ISRN Geometry

Volume 2013, Article ID 732432, 12 pages

http://dx.doi.org/10.1155/2013/732432

## On Twisted Products Finsler Manifolds

^{1}Faculty of Science, Department of Mathematics, Arak University, Arak 38156-8-8349, Iran^{2}Faculty of Science, Department of Mathematics, Qom University, Qom 3716146611, Iran

Received 16 May 2013; Accepted 10 June 2013

Academic Editors: I. Biswas and A. Borowiec

Copyright © 2013 E. Peyghan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

On the product of two Finsler manifolds , we consider the twisted metric which is constructed by using Finsler metrics and on the manifolds and , respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study C-reducible 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 and . Finally, we study locally dually flat twisted product Finsler manifold.

#### 1. Introduction

Twisted and warped product structures are widely used in geometry to construct new examples of semi-Riemannian manifolds with interesting curvature properties (see [1–3]). 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 [4], in Lagrangian submanifolds [5], in decomposition of curvature netted hypersurfaces [6], and so forth.

The notion of twisted product of Riemannian manifolds was mentioned first by Chen in [7] and was generalized for the pseudo-Riemannian case by Ponge and Reckziegel [8]. Chen extended the study of twisted product for CR-submanifolds in Kähler manifolds [9].

On the other hand, Finsler geometry is a natural extension of Riemannian geometry without the quadratic restriction. Therefore, it is natural to extend the construction of twisted product manifolds for Finsler geometry. In [10], Kozma-Peter-Shimada extended the construction of twisted product for the Finsler geometry.

Let and be two Finsler manifolds with Finsler metrics and , respectively, and let be a smooth function. On the product manifold , we consider the metric for all and , where is the slit tangent manifold . The manifold endowed with this metric, we call the twisted product of the manifolds and and denote it by . The function will be called the twisted function. In particular, if is constant on , then is called warped product manifold.

Let be a Finsler manifold. The second and third order derivatives of at are the symmetric trilinear forms and on , 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 and are scalar function on , is the angular metric, and [11]. If , then is called C-reducible Finsler metric, and if , then is called -like metric.

The geodesic curves of a Finsler metric on a smooth manifold are determined by the system of second-order differential equations , where the local functions are called the spray coefficients. is called a Berwald metric, if are quadratic in for any . Taking a trace of Berwald curvature yields mean Berwald curvature . Then is said to be isotropic mean Berwald metric if , where is the angular metric and is a scalar function on [12].

The second variation of geodesics gives rise to a family of linear maps at any point . is called the Riemann curvature in the direction . A Finsler metric is said to be of scalar flag curvature, if for some scalar function on the Riemann curvature is in the form . If , then 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 and find the relation between it and Riemmanian curvatures of its components and . In the cases that 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.

#### 2. Preliminary

Let be an -dimensional manifold. Denote by the tangent space at , by the tangent bundle of , and by the slit tangent bundle on [13]. A Finsler metric on is a function which has the following properties:(i)is on ;(ii) is positively 1-homogeneous on the fibers of tangent bundle ;(iii)for each , the following quadratic form on is positive definite: Let and . To measure the non-Euclidean feature of , define by The family is called the Cartan torsion. It is well known that if and only if is Riemannian [14].

For , define mean Cartan torsion by , where , and . By Deicke's theorem, is Riemannian if and only if .

Let be a Finsler manifold. For , define the Matsumoto torsion by , where where is the angular metric. In [15], it is proved that a Finsler metric on a manifold of dimension is a Randers metric if and only if , for all . A Randers metric on a manifold is just a Riemannian metric perturbed by a one form on such that .

A Finsler metric is called semi-C-reducible if its Cartan tensor is given by where and are scalar function on and with . In [11], Matsumoto-Shibata proved that every metric on a manifold of dimension is semi-C-reducible.

Given a Finsler manifold , then a global vector field is induced by on , which in a standard coordinate for is given by , where is called the spray associated to . In local coordinates, a curve is a geodesic if and only if its coordinates satisfy [16].

A Finsler metric on a manifold is said to be locally dually flat if at any point there is a coordinate system in which the spray coefficients are in the following form: where is a scalar function on satisfying for all . Such a coordinate system is called an adapted coordinate system. In [17], Shen proved that the Finsler metric on an open subset is dually flat if and only if it satisfies .

For a tangent vector , define and by and , where and are called the Berwald curvature and mean Berwald curvature, respectively. Then is called a Berwald metric and weakly Berwald metric if and , respectively [14]. It is proved that on a Berwald space, the parallel translation along any geodesic preserves the Minkowski functionals [18].

A Finsler metric is said to be isotropic Berwald metric and isotropic mean Berwald metric if its Berwald curvature and mean Berwald curvature are in the following form, respectively: where is a scalar function on [19].

The Riemann curvature 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 [20]. For a flag with flagpole , the flag curvature is defined by We say that a Finsler metric is of scalar curvature if for any , the flag curvature is a scalar function on the slit tangent bundle . If , then is said to be of constant flag curvature.

#### 3. Nonlinear Connection

Let and be two Finsler manifolds. Then the functions define a Finsler tensor field of type on and , respectively. Now let be a doubly warped Finsler manifold, , , , and . Then by using (13), we conclude that where , , , , , , and .

Now we consider spray coefficients of , , and as Taking into account the homogeneity of both and and using (15) and (16), we can conclude that and are positively homogeneous of degree two with respect to and , respectively. Hence from Euler theorem for homogeneous functions, we infer that By setting in (17), we have Direct calculations give us Putting these equations together in the previous equation and using (15) imply that Similarly, by setting in (17) and using (16), we obtain where , , , and . Therefore we have , where , , and are given by (17), (21), and (22), respectively.

Now, we put Then we have the following.

Lemma 1. *The coefficients defined by (23) satisfy in the following:
**
where
*

Next, kernel of the differential of the projection map
which is a well-defined subbundle of , is considered. Locally, is spanned by the natural vector fields , and it is called the *twisted vertical distribution* on . Then, using the functions given by (25)–(28), the nonholonomic vector fields are defined as follows:
which make it possible to construct a complementary vector subbundle to in as follows:
is called the *twisted horizontal distribution* on . Thus the tangent bundle of admits the decomposition
It is shown that is a nonlinear connection on . In the following, we compute the nonlinear connection of a twisted product Finsler manifold.

Proposition 2. *If is a twisted product Finsler manifold, then is the nonlinear connection on . Further, one has
*

*Definition 3. *Using decomposition (33), the twisted vertical morphism is defined by
where

For this projective morphism, the following hold:
From the previous equations, we conclude that
This mapping is called the *twisted vertical projective*.

*Definition 4. *Using decomposition (33), the doubly warped horizontal projective is defined by
or

For this projective morphism, the following hold: Thus we result that

*Definition 5. *Using decomposition (33), the twisted almost tangent structure is defined by
or

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 and as follows: Thus The Lie brackets of this basis is given by where

Therefore, we have the following.

Corollary 6. *Let be a twisted product Finsler manifold. Then
**
where
*

With a simple calculation, we have the following.

Corollary 7. *Let be a twisted product Finsler manifold. Then
**
where
**
where . Apart from , the functions are given by
*

Corollary 8. *Let be a twisted product Finsler manifold. Then
**
where
*

*Proof. *By using (55), we have
Since is a function with respect to , then by (25) and (30) we obtain
Interchanging , , and in the previous equation gives us
Putting these equation in (64) give us (57). In the similar way, we can prove the another relation.

By using (i) of (23) and (57)–(62), we can conclude the following.

Lemma 9. *Let be a twisted product Finsler manifold. Then , where and are defined by (55) and (i) of (23), respectively.*

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry, and it is first introduced by Finsler and emphasized by Cartan which measures a departure from a Riemannian manifold. More precisely, a Finsler metric reduces to a Riemannian metric if and only if it has vanishing Cartan torsion. The local components of Cartan tensor field of the twisted Finsler manifold is defined by From this definition, we conclude the following.

Lemma 10. *Let and be the local components of Cartan tensor field on and , respectively. Then one has
**
where
**
and .*

By using the Lemma 10, we can get the following.

Corollary 11. *Let be a twisted product Finsler manifold. Then is a Riemannian manifold if and only if and are Riemannian manifold.*

Various interesting special forms of Cartan tensors have been obtained by some Finslerians [11]. The Finsler spaces having such special forms have been called C-reducible, C2-like, semi-C-reducible, and so forth. In [21], Matsumoto introduced the notion of C-reducible Finsler metrics and proved that any Randers metric is C-reducible. Later on, Matsumoto-Hj proves that the converse is true too [15].

Here, we define the Matsumoto twisted tensor for a twisted product Finsler manifold as follows: where , , and . By attention to the previous equation and relations we obtain Contracting the previous equation in gives us Similarly, we obtain Therefore if , then we get ; that is, and are Riemannian manifolds. Thus we have the following.

Theorem 12. *There is not exist any C-reducible twisted product Finsler manifold.*

Now, we are going to consider semi-C-reducible twisted product Finsler manifold . Let be a semi-C-reducible twisted product Finsler manifold. Then we have where and and are scalar function on with . This equation gives us Contractiing the previous equation with implies that Therefore we have or . If , then is -like metric. But if , then ; that is, is Riemannian metric. In this case, with similar way, we conclude that is Riemannian metric. But, definition cannot be a Riemannian metric. Therefore we have the following.

Theorem 13. *Every semi-C-reducible twisted product Finsler manifold is a -like manifold.*

#### 4. Riemannian Curvature

The Riemannian curvature of twisted product Finsler manifold with respect to Berwald connection is given by

Lemma 14. *Let be a twisted product Finsler manifold. Then one has
**
where and are given by (50) and (78).*

*Proof. *By using (78), we have
By using Corollary 8 and Lemma 9, we obtain
Interchanging and in the previous equation implies that
Setting (81) and (82) in (80) gives us . In the similar way, we can obtain this relation for another indices.

Using (78), we can compute the Riemannian curvature of a twisted product Finsler manifold.

Lemma 15. *Let be a twisted product Finsler manifold. Then the coefficients of Riemannian curvature are as follows:
**
where
**
and denotes the interchange of indices , , and subtraction.*

By Theorem 18, we have the following.

Theorem 16. *Let be a flat twisted product Finsler manifold, and let be Riemannian. If is a function on , only, then is locally flat.*

Similarly, we get the following.

Theorem 17. *Let be a flat twisted product Finsler manifold, and let be Riemannian. If is a function on , only, then is a space of positive constant curvature .*

*Proof. *Since is Riemannain and is a function on , then by (94), we obtain
Since is flat, then . Thus the proof is complete.

Theorem 18. *Let be a twisted product Riemannian manifold, and let be a function on , only. Then is flat, if and only if is flat, and the Riemannian curvature of satisfies in the following equation:
*

#### 5. Twisted Product Finsler Manifolds with Non-Riemannian Curvature Properties

There are several important non-Riemannian quantities such as the Berwald curvature , the mean Berwald curvature and the Landsberg curvature [22]. 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 19. *Let be a twisted product Finsler manifold. Then the coefficients of Berwald curvature are as follows:
*

Let is a Berwald manifold. Then we have . By using (102), we get Multiplying this equation in , we obtain Thus if is not constant on , then we have . Also, from (101), we result that Differentiating this equation with respect to gives us Similarly we obtain Setting the last equation in (99) implies that ; that is, is Berwaldian. These explanations give us the following theorem.

Theorem 20. *Let be a twisted product Finsler manifold, and let be not constant on . Then is Berwaldian if and only if is Berwaldian, is Riemannian, and the equation is hold.*

But if is constant on , that is, , then we get the following.

Theorem 21. *Let be a twisted product Finsler manifold, and is constant on . Then is Berwaldian if and only if is Berwaldian and the Berwald curvature of satisfies in the following equation:
*

Here, we consider twisted product Finsler manifold of isotropic Berwald curvature.

Theorem 22. *Every isotropic Berwald twisted product Finsler manifold is a Berwald manifold.*

*Proof. *Let be an isotropic Berwald manifold. Then we have
where is a function on . Setting , , , and and using (103) imply that
Multiplying the previous equation in , we derive that . Thus we have ; that is, 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 23. *Let be a twisted product Finsler manifold. Then the coefficients of mean Berwald curvature are as follows:
**
where and are the coefficients of mean Berwald curvature of and , respectively.*

*Proof. *By definition and Lemma 19, we get the proof.

Theorem 24. *The twisted product Finsler manifold is weakly Berwald if and only if is weakly Berwald, , and the following hold:
*

*Proof. *If be a weakly Berwald manifold, then we have
Thus by using (114), we result that . This equation implies that
By setting these equations in (112) and (113), we conclude that and satisfies in (115).

Now, if is constant on , then (115) implies that . Thus we conclude the following.

Corollary 25. *Let be a twisted product Finsler manifold, and let be a function on , only. Then is weakly Berwald if and only if and are weakly Berwald manifolds and .*

Now, we consider twisted product Finsler manifolds with isotropic mean Berwald curvature. It is remarkable that as a consequence of Lemma 23, we have the following.

Lemma 26. *Twisted product Finsler manifold is isotropic mean Berwald manifold if and only if
*