometrical objects of twisted product Finsler manifold. In order to simplify the equations, we rewritten the basis of HT M • and V T M • as follows: Thus The Lie brackets of this basis is given by [ ∂ ∂y a , Therefore, we have the following.
With a simple calculation, we have the following.
where C kh i;j = ∂C kh i ∂y j . Apart from G c ab , the functions F c ab given by Proof. By using (3 .28) we have Since g ij is a function with respect to (x, y), then by (3 .14) and (3 .18) we obtain Interchanging i, j and h in the above equation gives us Putting these equation in (3 .35), give us (3 .29). In the similar way, we can prove the another relation.
Lemma 3. Let C k ij and C γ αβ be the local components of Cartan tensor field on M 1 and M 2 , respectively. Then we have and C k iβ = C k αj = C k αβ = C γ ij = C γ iβ = C γ αj = 0. By using the Lemma 3, we can get the following. Various interesting special forms of Cartan tensors have been obtained by some Finslerians [17]. The Finsler spaces having such special forms have been called C-reducible, C2-like, semi-C-reducible, and etc. In [15], Matsumoto introduced the notion of C-reducible Finsler metrics and proved that any Randers metric is C-reducible. Later on, Matsumoto-Hōjō proves that the converse is true too [16].
Here, we define the Matsumoto twisted tensor M abc for a twisted product Finsler manifold (M 1 × f M 2 , F ) as follows: where I a = g bc C abc , C abc = g cd C d ab and h ab = g ab − 1 F 2 y a y b . By attention to the above equation and relations Contracting the above equation in y j y k give us Therefore if M iβλ = M αjk = 0, then we get I i = I α = 0, i.e., (M 1 , F 1 ) and (M 2 , F 2 ) are Riemannian manifolds. Thus we have Theorem 1. There is not exist any C-reducible twisted product Finsler manifold.
Now, we are going to consider semi-C-reducible twisted product Finsler manifold (M 1 × f M 2 , F ). Let (M 1 × f M 2 , F ) be a semi-C-reducible twisted product Finsler manifold. Then we have where C 2 = I a I a and p and q are scalar function on M 1 × f M 2 with p + q = 1. This equation gives us Contraction the above equation with y j y k implies that Therefore we have p = 0 or I α = 0. If p = 0, then F is C2-like metric. But if p = 0, then I α = 0, i. e., F 2 is Riemannian metric. In this case, with similar way we conclude that F 1 is Riemannian metric. But, by definition F can not be a Riemannian metric. Therefore we have

Riemannian Curvature
The Riemannian curvature of twisted product Finsler manifold (M 1 × f M 2 , F ) with respect to Berwald connection is given by (4 .36) where R a cd and y b R a b cd are given by (3 .27) and (4 .36). Proof. By using (4 .36), we have (4 .37) By using Corollary 3 and Lemma 2, we obtain Interchanging i and j in the above equation imply that j denotes the interchange of indices i, j and subtraction.

Twisted Product Finsler manifolds With Non-Riemannian Curvature Properties
There are several important non-Riemannian quantities such as the Berwald curvature B, the mean Berwald curvature E and the Landsberg curvature L, etc [26]. They all vanish for Riemannian metrics, hence they are said to be non-Riemannian. In this section, we find some necessary and sufficient conditions under which a twisted product Riemannian manifold are Berwaldian, of isotropic Berwald curvature, of isotropic mean Berwald curvature. First, we prove the following.
Lemma 6. Let (M 1 × f M 2 , F ) be a twisted product Finsler manifold . Then the coefficients of Berwald curvature are as follows:  is Berwaldian and the Berwald curvature of (M 2 , F 2 ) satisfies in the following equation: Here, we consider twisted product Finsler manifold (M 1 × f M 2 , F ) of isotropic Berwald curvature.

Theorem 8. Every isotropic Berwald twisted product Finsler manifold
where c = c(x) is a function on M . Setting a = j, b = k, c = l, d = γ and using (5 .61) imply that cF −1 { 3 F 2 y j y k y l v γ − v γ (y j g kl + y k g jl + y l g jk )} = 0.
Multiplying the above equation in y j y k , we derive that cf 2 F 2 1 F 2 2 = 0. Thus we have c = 0, i.e., (M 1 × f M 2 ) is Berwaldian. Now, we are going to study twisted product Finsler manifold of isotropic mean Berwald curvature. For this work, we must compute the coefficients of mean Berwald curvature of a twisted product Finsler manifold. Lemma 7. Let (M 1 × f M 2 , F ) be a twisted product Finsler manifold. Then the coefficients of mean Berwald curvature are as follows: where E ij and E αβ are the coefficients of mean Berwald curvature of (M 1 , F 1 ) and (M 2 , F 2 ), respectively.
Proof. By definition and Lemma 6, we get the proof.
Proof. If (M 1 × f M 2 ) be a weakly Berwald manifold, then we have  Now, we consider twisted product Finsler manifolds with isotropic mean Berwald curvature. It is remarkable that as a consequence of Lemma 7, we have the following.
where c = c(x) is a scalar function on M .
Theorem 10. Every twisted product Finsler manifold (M 1 × f M 2 , F ) with isotropic mean Berwald curvature is a weakly Berwald manifold.
Proof. Suppose that F is isotropic mean Berwald twisted product Finsler metric. Then differentiating (5 .69) with respect v γ gives us c(n + 1)f 2 F −5 v γ y i = 0 Thus, we conclude that c = 0. This implies that F reduces to a weakly Berwald metric.

Locally Dually Flat Twisted Product Finsler Manifolds
In [1], Amari-Nagaoka introduced the notion of dually flat Riemannian metrics when they study the information geometry on Riemannian manifolds. Information geometry has emerged from investigating the geometrical structure of a family of probability distributions and has been applied successfully to various areas including statistical inference, control system theory and multi-terminal information theory. In Finsler geometry, Shen extends the notion of locally dually flatness for Finsler metrics [24]. Dually flat Finsler metrics form a special and valuable class of Finsler metrics in Finsler information geometry, which play a very important role in studying flat Finsler information structure [9] [28].
In this section, we study locally dually flat twisted product Finsler metrics. It is remarkable that, a Finsler metric F = F (x, y) on a manifold M is said to be locally dually flat if at any point there is a standard coordinate system (x a , y a ) in T M such that it satisfies (6 .70) In this case, the coordinate (x a ) is called an adapted local coordinate system. By using (6 .70) we can obtain the following lemma. Now, let F be a locally dually flat Finsler metric. Taking derivative with respect to v γ from (6 .71) yields f l = 0, which means that f is a constant function on M 1 . In this case, the relations (6 .71) and (6 .72) reduce to the following ∂ 2 F 2 1 ∂x k ∂y l y k = 2 (6 .74) By (6 .73), we deduce that F 1 is locally dually flat. Now, we assume that F 1 and F 2 are locally dually flat Finsler metrics. Then we have Therefore we can conclude the following.
(i) If F is locally dually flat then F 1 is locally dually flat, f is a function with respect (u α ) only and F 2 satisfy in (6 .74).
(ii) If F 1 and F 2 are locally dually flat. Then F is locally dually flat if and only if f is a function with respect (u α ) only and F 2 satisfies in (6 .77).
By Theorem 11, we conclude the following.
Corollary 6. There is not exist any locally dually flat proper twisted product Finsler manifold.