On Twisted Products Finsler Manifolds

Abstract

On the product of two Finsler manifolds 𝑀₁×𝑀₂, we consider the twisted metric 𝐹 which is constructed 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 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 𝑀₁ and 𝑀₂. 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 (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂) be two Finsler manifolds with Finsler metrics 𝐹₁ and 𝐹₂, respectively, and let 𝑓∶𝑀₁×𝑀₂→𝑅⁺ be a smooth function. On the product manifold 𝑀₁×𝑀₂, we consider the metric 𝐹𝑣₁,𝑣₂=𝐹²₁𝑣₁+𝑓²𝑥,𝑦𝐹²₂𝑣₂ (1) for all (𝑥,𝑦)∈𝑀₁×𝑀₂ and (𝑣₁,𝑣₂)∈𝑇𝑀^∘₁×𝑇𝑀^∘₂, where 𝑇𝑀^∘₁ is the slit tangent manifold 𝑇𝑀^∘₁=𝑇𝑀₁⧵{∘}. The manifold 𝑀₁×𝑀₂ endowed with this metric, we call the twisted product of the manifolds 𝑀₁ and 𝑀₂ and denote it by 𝑀₁×_𝑓𝑀₂. The function 𝑓 will be called the twisted function. In particular, if 𝑓 is constant on 𝑀₂, then 𝑀₁×_𝑓𝑀₂ is called warped product manifold.

Let (𝑀,𝐹) be a Finsler manifold. The second and third order derivatives of (1/2)𝐹²_𝑥 at 𝑦∈𝑇_𝑥𝑀₀ are the symmetric trilinear forms 𝐠_𝑦 and 𝐂_𝑦 on 𝑇_𝑥𝑀, which called the fundamental tensor and Cartan torsion, respectively. A Finsler metric is called semi-C-reducible if its Cartan tensor is given by 𝐶_𝑖𝑗𝑘=𝑝ℎ1+𝑛^𝑖𝑗𝐼_𝑘+ℎ_𝑗𝑘𝐼_𝑖+ℎ_𝑘𝑖𝐽_𝑗+𝑞𝐶²𝐼_𝑖𝐼_𝑗𝐼_𝑘, (2) where 𝑝=𝑝(𝑥,𝑦) and 𝑞=𝑞(𝑥,𝑦) are scalar function on 𝑇𝑀, ℎ_𝑖𝑗 is the angular metric, and 𝐶²=𝐼^𝑖𝐼_𝑖 [11]. 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 equations 𝑐^𝑖̈+2𝐺^𝑖(𝑐)̇=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_𝑥𝑀 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⁻¹𝐡, where 𝐡=ℎ_𝑖𝑗𝑑𝑥^𝑖⊗𝑑𝑥^𝑗 is the angular metric and 𝑐=𝑐(𝑥) is a scalar function on 𝑀 [12].

The second variation of geodesics gives rise to a family of linear maps 𝐑_𝑦=𝑅^𝑖_𝑘𝑑𝑥^𝑘⊗(𝜕/𝜕𝑥^𝑖)|_𝑥∶𝑇_𝑥𝑀→𝑇_𝑥𝑀 at any point 𝑦∈𝑇_𝑥𝑀. 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 𝑇𝑀₀ the Riemann curvature is in the form 𝑅^𝑖_𝑘=𝐊𝐹²ℎ^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 (𝑀₁×_𝑓𝑀₂,𝐹) and find the relation between it and Riemmanian curvatures of its components (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂). In the cases that (𝑀₁×_𝑓𝑀₂,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.

2. Preliminary

Let 𝑀 be an 𝑛-dimensional 𝐶^∞ manifold. Denote by 𝑇_𝑥𝑀 the tangent space at 𝑥∈𝑀, by 𝑇𝑀=∪_𝑥∈𝑀𝑇_𝑥𝑀 the tangent bundle of 𝑀, and by 𝑇𝑀^∘=𝑇𝑀⧵{0} the slit tangent bundle on 𝑀 [13]. A Finsler metric on 𝑀 is a function 𝐹∶𝑇𝑀→[0,∞) which has the following properties:(i)𝐹is 𝐶^∞ on 𝑇𝑀^∘;(ii)𝐹 is positively 1-homogeneous on the fibers of tangent bundle 𝑇𝑀;(iii)for each 𝑦∈𝑇_𝑥𝑀, the following quadratic form 𝐠_𝑦 on 𝑇_𝑥𝑀 is positive definite: 𝐠_𝑦()1𝑢,𝑣∶=2𝜕²𝐹𝜕𝑠𝜕𝑡²𝑦+𝑠𝑢+𝑡𝑣∣_𝑠,𝑡=0,𝑢,𝑣∈𝑇_𝑥𝑀. (3)Let 𝑥∈𝑀 and 𝐹_𝑥∶=𝐹∣_𝑇𝑥𝑀. To measure the non-Euclidean feature of 𝐹_𝑥, define 𝐂_𝑦∶𝑇_𝑥𝑀⊗𝑇_𝑥𝑀⊗𝑇_𝑥𝑀→ℝ by 𝐂_𝑦()1𝑢,𝑣,𝑤∶=2𝑑𝐠𝑑𝑡^{𝑦+𝑡𝑤}(𝑢,𝑣)∣_𝑡=0,𝑢,𝑣,𝑤∈𝑇_𝑥𝑀. (4) The family 𝐂∶={𝐂_𝑦}_{𝑦∈𝑇𝑀}^∘ is called the Cartan torsion. It is well known that 𝐂=0 if and only if 𝐹 is Riemannian [14].

For 𝑦∈𝑇_𝑥𝑀^∘, define mean Cartan torsion 𝐈_𝑦 by 𝐈_𝑦(𝑢)∶=𝐼_𝑖(𝑦)𝑢^𝑖, where 𝐼_𝑖∶=𝑔^𝑗𝑘𝐶_𝑖𝑗𝑘, 𝐶_𝑖𝑗𝑘=(1/2)(𝜕𝑔_𝑖𝑗/𝜕𝑦^𝑘) and 𝑢=𝑢^𝑖(𝜕/𝜕𝑥^𝑖)∣_𝑥. By Deicke's theorem, 𝐹 is Riemannian if and only if 𝐈_𝑦=0.

Let (𝑀,𝐹) be a Finsler manifold. For 𝑦∈𝑇_𝑥𝑀^∘, define the Matsumoto torsion 𝐌_𝑦∶𝑇_𝑥𝑀⊗𝑇_𝑥𝑀⊗𝑇_𝑥𝑀→ℝ by 𝐌_𝑦(𝑢,𝑣,𝑤)∶=𝑀_𝑖𝑗𝑘(𝑦)𝑢^𝑖𝑣^𝑗𝑤^𝑘, where 𝑀_𝑖𝑗𝑘∶=𝐶_𝑖𝑗𝑘−1𝐼𝑛+1^𝑖ℎ_𝑗𝑘+𝐼_𝑗ℎ_𝑖𝑘+𝐼_𝑘ℎ_𝑖𝑗, (5) where ℎ_𝑖𝑗∶=𝐹𝐹_{𝑦𝑖𝑦𝑗} is the angular metric. In [15], 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, for all 𝑦∈𝑇𝑀₀. A Randers metric 𝐹=𝛼+𝛽 on a manifold 𝑀 is just a Riemannian metric 𝛼=𝑎_𝑖𝑗𝑦^𝑖𝑦^𝑗 perturbed 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 𝐶_𝑖𝑗𝑘=𝑝ℎ1+𝑛^𝑖𝑗𝐼_𝑘+ℎ_𝑗𝑘𝐼_𝑖+ℎ_𝑘𝑖𝐼_𝑗+𝑞𝐶²𝐼_𝑖𝐼_𝑗𝐼_𝑘, (6) where 𝑝=𝑝(𝑥,𝑦) and 𝑞=𝑞(𝑥,𝑦) are scalar function on 𝑇𝑀 and 𝐶²=𝐼^𝑖𝐼_𝑖 with 𝑝+𝑞=1. In [11], 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 𝑇𝑀^∘, which in a standard coordinate (𝑥^𝑖,𝑦^𝑖) for 𝑇𝑀^∘ is given by 𝐆=𝑦^𝑖(𝜕/𝜕𝑥^𝑖)−2𝐺^𝑖(𝑥,𝑦)(𝜕/𝜕𝑦^𝑖), where 𝐺^𝑖1∶=4𝑔^𝑖𝑙𝜕²𝐹²𝜕𝑥^𝑘𝜕𝑦^𝑙𝑦^𝑘−𝜕𝐹²𝜕𝑥^𝑙,𝑦∈𝑇_𝑥𝑀. (7)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)) satisfy 𝑐^𝑖̈+2𝐺^𝑖(𝑐)̇=0 [16].

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: 𝐺^𝑖1=−2𝑔^𝑖𝑗𝐻_𝑦𝑗, (8) where 𝐻=𝐻(𝑥,𝑦) is a 𝐶^∞ scalar function on 𝑇𝑀^∘ satisfying 𝐻(𝑥,𝜆𝑦)=𝜆³𝐻(𝑥,𝑦) for all 𝜆>0. Such a coordinate system is called an adapted coordinate system. In [17], Shen proved that the Finsler metric 𝐹 on an open subset 𝑈⊂ℝ^𝑛 is dually flat if and only if it satisfies (𝐹²)_{𝑥𝑘𝑦𝑙}𝑦^𝑘=2(𝐹²)_𝑥𝑙.

For a tangent vector 𝑦∈𝑇_𝑥𝑀^∘, define 𝐁_𝑦∶𝑇_𝑥𝑀⊗𝑇_𝑥𝑀⊗𝑇_𝑥𝑀→𝑇_𝑥𝑀 and 𝐄_𝑦∶𝑇_𝑥𝑀⊗𝑇_𝑥𝑀→ℝ by 𝐁_𝑦(𝑢,𝑣,𝑤)∶=𝐵^𝑖_𝑗𝑘𝑙(𝑦)𝑢^𝑗𝑣^𝑘𝑤^𝑙(𝜕/𝜕𝑥^𝑖)∣_𝑥 and 𝐄_𝑦(𝑢,𝑣)∶=𝐸_𝑗𝑘(𝑦)𝑢^𝑗𝑣^𝑘, where 𝐵^𝑖_𝑗𝑘𝑙𝜕∶=³𝐺^𝑖𝜕𝑦^𝑗𝜕𝑦^𝑘𝜕𝑦^𝑙,𝐸_𝑗𝑘1∶=2𝐵^𝑚_𝑗𝑘𝑚. (9)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 [14]. It is proved that on a Berwald space, the parallel translation along any geodesic preserves the Minkowski functionals [18].

A Finsler metric 𝐹 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: 𝐵^𝑖_𝑗𝑘𝑙=𝑐𝐹_{𝑦𝑗𝑦𝑘}𝛿^𝑖_𝑙+𝐹_{𝑦𝑘𝑦𝑙}𝛿^𝑖_𝑗+𝐹_{𝑦𝑙𝑦𝑗}𝛿^𝑖_𝑘+𝐹_{𝑦𝑗𝑦𝑘𝑦𝑙}𝑦^𝑖,𝐸_𝑖𝑗=12(𝑛+1)𝑐𝐹⁻¹ℎ_𝑖𝑗, (10) where 𝑐=𝑐(𝑥) is a scalar function on 𝑀 [19].

The Riemann curvature 𝐑_𝑦=𝑅^𝑖_𝑘𝑑𝑥^𝑘⊗(𝜕/𝜕𝑥^𝑖)∣_𝑥∶𝑇_𝑥𝑀→𝑇_𝑥𝑀 is a family of linear maps on tangent spaces defined by 𝑅^𝑖_𝑘=2𝜕𝐺^𝑖𝜕𝑥^𝑘−𝑦^𝑗𝜕²𝐺^𝑖𝜕𝑥^𝑗𝜕𝑦^𝑘+2𝐺^𝑗𝜕²𝐺^𝑖𝜕𝑦^𝑗𝜕𝑦^𝑘−𝜕𝐺^𝑖𝜕𝑦^𝑗𝜕𝐺^𝑗𝜕𝑦^𝑘. (11) 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 𝑃=span{𝑦,𝑢}⊂𝑇_𝑥𝑀 with flagpole 𝑦, the flag curvature 𝐊=𝐊(𝑃,𝑦) is defined by 𝐊𝐠𝑃,𝑦∶=^𝑦𝑢,𝐑_𝑦(𝑢)𝐠_𝑦𝑦,𝑦𝐠_𝑦(𝑢,𝑢)−𝐠_𝑦𝑦,𝑢². (12) We say that a Finsler metric 𝐹 is of scalar curvature if for any 𝑦∈𝑇_𝑥𝑀, the flag curvature 𝐊=𝐊(𝑥,𝑦) is a scalar function on the slit tangent bundle 𝑇𝑀^∘. If 𝐊=constant, then 𝐹 is said to be of constant flag curvature.

3. Nonlinear Connection

Let (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂) be two Finsler manifolds. Then the functions (i)𝑔_𝑖𝑗=1𝑥,𝑦2𝜕²𝐹²₁𝑥,𝑦𝜕𝑦^𝑖𝜕𝑦^𝑗,(ii)𝑔_𝛼𝛽()=1𝑢,𝑣2𝜕²𝐹²₂(𝑢,𝑣)𝜕𝑣^𝛼𝜕𝑣^𝛽 (13) define a Finsler tensor field of type (0,2) on 𝑇𝑀^∘₁ and 𝑇𝑀^∘₂, respectively. Now let (𝑀₁×_𝑓𝑀₂,𝐹) be a doubly warped Finsler manifold, 𝐱=(𝑥,𝑢)∈𝑀, 𝐲=(𝑦,𝑣)∈𝑇_𝐱𝑀, 𝑀=𝑀₁×𝑀₂, and 𝑇_𝐱𝑀=𝑇_𝑥𝑀₁⊕𝑇_𝑢𝑀₂. Then by using (13), we conclude that 𝐠_𝑎𝑏=1𝑥,𝑢,𝑦,𝑣2𝜕²𝐹²𝑥,𝑢,𝑦,𝑣𝜕𝐲^𝑎𝐲^𝑏=𝑔_𝑖𝑗00𝑓²𝑔_𝛼𝛽, (14) where 𝐲^𝑎=(𝑦^𝑖,𝑣^𝛼), 𝐠_𝑖𝑗=𝑔_𝑖𝑗, 𝐠_𝛼𝛽=𝑓²𝑔_𝛼𝛽, 𝐠_𝑖𝛽=𝐠_𝛼𝑗=0, 𝑖,𝑗,…∈{1,…,𝑛₁}, 𝛼,𝛽,…∈{1,…,𝑛₂}, and 𝑎,𝑏,…∈{1,…,𝑛₁+𝑛₂}.

Now we consider spray coefficients of 𝐹₁, 𝐹₂, and 𝐹 as 𝐺^𝑖=1𝑥,𝑦4𝑔^𝑖ℎ𝜕²𝐹²₁𝜕𝑦^ℎ𝜕𝑥^𝑗𝑦^𝑗−𝜕𝐹²₁𝜕𝑥^ℎ,𝐺𝑥,𝑦_𝛼()=1𝑢,𝑣4𝑔^𝛼𝛾𝜕²𝐹²₂𝜕𝑣^𝛾𝜕𝑢^𝛽𝑣^𝛽−𝜕𝐹²₂𝜕𝑢^𝛾(),𝐆𝑢,𝑣_𝑎()=1𝐱,𝐲4𝐠^𝑎𝑏𝜕²𝐹²𝜕𝐲^𝑏𝜕𝐱^𝑐𝐲^𝑐−𝜕𝐹²𝜕𝐱^𝑏(𝐱,𝐲). (15)(16)(17) Taking into account the homogeneity of both 𝐹²₁ and 𝐹²₂ and using (15) and (16), 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 𝜕𝐺^𝑖𝜕𝑦^𝑗𝑦^𝑗=2𝐺^𝑖,𝜕𝐺^𝛼𝜕𝑣^𝛽𝑣^𝛽=2𝐺^𝛼. (18) By setting 𝑎=𝑖 in (17), we have 𝐆^𝑖=1𝑥,𝑢,𝑦,𝑣4𝐠^𝑖ℎ𝜕²𝐹²𝜕𝑦^ℎ𝜕𝑥^𝑗𝑦^𝑗+𝜕²𝐹²𝜕𝑦^ℎ𝜕𝑢^𝛼𝑣^𝛼−𝜕𝐹²𝜕𝑥^ℎ. (19) Direct calculations give us 𝜕𝐹²𝜕𝑥^ℎ=𝜕𝐹²₁𝜕𝑥^ℎ+𝜕𝑓²𝜕𝑥^ℎ𝐹²₂,𝜕²𝐹²𝜕𝑦^ℎ𝜕𝑥^𝑗=𝜕²𝐹²₁𝜕𝑦^ℎ𝜕𝑥^𝑗,𝜕²𝐹²𝜕𝑦^ℎ𝜕𝑢^𝛼=0. (20) Putting these equations together 𝐠^𝑖ℎ=𝑔^𝑖ℎ in the previous equation and using (15) imply that 𝐆^𝑖𝑥,𝑢,𝑦,𝑣=𝐺^𝑖−1𝑥,𝑦2𝑓𝑓^𝑖𝐹²₂. (21) Similarly, by setting 𝑎=𝛼 in (17) and using (16), we obtain 𝐆^𝛼𝑥,𝑢,𝑦,𝑣=𝐺^𝛼(𝑢,𝑣)+𝑓⁻¹𝑓_𝑗𝑣^𝛼𝑦^𝑗+𝑓_𝜆𝑣^𝛼𝑣^𝜆−12𝑓_𝛾𝑔^𝛼𝛾𝐹²₂, (22) where 𝑓_𝑖=𝜕𝑓/𝜕𝑥^𝑖, 𝑓_𝛾=𝜕𝑓/𝜕𝑢^𝛾, 𝑓^𝑖=𝑔^𝑖ℎ𝑓_ℎ, and 𝑓^𝛾=𝑔^𝜆𝛾𝑓_𝜆. Therefore we have 𝐆^𝑎=(𝐆^𝑖,𝐆^𝛼), where 𝐆^𝑎, 𝐆^𝑖, and 𝐆^𝛼 are given by (17), (21), and (22), respectively.

Now, we put (i)𝐆^𝑎_𝑏∶=𝜕𝐆^𝑎𝜕𝐲^𝑏,(ii)𝐺^𝑖_𝑗∶=𝜕𝐺^𝑖𝜕𝑦^𝑗,(iii)𝐺^𝛼_𝛽∶=𝜕𝐺^𝛼𝜕𝑣^𝛽. (23) Then we have the following.

Lemma 1. The coefficients 𝐆^𝑎_𝑏 defined by (23) satisfy in the following: 𝐆^𝑎_𝑏=𝐆𝑥,𝑢,𝑦,𝑣^𝑖𝑗𝑥,𝑢,𝑦,𝑣𝐆^𝛼_𝑗𝐆𝑥,𝑢,𝑦,𝑣_𝑖𝛽𝑥,𝑢,𝑦,𝑣𝐆^𝛼_𝛽𝑥,𝑢,𝑦,𝑣, (24) where 𝐆^𝑖_𝑗𝑥,𝑢,𝑦,𝑣∶=𝜕𝐆^𝑖𝜕𝑦^𝑗=𝐺^𝑖_𝑗+𝐶_𝑗^𝑖ℎ𝑓𝑓_ℎ𝐹²₂,𝐆^𝑖_𝛽𝑥,𝑢,𝑦,𝑣∶=𝜕𝐆^𝑖𝜕𝑣^𝛽=−𝑓𝑓^𝑖𝑣_𝛽,𝐆^𝛼_𝑗𝑥,𝑢,𝑦,𝑣∶=𝜕𝐆^𝛼𝜕𝑦^𝑗=𝑓⁻¹𝑓_𝑗𝑣^𝛼,𝐆^𝛼_𝛽𝑥,𝑢,𝑦,𝑣∶=𝜕𝐆^𝛼𝜕𝑣^𝛽=𝐺^𝛼_𝛽+𝑓⁻¹𝐶_𝛽^𝛼𝛾𝑓_𝛾𝐹²₂+𝑓_𝑗𝑦^𝑗𝛿^𝛼_𝛽−𝑓^𝛼𝑣_𝛽+𝑓_𝛽𝑣^𝛼+𝑓_𝛾𝑣^𝛾𝛿^𝛼_𝛽. (25)(26)(27)(28)

Next, 𝑉𝑇𝑀^∘ kernel of the differential of the projection map 𝜋∶=𝜋₁,𝜋₂∶𝑇𝑀^∘₁⊕𝑇𝑀^∘₂⟶𝑀₁×𝑀₂, (29) which is a well-defined subbundle of 𝑇𝑇𝑀^∘, is considered. Locally, Γ(𝑉𝑇𝑀^∘) is spanned by the natural vector fields {𝜕/𝜕𝑦¹,…,𝜕/𝜕𝑦^𝑛1,𝜕/𝜕𝑣¹,…,𝜕/𝜕𝑣^𝑛2}, and it is called the twisted vertical distribution on 𝑇𝑀^∘. Then, using the functions given by (25)–(28), the nonholonomic vector fields are defined as follows: 𝛿^𝑡𝛿^𝑡𝑥^𝑖𝜕∶=𝜕𝑥^𝑖−𝐆^𝑗_𝑖𝜕𝜕𝑦^𝑗−𝐆^𝛽_𝑖𝜕𝜕𝑣^𝛽,𝛿^𝑡𝛿^𝑡𝑢^𝛼𝜕∶=𝜕𝑢^𝛼−𝐆^𝑗_𝛼𝜕𝜕𝑦^𝑗−𝐆^𝛽_𝛼𝜕𝜕𝑣^𝛽, (30)(31) which make it possible to construct a complementary vector subbundle 𝐻𝑇𝑀^∘ to 𝑉𝑇𝑀^∘ in 𝑇𝑇𝑀^∘ as follows: 𝐻𝑇𝑀^∘𝛿∶=span^𝑡𝛿^𝑡𝑥¹𝛿,…,^𝑡𝛿^𝑡𝑥^𝑛1,𝛿^𝑡𝛿^𝑡𝑢¹𝛿,…,^𝑡𝛿^𝑡𝑢^𝑛2. (32)𝐻𝑇𝑀^∘ is called the twisted horizontal distribution on 𝑇𝑀^∘. Thus the tangent bundle of 𝑇𝑀^∘ admits the decomposition 𝑇𝑇𝑀^∘=𝐻𝑇𝑀^∘⊕𝑉𝑇𝑀^∘. (33) It is shown that 𝐆∶=(𝐆^𝑎_𝑏) is a nonlinear connection on 𝑇𝑀=𝑇𝑀₁⊕𝑇𝑀₂. In the following, we compute the nonlinear connection of a twisted product Finsler manifold.

Proposition 2. If (𝑀₁×_𝑓𝑀₂,𝐹) is a twisted product Finsler manifold, then 𝐆=(𝐆^𝑎_𝑏) is the nonlinear connection on 𝑇𝑀. Further, one has 𝜕𝐆^𝑖_𝑗𝜕𝑦^𝑘𝑦^𝑘+𝜕𝐆^𝑖_𝑗𝜕𝑣^𝛾𝑣^𝛾=𝐆^𝑖_𝑗,𝜕𝐆^𝑖_𝛽𝜕𝑦^𝑘𝑦^𝑘+𝜕𝐆^𝑖_𝛽𝜕𝑣^𝛾𝑣^𝛾=𝐆^𝑖_𝛽,𝜕𝐆^𝛼_𝑗𝜕𝑦^𝑘𝑦^𝑘+𝜕𝐆^𝛼_𝑗𝜕𝑣^𝛾𝑣^𝛾=𝐆^𝛼_𝑗,𝜕𝐆^𝛼_𝛽𝜕𝑦^𝑘𝑦^𝑘+𝜕𝐆^𝛼_𝛽𝜕𝑣^𝛾𝑣^𝛾=𝐆^𝛼_𝛽. (34)

Definition 3. Using decomposition (33), the twisted vertical morphism 𝑣^𝑡∶𝑇𝑇𝑀^∘→𝑉𝑇𝑀^∘ is defined by 𝑣^𝑡𝜕∶=𝜕𝑦^𝑖⊗𝛿^𝑡𝑦^𝑖+𝜕𝜕𝑣^𝛼⊗𝛿^𝑡𝑣^𝛼, (35) where 𝛿^𝑡𝑦^𝑖∶=𝑑𝑦^𝑖+𝐆^𝑖_𝑗𝑑𝑥^𝑗+𝐆^𝑖_𝛽𝑑𝑢^𝛽,𝛿^𝑡𝑣^𝛼∶=𝑑𝑣^𝛼+𝐆^𝛼_𝑗𝑑𝑥^𝑗+𝐆^𝛼_𝛽𝑑𝑢^𝛽. (36)

For this projective morphism, the following hold: 𝑣^𝑡𝜕𝜕𝑦^𝑖=𝜕𝜕𝑦^𝑖,𝑣^𝑡𝜕𝜕𝑣^𝛼=𝜕𝜕𝑣^𝛼,𝑣^𝑡𝛿^𝑡𝛿^𝑡𝑥^𝑖=0,𝑣^𝑡𝛿^𝑡𝛿^𝑡𝑢^𝑖=0. (37) From the previous equations, we conclude that 𝑣^𝑡2=𝑣^𝑡,ker𝑣^𝑡=𝐻𝑇𝑀^∘. (38) This mapping is called the twisted vertical projective.

Definition 4. Using decomposition (33), the doubly warped horizontal projective ℎ^𝑡∶𝑇𝑇𝑀^∘→𝐻𝑇𝑀^∘ is defined by ℎ^𝑡=𝑖𝑑−𝑣^𝑡 (39) or ℎ^𝑡=𝛿^𝑡𝛿^𝑡𝑥^𝑖⊗𝑑𝑥^𝑖+𝛿^𝑡𝛿^𝑡𝑢^𝛼⊗𝑑𝑢^𝛼. (40)

For this projective morphism, the following hold: ℎ^𝑡𝛿^𝑡𝛿^𝑡𝑥^𝑖=𝛿^𝑡𝛿^𝑡𝑥^𝑖,ℎ^𝑡𝛿^𝑡𝛿^𝑡𝑢^𝛼=𝛿^𝑡𝛿^𝑡𝑢^𝛼,ℎ^𝑡𝜕𝜕𝑦^𝑖=0,ℎ^𝑡𝜕𝜕𝑣^𝛼=0. (41) Thus we result that ℎ^𝑡2=ℎ^𝑡,kerℎ^𝑡=𝑉𝑇𝑀^∘. (42)

Definition 5. Using decomposition (33), the twisted almost tangent structure 𝐽^𝑡∶𝐻𝑇𝑀^∘→𝑉𝑇𝑀^∘ is defined by 𝐽^𝑡∶𝜕𝜕𝑦^𝑖⊗𝑑𝑥^𝑖+𝜕𝜕𝑣^𝛼⊗𝑑𝑢^𝛼, (43) or 𝐽^𝑡𝛿^𝑡𝛿^𝑡𝑥^𝑖=𝜕𝜕𝑦^𝑖,𝐽^𝑡𝛿^𝑡𝛿^𝑡𝑢^𝛼=𝜕𝜕𝑣^𝛼,𝐽^𝑡𝜕𝜕𝑦^𝑖=𝐽^𝑡𝜕𝜕𝑣^𝛼=0. (44)

Thus we result that 𝐽^𝑡2=0,ker𝐽^𝑡=𝐼𝑚𝐽^𝑡=𝑉𝑇𝑀^∘. (45) Here, we introduce some geometrical objects of twisted product Finsler manifold. In order to simplify the equations, we rewritten the basis of 𝐻𝑇𝑀^∘ and 𝑉𝑇𝑀^∘ as follows: 𝛿^𝑡𝛿^𝑡𝐱^𝑎=𝛿^𝑡𝛿^𝑡𝑥^𝑖𝛿^𝑖_𝑎+𝛿^𝑡𝛿^𝑡𝑢^𝛼𝛿^𝛼_𝑎,𝜕𝜕𝐲^𝑎=𝜕𝜕𝑦^𝑖𝛿^𝑖_𝑎+𝜕𝜕𝑣^𝛼𝛿^𝛼_𝑎. (46) Thus 𝑇𝑇𝑀^∘𝛿=span^𝑡𝛿^𝑡𝐱^𝑎,𝜕𝜕𝐲^𝑎. (47) The Lie brackets of this basis is given by 𝛿^𝑡𝛿^𝑡𝐱^𝑎,𝛿^𝑡𝛿^𝑡𝐱^𝑏=𝐑^𝑐_𝑎𝑏𝜕𝜕𝐲^𝑐,𝛿^𝑡𝛿^𝑡𝐱^𝑎,𝜕𝜕𝐲^𝑏=𝐆^𝑐_𝑎𝑏𝜕𝜕𝐲^𝑐,𝜕𝜕𝐲^𝑎,𝜕𝜕𝐲^𝑏=0, (48) where (i)𝐑^𝑐_𝑎𝑏=𝛿^𝑡𝐆^𝑐_𝑎𝛿^𝑡𝐱^𝑏−𝛿^𝑡𝐆^𝑐_𝑏𝛿^𝑡𝐱^𝑎,(ii)𝐆^𝑐_𝑎𝑏=𝜕𝐆^𝑐_𝑎𝜕𝐲^𝑏. (49)(50)

Therefore, we have the following.

Corollary 6. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then 𝐑^𝑐_𝑎𝑏=𝐑^𝑘_𝑖𝑗,𝐑^𝑘_𝑖𝛽,𝐑^𝑘_𝛼𝑗,𝐑^𝑘_𝛼𝛽,𝐑^𝛾_𝑖𝑗,𝐑^𝛾_𝑖𝛽,𝐑^𝛾_𝛼𝑗,𝐑^𝛾_𝛼𝛽, (51) where 𝐑^𝑘_𝑖𝑗𝛿∶=^𝑡𝐆^𝑘_𝑖𝛿^𝑡𝑥^𝑗−𝛿^𝑡𝐆^𝑘_𝑗𝛿^𝑡𝑥^𝑖,𝐑^𝑘_𝑖𝛽𝛿∶=^𝑡𝐆^𝑘_𝑖𝛿^𝑡𝑢^𝛽−𝛿^𝑡𝐆^𝑘_𝛽𝛿^𝑡𝑥^𝑖,𝐑^𝑘_𝛼𝑗𝛿∶=^𝑡𝐆^𝑘_𝛼𝛿^𝑡𝑥^𝑗−𝛿^𝑡𝐆^𝑘_𝑗𝛿^𝑡𝑢^𝛼,𝐑^𝑘_𝛼𝛽𝛿∶=^𝑡𝐆^𝑘_𝛼𝛿^𝑡𝑢^𝛽−𝛿^𝑡𝐆^𝑘_𝛽𝛿^𝑡𝑢^𝛼,𝐑^𝛾_𝑖𝑗𝛿∶=^𝑡𝐆^𝛾_𝑖𝛿^𝑡𝑥^𝑗−𝛿^𝑡𝐆^𝛾_𝑗𝛿^𝑡𝑥^𝑖,𝐑^𝛾_𝑖𝛽𝛿∶=^𝑡𝐆^𝛾_𝑖𝛿^𝑡𝑢^𝛽−𝛿^𝑡𝐆^𝛾_𝛽𝛿^𝑡𝑥^𝑖,𝐑^𝛾_𝛼𝑗𝛿∶=^𝑡𝐆^𝛾_𝛼𝛿^𝑡𝑥^𝑗−𝛿^𝑡𝐆^𝛾_𝑗𝛿^𝑡𝑢^𝛼,𝐑^𝛾_𝛼𝛽𝛿∶=^𝑡𝐆^𝛾_𝛼𝛿^𝑡𝑢^𝛽−𝛿^𝑡𝐆^𝛾_𝛽𝛿^𝑡𝑢^𝛼. (52)

With a simple calculation, we have the following.

Corollary 7. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then 𝐆^𝑐_𝑎𝑏=𝐆^𝑘_𝑖𝑗,𝐆^𝑘_𝑖𝛽,𝐆^𝑘_𝛼𝑗,𝐆^𝑘_𝛼𝛽,𝐆^𝛾_𝑖𝑗,𝐆^𝛾_𝑖𝛽,𝐆^𝛾_𝛼𝑗,𝐆^𝛾_𝛼𝛽, (53) where 𝐆^𝛾_𝛼𝛽=𝜕𝐆^𝛾_𝛼𝜕𝑣^𝛽=𝐺^𝛾_𝛼𝛽+𝑓⁻¹𝐶^𝛾𝜆_𝛼;𝛽𝑓_𝜆𝐹²₂+2𝐶_𝛼^𝛾𝜆𝑓_𝜆𝑣_𝛽+2𝐶_𝛽^𝛾𝜆𝑓_𝜆𝑣_𝛼−𝑓^𝛾𝑔_𝛼𝛽+𝑓_𝛽𝛿^𝛾_𝛼+𝑓_𝛼𝛿^𝛾_𝛽=𝐆^𝛾_𝛽𝛼,𝐆^𝑘_𝑖𝑗=𝜕𝐆^𝑘_𝑖𝜕𝑦^𝑗=𝐺^𝑘_𝑖𝑗+𝐶^𝑘ℎ_𝑖;𝑗𝑓𝑓_ℎ𝐹²₂=𝐆^𝑘_𝑗𝑖,𝐆^𝑘_𝑖𝛽=𝜕𝐆^𝑘_𝑖𝜕𝑣^𝛽=2𝐶_𝑖^𝑘ℎ𝑓𝑓_ℎ𝑣_𝛽=𝐆^𝑘_𝛽𝑖,𝐆^𝑘_𝛼𝛽=𝜕𝐆^𝑘_𝛼𝜕𝑣^𝛽=−𝑓𝑓^𝑘𝑔_𝛼𝛽=𝐆^𝑘_𝛽𝛼,𝐆^𝛾_𝑖𝛽=𝜕𝐆^𝛾_𝑖𝜕𝑣^𝛽=𝑓⁻¹𝑓_𝑖𝛿^𝛾_𝛽=𝐆^𝛾_𝛽𝑖,𝐆^𝛾_𝑖𝑗=𝜕𝐆^𝛾_𝑖𝜕𝑦^𝑗=𝐆^𝛾_𝑗𝑖=0, (54) where 𝐶^𝑘ℎ_𝑖;𝑗=𝜕𝐶_𝑖^𝑘ℎ/𝜕𝑦^𝑗. Apart from 𝐆^𝑐_𝑎𝑏, the functions 𝐅^𝑐_𝑎𝑏 are given by 𝐅^𝑐_𝑎𝑏=12𝐠^𝑐𝑒𝛿^𝑡𝐠_𝑒𝑎𝛿^𝑡𝐱^𝑏+𝛿^𝑡𝐠_𝑒𝑏𝛿^𝑡𝐱^𝑎−𝛿^𝑡𝐠_𝑎𝑏𝛿^𝑡𝐱^𝑒. (55)

Corollary 8. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then 𝐅^𝑐_𝑎𝑏=𝐅^𝑘_𝑖𝑗,𝐅^𝑘_𝑖𝛽,𝐅^𝑘_𝛼𝑗,𝐅^𝑘_𝛼𝛽,𝐅^𝛾_𝑖𝑗,𝐅^𝛾_𝑖𝛽,𝐅^𝛾_𝛼𝑗,𝐅^𝛾_𝛼𝛽, (56) where 𝐅^𝑘_𝑖𝑗=𝐹^𝑘_𝑖𝑗−𝑀^𝑟_𝑗𝐶^𝑘_𝑖𝑟+𝑀^𝑟_𝑖𝐶^𝑘_𝑗𝑟−𝑀^𝑟_ℎ𝐶_𝑖𝑗𝑟𝑔^𝑘ℎ,𝐅^𝑘_𝑖𝛽=−𝐆^𝑟_𝛽𝐶^𝑘_𝑖𝑟=𝐅^𝑘_𝛽𝑖,𝐅^𝑘_𝛼𝛽=−𝑓𝑓^𝑘𝑔_𝛼𝛽+𝑓²𝑔^𝑘ℎ𝐆^𝜆_ℎ𝐶_𝛼𝛽𝜆,𝐅^𝛾_𝑖𝑗=𝑓⁻²𝑔^𝛾𝜆𝐆^𝑟_𝜆𝐶_𝑖𝑗𝑟,𝐅^𝛾_𝑖𝛽=𝑓⁻¹𝑓_𝑖𝛿^𝛾_𝛽−𝐆^𝛼_𝑖𝐶^𝛾_𝛼𝛽=𝐅^𝛾_𝛽𝑖,𝐅^𝛾_𝛼𝛽=𝐹^𝛾_𝛼𝛽+𝑁^𝛾_𝛼𝛽−𝑀^𝜇_𝛽𝐶^𝛾_𝛼𝜇+𝑀^𝜇_𝛼𝐶^𝛾_𝛽𝜇−𝑀^𝜇_𝜆𝐶_𝛼𝛽𝜇𝑔^𝛾𝜆𝐹_𝑘𝑖𝑗,=12𝑔^𝑘ℎ𝛿𝑔_ℎ𝑖𝛿𝑥^𝑗+𝛿𝑔_ℎ𝑗𝛿𝑥^𝑖−𝛿𝑔_𝑖𝑗𝛿𝑥^ℎ,𝐹^𝛾_𝛼𝛽=12𝑔^𝛾𝜆𝛿𝑔_𝜆𝛼𝛿𝑢^𝛽+𝛿𝑔_𝜆𝛽𝛿𝑢^𝛼−𝛿𝑔_𝛼𝛽𝛿𝑢^𝜆,𝑀^𝑟_𝑖=𝐶_𝑖^𝑟ℎ𝑓𝑓_ℎ𝐹²₂,𝑀^𝜇_𝛼=𝑓⁻¹𝐶_𝛼^𝜇𝛾𝑓_𝛾𝐹²₂+𝑓_𝑟𝑦^𝑟𝛿^𝜇_𝛼+𝑓_𝛾𝑣^𝛾𝛿^𝜇_𝛼−𝑔^𝜇𝛾𝑓_𝛾𝑣_𝛼+𝑓_𝛼𝑣^𝜇,𝑁^𝛾_𝛼𝛽=𝑓⁻¹𝑓_𝛽𝛿^𝛾_𝛼+𝑓_𝛼𝛿^𝛾_𝛽−𝑓_𝜆𝑔^𝛾𝜆𝑔_𝛼𝛽. (57)(58)(59)(60)(61)(62)(63)

Proof. By using (55), we have 𝐅^𝑘_𝑖𝑗=12𝑔^𝑘ℎ𝛿^𝑡𝑔_ℎ𝑖𝛿^𝑡𝑥^𝑗+𝛿^𝑡𝑔_ℎ𝑗𝛿^𝑡𝑥^𝑖−𝛿^𝑡𝑔_𝑖𝑗𝛿^𝑡𝑥^ℎ. (64) Since 𝑔_𝑖𝑗 is a function with respect to (𝑥,𝑦), then by (25) and (30) we obtain 𝛿^𝑡𝑔_ℎ𝑖𝛿^𝑡𝑥^𝑗=𝛿𝑔_ℎ𝑖𝛿𝑥^𝑗−2𝑀^𝑟_𝑗𝐶_ℎ𝑖𝑟. (65) Interchanging 𝑖, 𝑗, and ℎ in the previous equation gives us 𝛿^𝑡𝑔_ℎ𝑗𝛿^𝑡𝑥^𝑖=𝛿𝑔_ℎ𝑗𝛿𝑥^𝑖−2𝑀^𝑟_𝑖𝐶_ℎ𝑗𝑟,𝛿^𝑡𝑔_𝑖𝑗𝛿^𝑡𝑥^ℎ=𝛿𝑔_𝑖𝑗𝛿𝑥^ℎ−2𝑀^𝑟_ℎ𝐶_𝑖𝑗𝑟. (66) 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 (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then 𝐲^𝑐𝐅^𝑎_𝑏𝑐=𝐆^𝑎_𝑏, where 𝐅^𝑎_𝑏𝑐 and 𝐆^𝑎_𝑏 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 (𝑀₁×_𝑓𝑀₂,𝐹) is defined by 𝐂^𝑎_𝑏𝑐=12𝐠^𝑎𝑒𝜕𝐠_𝑏𝑒𝜕𝐲^𝑐. (67) From this definition, we conclude the following.

Lemma 10. Let 𝐶^𝑘_𝑖𝑗 and 𝐶^𝛾_𝛼𝛽 be the local components of Cartan tensor field on 𝑀₁ and 𝑀₂, respectively. Then one has 𝐂^𝑐_𝑎𝑏=𝐂^𝑘_𝑖𝑗,𝐂^𝑘_𝑖𝛽,𝐂^𝑘_𝛼𝑗,𝐂^𝑘_𝛼𝛽,𝐂^𝛾_𝑖𝑗,𝐂^𝛾_𝑖𝛽,𝐂^𝛾_𝛼𝑗,𝐂^𝛾_𝛼𝛽, (68) where 𝐂^𝑘_𝑖𝑗=12𝑔^𝑘ℎ𝜕𝑔_𝑖𝑗𝜕𝑦^ℎ=𝐶^𝑘_𝑖𝑗,𝐂^𝛾_𝛼𝛽=12𝑔^𝛾𝜆𝜕𝑔_𝛼𝛽𝜕𝑣^𝜆=𝐶^𝛾_𝛼𝛽, (69) and 𝐂^𝑘_𝑖𝛽=𝐂^𝑘_𝛼𝑗=𝐂^𝑘_𝛼𝛽=𝐂^𝛾_𝑖𝑗=𝐂^𝛾_𝑖𝛽=𝐂^𝛾_𝛼𝑗=0.

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

Corollary 11. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then (𝑀₁×_𝑓𝑀₂,𝐹) is a Riemannian manifold if and only if (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂) 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-Hojo proves that the converse is true too [15].

Here, we define the Matsumoto twisted tensor 𝐌_𝑎𝑏𝑐 for a twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,𝐹) as follows: 𝐌_𝑎𝑏𝑐=𝐂_𝑎𝑏𝑐−1𝐈𝑛+1^𝑎𝐡_𝑏𝑐+𝐈_𝑏𝐡_𝑎𝑐+𝐈_𝑐𝐡_𝑎𝑏, (70) where 𝐈_𝑎=𝐠^𝑏𝑐𝐂_𝑎𝑏𝑐, 𝐂_𝑎𝑏𝑐=𝐠_𝑐𝑑𝐂^𝑑_𝑎𝑏, and 𝐡_𝑎𝑏=𝐠_𝑎𝑏−(1/𝐹²)𝐲_𝑎𝐲_𝑏. By attention to the previous equation and relations 𝐂_𝑖𝑗𝑘=𝐶_𝑖𝑗𝑘,𝐂_𝛼𝛽𝛾=𝑓²𝐶_𝛼𝛽𝛾, (71) we obtain 𝐌_𝛼𝑗𝑘1=−𝐼𝑛+1^𝛼𝑔_𝑗𝑘−1𝐹²𝑦_𝑗𝑦_𝑘−𝑓²𝐹²𝑣_𝛼𝐼_𝑗𝑦_𝑘+𝐼_𝑘𝑦_𝑗. (72) Contracting the previous equation in 𝑦^𝑗𝑦^𝑘 gives us 𝑦^𝑗𝑦^𝑘𝐌_𝛼𝑗𝑘𝑓=−²𝐹²₁𝐹²₂(𝑛+1)𝐹²𝐼_𝛼. (73) Similarly, we obtain 𝑣^𝜆𝑣^𝛽𝐌_𝑖𝛽𝜆𝑓=−²𝐹²₁𝐹²₂(𝑛+1)𝐹²𝐼_𝑖. (74) Therefore if 𝐌_𝑖𝛽𝜆=𝐌_𝛼𝑗𝑘=0, then we get 𝐼_𝑖=𝐼_𝛼=0; that is, (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂) 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 (𝑀₁×_𝑓𝑀₂,𝐹). Let (𝑀₁×_𝑓𝑀₂,𝐹) be a semi-C-reducible twisted product Finsler manifold. Then we have 𝐂_𝑎𝑏𝑐=𝑝𝐈𝑛+1^𝑎𝐡_𝑏𝑐+𝐈_𝑏𝐡_𝑎𝑐+𝐈_𝑐𝐡_𝑎𝑏+𝑞𝐂²𝐈_𝑎𝐈_𝑏𝐈_𝑐, (75) where 𝐂²=𝐈^𝑎𝐈_𝑎 and 𝑝 and 𝑞 are scalar function on 𝑀₁×_𝑓𝑀₂ with 𝑝+𝑞=1. This equation gives us 0=𝐂_𝛼𝑗𝑘=𝑝𝐼𝑛+1^𝛼𝑔_𝑗𝑘−1𝐹²𝑦_𝑗𝑦_𝑘−𝑓²𝐹²𝑣_𝛼𝐼_𝑗𝑦_𝑘+𝐼_𝑘𝑦_𝑗+𝑞𝐂²𝐼_𝛼𝐼_𝑗𝐼_𝑘. (76) Contractiing the previous equation with 𝑦^𝑗𝑦^𝑘 implies that 𝑝𝑓²𝐹²₁𝐹²₂𝐼_𝛼=0. (77) 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; that is, 𝐹₂ is Riemannian metric. In this case, with similar way, we conclude that 𝐹₁ is Riemannian metric. But, definition 𝐹 cannot be a Riemannian metric. Therefore we have the following.

Theorem 13. Every semi-C-reducible twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,𝐹) is a 𝐶2-like manifold.

4. Riemannian Curvature

The Riemannian curvature of twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,𝐹) with respect to Berwald connection is given by 𝐑^𝑎_𝑏𝑐𝑑=𝛿^𝑡𝐅^𝑎_𝑏𝑐𝛿^𝑡𝐱^𝑑−𝛿^𝑡𝐅^𝑎_𝑏𝑑𝛿^𝑡𝐱^𝑐+𝐅^𝑎_𝑑𝑒𝐅^𝑒_𝑏𝑐−𝐅^𝑎_𝑐𝑒𝐅^𝑒_𝑏𝑑. (78)

Lemma 14. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then one has 𝐑^𝑎_𝑐𝑑=𝐲^𝑏𝐑^𝑎_𝑏𝑐𝑑, (79) where 𝐑^𝑎_𝑐𝑑 and 𝐲^𝑏𝐑^𝑎_𝑏𝑐𝑑 are given by (50) and (78).

Proof. By using (78), we have 𝐲^𝑏𝐑^𝑖_𝑏𝑘𝑙=𝐲^𝑏𝛿^𝑡𝐅^𝑖_𝑏𝑘𝛿^𝑡𝐱^𝑙−𝐲^𝑏𝛿^𝑡𝐅^𝑖_𝑏𝑙𝛿^𝑡𝐱^𝑘+𝐲^𝑏𝐅^𝑖_𝑙𝑒𝐅^𝑒_𝑏𝑘.−𝐲^𝑏𝐅^𝑖_𝑘𝑒𝐅^𝑒_𝑏𝑙. (80) By using Corollary 8 and Lemma 9, we obtain 𝐲^𝑏𝛿^𝑡𝐅^𝑖_𝑏𝑘𝛿^𝑡𝐱^𝑙=𝛿^𝑡𝐆^𝑖_𝑘𝛿^𝑡𝑥^𝑙+𝐅^𝑖_𝑗𝑘𝐆^𝑗_𝑙+𝐅^𝑖_𝛽𝑘𝐆^𝛽_𝑙,𝐲^𝑏𝐅^𝑖_𝑙𝑒𝐅^𝑒_𝑏𝑘=𝐅^𝑖_𝑙ℎ𝐆^ℎ_𝑘+𝐅^𝑖_𝑙𝛾𝐆^𝛾_𝑘. (81) Interchanging 𝑖 and 𝑗 in the previous equation implies that 𝐲^𝑏𝛿^𝑡𝐅^𝑖_𝑏𝑙𝛿^𝑡𝐱^𝑘=𝛿^𝑡𝐆^𝑖_𝑙𝛿^𝑡𝑥^𝑘+𝐅^𝑖_𝑗𝑙𝐆^𝑗_𝑘+𝐅^𝑖_𝛽𝑙𝐆^𝛽_𝑘,𝐲^𝑏𝐅^𝑖_𝑘𝑒𝐅^𝑒_𝑏𝑙=𝐅^𝑖_𝑘ℎ𝐆^ℎ_𝑙+𝐅^𝑖_𝑘𝛾𝐆^𝛾_𝑙. (82) Setting (81) and (82) in (80) gives us 𝐲^𝑏𝐑^𝑖_𝑏𝑘𝑙=𝐑^𝑖_𝑘𝑙. 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 (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then the coefficients of Riemannian curvature are as follows: 𝐑^𝑖_𝑗𝑘𝑙=𝑅^𝑖_𝑗𝑘𝑙−𝑀^𝑟_𝑙𝜕𝐹^𝑖_𝑗𝑘𝜕𝑦^𝑟+𝛿^𝑡𝑀^𝑖_𝑗𝑘𝛿^𝑡𝑥^𝑙+𝐹^𝑖_𝑙ℎ𝑀^ℎ_𝑗𝑘+𝑀^𝑖_𝑙ℎ𝐹^ℎ_𝑗𝑘−𝑀^𝑖_𝑙ℎ𝑀^ℎ_𝑗𝑘+𝑓⁻²𝑔^𝛼𝛾𝐆^𝑟_𝛼𝐆^𝑚_𝛾𝐶^𝑖_𝑙𝑟𝐶_𝑗𝑘𝑚−ℭ^𝑘_𝑙,𝐑^𝑖_𝛼𝑘𝑙=−𝛿^𝑡𝛿^𝑡𝑥^𝑙𝐆^𝑟_𝛼𝐶^𝑖_𝑘𝑟−𝐹^𝑖_𝑟𝑙−𝑀^𝑖_𝑟𝑙𝐆^𝑚_𝛼𝐶^𝑟_𝑘𝑚−𝑓⁻¹𝐆^𝑟_𝛽𝐶^𝑖_𝑙𝑟𝑓_𝑘𝛿^𝛽_𝛼+𝐆^𝑟_𝛽𝐆^𝜇_𝑘𝐶^𝑖_𝑙𝑟𝐶^𝛽_𝛼𝜇−ℭ^𝑘_𝑙,𝐑^𝑖_𝑗𝛽𝜆=−𝛿^𝑡𝛿^𝑡𝑢^𝜆𝐆^𝑟_𝛽𝐶^𝑖_𝑗𝑟+𝐆^𝑚_𝜆𝐆^𝑙_𝛽𝐶^𝑖_𝑟𝑚𝐶^𝑟_𝑗𝑙−𝑓^𝑖𝑔_𝛼𝜆−𝑓𝐆^𝜇_ℎ𝑔^𝑖ℎ𝐶_𝛼𝜆𝜇𝑓_𝑗𝛿^𝛼_𝛽−𝑓𝐆^𝜈_𝑗𝐶^𝛼_𝛽𝜈−ℭ^𝛽_𝜆,𝐑^𝑖_𝛼𝛽𝑙=𝛿^𝑡𝛿^𝑡𝑢^𝛽𝐆^𝑟_𝛼𝐶^𝑖_𝑙𝑟−𝛿^𝑡𝛿^𝑡𝑥^𝑙𝑓𝑓^𝑖𝑔_𝛼𝛽−𝑓𝐆^𝜆_ℎ𝑔^𝑖ℎ𝐶_𝛼𝛽𝜆−𝐆^𝑚_𝛽𝐆^𝑠_𝛼𝐶^𝑖_𝑟𝑚𝐶^𝑟_𝑙𝑠+𝑓^𝑖𝑔_𝜇𝛽−𝑓𝐆^𝜆_ℎ𝑔^𝑖ℎ𝐶_𝜇𝛽𝜆×𝑓_𝑙𝛿^𝜇_𝛼−𝑓𝐆^𝜈_𝑙𝐶^𝜇_𝛼𝜈−𝑓𝑔^𝑟ℎ𝐹^𝑖_𝑟𝑙−𝑀^𝑖_𝑟𝑙×𝑓_ℎ𝑔_𝛼𝛽−𝑓𝐆^𝜆_ℎ𝐶_𝛼𝛽𝜆−𝐆^𝑟_𝜇𝐶^𝑖_𝑙𝑟𝐹^𝜇_𝛼𝛽+𝑁^𝜇_𝛼𝛽−𝑀^𝜇_𝛼𝛽,𝐑^𝑖_𝑗𝛽𝑙𝛿=−^𝑡𝛿^𝑡𝑥^𝑙𝐆^𝑟_𝛽𝐶^𝑖_𝑗𝑟−𝛿^𝑡𝛿^𝑡𝑢^𝛽𝐹^𝑖_𝑗𝑙−𝑀^𝑖_𝑗𝑙−𝐹^𝑖_𝑙𝑟−𝑀^𝑖_𝑙𝑟𝐆^𝑠_𝛽𝐶^𝑟_𝑗𝑠−𝑓⁻¹𝐆^𝑟_𝛼𝐶^𝑖_𝑙𝑟×𝑓_𝑗𝛿^𝛼_𝛽−𝑓𝐆^𝜇_𝑗𝐶^𝛼_𝛽𝜇+𝐆^𝑠_𝛽𝐶^𝑖_𝑟𝑠𝐹^𝑟_𝑗𝑙−𝑀^𝑟_𝑗𝑙+𝑓⁻¹𝐆^𝑟_𝜇𝐶_𝑗𝑙𝑟𝑓^𝑖𝛿^𝜇_𝛽−𝑓𝐆^𝜆_ℎ𝑔^𝑖ℎ𝐶^𝜇_𝛽𝜆,𝐑^𝑖_𝛼𝛽𝜆=−𝛿^𝑡𝛿^𝑡𝑢^𝜆𝑓𝑓^𝑖𝑔_𝛼𝛽−𝑓²𝑔^𝑖ℎ𝐆^𝜇_ℎ𝐶_𝛼𝛽𝜇+𝑓𝐆^𝑠_𝜆𝐶^𝑖_𝑟𝑠×𝑓^𝑟𝑔_𝛼𝛽−𝑓𝐆^𝜇_𝑙𝐶_𝛼𝛽𝜇𝑔^𝑟𝑙−𝑓𝑓^𝑖𝑔_𝜆𝜇−𝑓𝑔^𝑖ℎ𝐆^𝜅_ℎ𝐶_𝜆𝜇𝜅×𝐹^𝜇_𝛼𝛽+𝑁^𝜇_𝛼𝛽−𝑀^𝜇_𝛼𝛽−ℭ^𝛽_𝜆,𝐑^𝛾_𝑗𝑘𝑙=𝛿^𝑡𝛿^𝑡𝑥^𝑙𝑓⁻²𝑔^𝛾𝜆𝐆^𝑟_𝜆𝐶_𝑗𝑘𝑟+𝑓⁻²𝑔^𝛾𝜆𝐆^𝑠_𝜆𝐶_𝑙𝑟𝑠×𝐹^𝑟_𝑗𝑘−𝑀^𝑟_𝑗𝑘+𝑓⁻³𝐆^𝑟_𝜇𝐶_𝑗𝑘𝑟𝑓_𝑙𝑔^𝛾𝜇−𝑓𝐆^𝛼_𝑙𝐶_𝛼^𝛾𝜇−ℭ^𝑘_𝑙,𝐑^𝛾_𝑗𝛽𝑙=𝛿^𝑡𝛿^𝑡𝑥^𝑙𝑓⁻¹𝑓_𝑗𝛿^𝛾_𝛽−𝑓𝐆^𝛼_𝑗𝐶^𝛾_𝛼𝛽−𝛿^𝑡𝛿^𝑡𝑢^𝛽𝑓⁻²𝑔^𝛾𝜆𝐆^𝑟_𝜆𝐶_𝑗𝑙𝑟−𝑓⁻²𝑔^𝛾𝜆𝐆^𝑠_𝜆𝐆^𝑚_𝛽𝐶^ℎ_𝑙𝑠𝐶_ℎ𝑗𝑚+𝑓⁻²𝑓_𝑙𝛿^𝛾_𝜇−𝑓𝐆^𝛼_𝑙𝐶^𝛾_𝜇𝛼𝑓_𝑗𝛿^𝜇_𝛽−𝑓𝐆^𝜈_𝑗𝐶^𝜇_𝛽𝜈−𝑓⁻¹𝑓_𝑟𝛿^𝛾_𝛽−𝑓𝐆^𝛼_𝑟𝐶^𝛾_𝛽𝛼𝐹^𝑟_𝑗𝑙−𝑀^𝑟_𝑗𝑙−𝑓⁻²𝑔^𝜇𝜆𝐆^𝑟_𝜆𝐶_𝑗𝑙𝑟𝐹^𝛾_𝛽𝜇+𝑁^𝛾_𝛽𝜇−𝑀^𝛾_𝛽𝜇,𝐑^𝛾_𝛼𝛽𝑙=𝛿^𝑡𝛿^𝑡𝑥^𝑙𝐹^𝛾_𝛼𝛽+𝑁^𝛾_𝛼𝛽−𝑀^𝛾_𝛼𝛽−𝛿^𝑡𝛿^𝑡𝑢^𝛽𝑓⁻¹𝑓_𝑙𝛿^𝛾_𝛼−𝐆^𝜇_𝑙𝐶^𝛾_𝛼𝜇−𝑓⁻¹𝑔^𝛾𝜆𝐆^𝑠_𝜆𝐶^ℎ_𝑙𝑠𝑓_ℎ𝑔_𝛼𝛽−𝑓𝐆^𝜇_ℎ𝐶_𝛼𝛽𝜇+𝑓⁻¹𝑓_𝑙𝛿^𝛾_𝜇−𝑓𝐆^𝜅_𝑙𝐶^𝛾_𝜇𝜅𝐹^𝜇_𝛼𝛽+𝑁^𝜇_𝛼𝛽−𝑀^𝜇_𝛼𝛽+𝑓⁻¹𝐆^𝑠_𝛼𝐶^𝑟_𝑙𝑠𝑓_𝑟𝛿^𝛾_𝛽−𝑓𝐆^𝜅_𝑟𝐶^𝛾_𝛽𝜅−𝑓⁻¹𝑓_𝑙𝛿^𝜇_𝛼−𝑓𝐆^𝜅_𝑙𝐶^𝜇_𝛼𝜅𝐹^𝛾_𝛽𝜇+𝑁^𝛾_𝛽𝜇−𝑀^𝛾_𝛽𝜇,𝐑^𝛾_𝑗𝛽𝜆=𝛿^𝑡𝛿^𝑡𝑢^𝜆𝑓⁻¹𝑓_𝑗𝛿^𝛾_𝛽−𝐆^𝛼_𝑗𝐶^𝛾_𝛼𝛽+𝑓⁻¹𝐹^𝛾_𝛼𝜆+𝑁^𝛾_𝛼𝜆−𝑀^𝛾_𝛼𝜆𝑓_𝑗𝛿^𝛼_𝛽−𝑓𝐆^𝜈_𝑗𝐶^𝛼_𝛽𝜈−𝑓⁻¹𝐆^𝑚_𝛽𝐶^𝑟_𝑗𝑚𝑓_𝑟𝛿^𝛾_𝜆−𝑓𝐆^𝛼_𝑟𝐶^𝛾_𝜆𝛼−ℭ^𝛽_𝜆,𝐑^𝛾_𝛼𝑘𝑙=𝛿^𝑡𝛿^𝑡𝑥^𝑙𝑓⁻¹𝑓_𝑘𝛿^𝛾_𝛼−𝐆^𝜇_𝑘𝐶^𝛾_𝛼𝜇+𝑓⁻²𝑓_𝑙𝛿^𝛾_𝛽−𝑓𝐆^𝜅_𝑙𝐶^𝛾_𝛽𝜅𝑓_𝑘𝛿^𝛽_𝛼−𝑓𝐆^𝜈_𝑘𝐶^𝛽_𝛼𝜈−𝑓⁻²𝐆^𝑠_𝜇𝐆^𝑚_𝛼𝑔^𝛾𝜇𝐶^ℎ_𝑙𝑠𝐶_ℎ𝑘𝑚−ℭ^𝑘_𝑙,𝐑^𝛾_𝛼𝛽𝜆=𝑅^𝛾_𝛼𝛽𝜆−𝑀^𝜅_𝜆𝜕𝐹^𝛾_𝛼𝛽𝜕𝑣^𝜅+𝛿^𝑡𝑀^𝛾_𝛼𝛽𝛿^𝑡𝑢^𝜆+𝐹^𝛾_𝜆𝜇𝑀^𝜇_𝛼𝛽+𝑀^𝛾_𝜆𝜇𝐹^𝜇_𝛼𝛽−𝑀^𝛾_𝜆𝜇𝑀^𝜇_𝛼𝛽+𝛿^𝑡𝑁^𝛾_𝛼𝜆𝛿^𝑡𝑢^𝛽+𝐹^𝛾_𝛽𝜇𝑁^𝜇_𝛼𝜆+𝑁^𝛾_𝛽𝜇𝐹^𝜇_𝛼𝜆+𝑁^𝛾_𝛽𝜇𝑁^𝜇_𝛼𝜆+𝑁^𝛾_𝜆𝜇𝑀^𝜇_𝛼𝛽+𝑁^𝜇_𝛼𝛽𝑀^𝛾_𝜆𝜇+𝑔^𝑟𝑠𝛿^𝛾_𝜆𝑓_𝑠−𝑓𝑔^𝑟𝑠𝐆^𝜅_𝑠𝐶^𝛾_𝜆𝜅×𝑔_𝛼𝛽𝑓_𝑟−𝑓𝐆^𝜇_𝑟𝐶_𝛼𝛽𝜇−ℭ^𝛽_𝜆, (83)(84)(85)(86)(87)(88)(89)(90)(91)(92)(93)(94) where 𝑀^𝑖_𝑗𝑘=𝑀^𝑟_𝑘𝐶^𝑖_𝑗𝑟+𝑀^𝑟_𝑗𝐶^𝑖_𝑘𝑟−𝑀^𝑟_ℎ𝑔^𝑖ℎ𝐶_𝑗𝑘𝑟,𝑀^𝛾_𝛼𝛽=𝑀^𝜇_𝛽𝐶^𝛾_𝛼𝜇+𝑀^𝜇_𝛼𝐶^𝛾_𝛽𝜇−𝑀^𝜇_𝜈𝑔^𝛾𝜈𝐶_𝛼𝛽𝜇,𝑁^𝛾_𝛼𝛽=𝑓⁻¹𝑓_𝛽𝛿^𝛾_𝛼+𝑓_𝛼𝛿^𝛾_𝛽−𝑓_𝜆𝑔^𝛾𝜆𝑔_𝛼𝛽 (95) and ℭ^𝑖_𝑗 denotes the interchange of indices 𝑖, 𝑗, and subtraction.

By Theorem 18, we have the following.

Theorem 16. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a flat twisted product Finsler manifold, and let (𝑀₁,𝐹₁) be Riemannian. If 𝑓 is a function on 𝑀₂, only, then (𝑀₁,𝐹₁) is locally flat.

Similarly, we get the following.

Theorem 17. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a flat twisted product Finsler manifold, and let (𝑀₂,𝐹₂) be Riemannian. If 𝑓 is a function on 𝑀₁, only, then (𝑀₂,𝐹₂) is a space of positive constant curvature ||𝑔𝑟𝑎𝑑𝑓||².

Proof. Since 𝑀₂ is Riemannain and 𝑓 is a function on 𝑀₁, then by (94), we obtain 𝐑^𝛾_𝛼𝛽𝜆=𝑅^𝛾_𝛼𝛽𝜆+𝑔𝑟𝑎𝑑𝑓²𝛿^𝛾_𝜆𝑔_𝛼𝛽−𝛿^𝛾_𝛽𝑔_𝛼𝜆. (96) Since (𝑀₁×_𝑓𝑀₂,𝐹) is flat, then 𝐑^𝛾_𝛼𝛽𝜆=0. Thus the proof is complete.

Theorem 18. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Riemannian manifold, and let 𝑓 be a function on 𝑀₂, only. Then (𝑀₁×_𝑓𝑀₂,𝐹) is flat, if and only if (𝑀₁,𝐹₁) is flat, and the Riemannian curvature of (𝑀₂,𝐹₂) satisfies in the following equation: 𝑅^𝛾_𝛼𝛽𝜆=𝛿^𝑡𝑁^𝛾_𝛼𝜆𝛿^𝑡𝑢^𝛽+𝐹^𝛾_𝛽𝜇𝑁^𝜇_𝛼𝜆+𝑁^𝛾_𝛽𝜇𝐹^𝜇_𝛼𝜆+𝑁^𝛾_𝛽𝜇𝑁^𝜇_𝛼𝜆−ℭ^𝛽_𝜆. (97)

5. 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 [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 (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then the coefficients of Berwald curvature are as follows: 𝐁^𝛾_𝛼𝛽𝜆=𝐵^𝛾_𝛼𝛽𝜆+𝑓⁻¹𝐶^𝛾𝜈_{𝜆;𝛼;𝛽}𝑓_𝜈𝐹²₂+2𝐶^𝛾𝜈_𝛼;𝛽𝑓_𝜈𝑣_𝜆+2𝐶^𝛾𝜈_𝛼;𝜆𝑓_𝜈𝑣_𝛽+2𝐶_𝛼^𝛾𝜈𝑓_𝜈𝑔_𝜆𝛽+2𝐶^𝛾𝜈_𝜆;𝛽𝑓_𝜈𝑣_𝛼+2𝐶_𝛽^𝛾𝜈𝑓_𝜈𝑔_𝜆𝛼+2𝐶_𝜆^𝛾𝜈𝑓_𝜈𝑔_𝛼𝛽−2𝐶_𝛼𝛽𝜆𝑓^𝛾,𝐁^𝑘_𝑖𝑗𝑙=𝐵^𝑘_𝑖𝑗𝑙+𝑓𝐶^𝑘ℎ_{𝑙;𝑗;𝑖}𝑓_ℎ𝐹²₂,𝐁^𝑘_𝑖𝛽𝑙=2𝑓𝐶^𝑘ℎ_𝑖;𝑙𝑓_ℎ𝑣_𝛽,𝐁^𝑘_𝛼𝛽𝑙=2𝑓𝑔_𝛼𝛽𝐶_𝑙^𝑘ℎ𝑓_ℎ,𝐁^𝑘_𝛼𝛽𝜆=−2𝑓𝐶_𝛼𝛽𝜆𝑓^𝑘,𝐁^𝛾_𝑖𝛽𝜆=𝐁^𝛾_𝑖𝑗𝜆=𝐁^𝛾_𝑖𝑗𝑘=0. (98)(99)(100)(101)(102)(103)

Let (𝑀₁×_𝑓𝑀₂,𝐹) is a Berwald manifold. Then we have 𝐁^𝑑_𝑎𝑏𝑐=0. By using (102), we get 𝐶_𝛼𝛽𝜆𝑓^𝑘=0. (104) Multiplying this equation in 𝑔_𝑘𝑟, we obtain 𝐶_𝛼𝛽𝜆𝑓_𝑟=0. (105) Thus if 𝑓 is not constant on 𝑀₁, then we have 𝐶_𝛼𝛽𝜆=0. Also, from (101), we result that 𝐶_𝑙^𝑘ℎ𝑓_ℎ=0. (106) Differentiating this equation with respect to 𝑦^𝑗 gives us 𝐶^𝑘ℎ_𝑙;𝑗𝑓_ℎ=0. (107) Similarly we obtain 𝐶^𝑘ℎ_{𝑙;𝑗;𝑖}𝑓_ℎ=0. (108) Setting the last equation in (99) implies that 𝐵^𝑘_𝑖𝑗𝑙=0; that is, (𝑀₁,𝐹₁) is Berwaldian. These explanations give us the following theorem.

Theorem 20. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold, and let 𝑓 be not constant on 𝑀₁. Then (𝑀₁×_𝑓𝑀₂,𝐹) is Berwaldian if and only if (𝑀₁,𝐹₁) is Berwaldian, (𝑀₂,𝐹₂) is Riemannian, and the equation 𝐶_𝑙^𝑘ℎ𝑓_ℎ=0 is hold.

But if 𝑓 is constant on 𝑀₁, that is, 𝑓_𝑖=0, then we get the following.

Theorem 21. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold, and 𝑓 is constant on 𝑀₁. Then (𝑀₁×_𝑓𝑀₂,𝐹) is Berwaldian if and only if (𝑀₁,𝐹₁) is Berwaldian and the Berwald curvature of (𝑀₂,𝐹₂) satisfies in the following equation: 𝐵^𝛾_𝛼𝛽𝜆=−𝑓⁻¹𝐶^𝛾𝜈_{𝛽;𝛼;𝜆}𝑓_𝜈𝐹²₂+2𝐶^𝛾𝜈_𝛽;𝛼𝑓_𝜈𝑣_𝜆+2𝐶^𝛾𝜈_𝜆;𝛼𝑓_𝜈𝑣_𝛽+2𝐶_𝛼^𝛾𝜈𝑓_𝜈𝑔_𝜆𝛽+2𝐶^𝛾𝜈_𝛽;𝜆𝑓_𝜈𝑣_𝛼+2𝐶_𝛽^𝛾𝜈𝑓_𝜈𝑔_𝜆𝛼+2𝐶_𝜆^𝛾𝜈𝑓_𝜈𝑔_𝛼𝛽−2𝑔^𝛾𝜈𝐶_𝛼𝛽𝜆𝑓_𝜈. (109)

Here, we consider twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,𝐹) of isotropic Berwald curvature.

Theorem 22. Every isotropic Berwald twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,𝐹) is a Berwald manifold.

Proof. Let (𝑀₁×_𝑓𝑀₂,𝐹) be an isotropic Berwald manifold. Then we have 𝐁^𝑑_𝑎𝑏𝑐=𝑐𝐹⁻¹𝐡^𝑑_𝑎𝐡_𝑏𝑐+𝐡^𝑑_𝑏𝐡_𝑎𝑐+𝐡^𝑑_𝑐𝐡_𝑎𝑏+2𝐂_𝑎𝑏𝑐𝐲^𝑑, (110) where 𝑐=𝑐(𝐱) is a function on 𝑀. Setting 𝑎=𝑗, 𝑏=𝑘, 𝑐=𝑙, and 𝑑=𝛾 and using (103) imply that 𝑐𝐹⁻¹3𝐹²𝑦_𝑗𝑦_𝑘𝑦_𝑙𝑣^𝛾−𝑣^𝛾𝑦_𝑗𝑔_𝑘𝑙+𝑦_𝑘𝑔_𝑗𝑙+𝑦_𝑙𝑔_𝑗𝑘=0. (111) Multiplying the previous equation in 𝑦^𝑗𝑦^𝑘, we derive that 𝑐𝑓²𝐹²₁𝐹²₂=0. Thus we have 𝑐=0; that is, (𝑀₁×_𝑓𝑀₂) 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 (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then the coefficients of mean Berwald curvature are as follows: 𝐄_𝛼𝛽=𝐸_𝛼𝛽+𝑓𝑔_𝛼𝛽𝐼^ℎ𝑓_ℎ+12𝑓𝐼^𝜈_;𝛼;𝛽𝑓_𝜈𝐹²₂+𝑓⁻¹𝑓_𝜈×𝐶^𝛾𝜈_𝛼;𝛽𝑣_𝛾+𝐼^𝜈_;𝛼𝑣_𝛽+𝐼^𝜈_;𝛽𝑣_𝛼+𝐶^𝜈_𝛼𝛽+𝐼^𝜈𝑔_𝛼𝛽,𝐄_𝑖𝑗=𝐸_𝑖𝑗+12𝑓𝐼^ℎ_;𝑗;𝑖𝑓_ℎ𝐹²₂,𝐄_𝑖𝛽=𝑓𝐼^ℎ_;𝑖𝑓_ℎ𝑣_𝛽, (112)(113)(114) where 𝐸_𝑖𝑗 and 𝐸_𝛼𝛽 are the coefficients of mean Berwald curvature of (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂), respectively.

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

Theorem 24. The twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,𝐹) is weakly Berwald if and only if (𝑀₁,𝐹₁) is weakly Berwald, 𝐼^ℎ𝑓_ℎ=0, and the following hold: 𝐸_𝛼𝛽1=−2𝑓𝐼^𝜈_;𝛼;𝛽𝑓_𝜈𝐹²₂−𝑓⁻¹𝑓_𝜈×𝐶^𝛾𝜈_𝛼;𝛽𝑣_𝛾+𝐼^𝜈_;𝛼𝑣_𝛽+𝐼^𝜈_;𝛽𝑣_𝛼+𝐶^𝜈_𝛼𝛽+𝐼^𝜈𝑔_𝛼𝛽. (115)

Proof. If (𝑀₁×_𝑓𝑀₂) be a weakly Berwald manifold, then we have 𝐄_𝛼𝛽=𝐄_𝑖𝑗=𝐄_𝑖𝛽=0. (116) Thus by using (114), we result that 𝐼^ℎ_;𝑖𝑓_ℎ=0. This equation implies that 𝐼^ℎ_;𝑗;𝑖𝑓_ℎ=0,𝐼^ℎ𝑓_ℎ=0. (117) By setting these equations in (112) and (113), we conclude that 𝐸_𝑖𝑗=0 and 𝐸_𝛼𝛽 satisfies in (115).

Now, if 𝑓 is constant on 𝑀₂, then (115) implies that 𝐸_𝛼𝛽=0. Thus we conclude the following.

Corollary 25. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold, and let 𝑓 be a function on 𝑀₁, only. Then (𝑀₁×_𝑓𝑀₂,𝐹) is weakly Berwald if and only if (𝑀₁,𝐹₁) and (𝑀₂,𝐹₂) are weakly Berwald manifolds and 𝐼^ℎ𝑓_ℎ=0.

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 (𝑀₁×_𝑓𝑀₂,𝐹) is isotropic mean Berwald manifold if and only if 𝐸_𝛼𝛽+𝑓𝑔_𝛼𝛽𝐼^ℎ𝑓_ℎ+12𝑓𝐼^𝜈_;𝛼;𝛽𝑓_𝜈𝐹²₂+𝑓⁻¹𝑓_𝜈×𝐶^𝛾𝜈_𝛼;𝛽𝑣_𝛾+𝐼^𝜈_;𝛼𝑣_𝛽+𝐼^𝜈_;𝛽𝑣_𝛼+𝐶^𝜈_𝛼𝛽+𝐼^𝜈𝑔_𝛼𝛽−𝑛+12𝑐𝑓²𝐹⁻¹𝑔_𝛼𝛽−𝑓²𝐹²𝑣_𝛼𝑣_𝛽𝐸=0,_𝑖𝑗+12𝑓𝐼^ℎ_;𝑗;𝑖𝑓_ℎ𝐹²₂−𝑛+12𝑐𝐹⁻¹𝑔_𝑖𝑗−1𝐹²𝑦_𝑖𝑦_𝑗𝑐()𝐹=0,𝑛+1⁻³𝑦_𝑖+𝑓𝐼^ℎ_;𝑖𝑓_ℎ=0, (118)(119)(120) where 𝑐=𝑐(𝐱) is a scalar function on 𝑀.

Theorem 27. Every twisted product Finsler manifold (𝑀₁×_𝑓𝑀₂,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 (120) with respect to 𝑣^𝛾 gives us 𝑐(𝑛+1)𝑓²𝐹⁻⁵𝑣_𝛾𝑦_𝑖=0. (121) Thus, we conclude that 𝑐=0. This implies that 𝐹 reduces to a weakly Berwald metric.

6. Locally Dually Flat Twisted Product Finsler Manifolds

In [23], Amari and 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 multiterminal information theory. In Finsler geometry, Shen extends the notion of locally dually flatness for Finsler metrics [17]. 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 [24, 25].

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 𝑇𝑀 such that it satisfies 𝜕²𝐹²𝜕𝐱^𝑏𝜕𝐲^𝑎𝐲^𝑏=2𝜕𝐹²𝜕𝐱^𝑎. (122) In this case, the coordinate (𝐱^𝑎) is called an adapted local coordinate system. By using (122), we can obtain the following lemma.

Lemma 28. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold. Then 𝐹 is locally dually flat if and only if 𝐹₁ and 𝐹₂ satisfy in the following equations: 𝜕²𝐹²₁𝜕𝑥^𝑘𝜕𝑦^𝑙𝑦^𝑘=2𝜕𝐹²₁𝜕𝑥^𝑙+4𝑓𝑓_𝑙𝐹²₂,4𝑓_𝑘𝑣_𝛽𝑦^𝑘𝜕+𝑓²𝐹²₂𝜕𝑢^𝛼𝜕𝑣^𝛽𝑣^𝛼+4𝑓_𝛼𝑣_𝛽𝑣^𝛼=2𝑓𝜕𝐹²₂𝜕𝑢^𝛽+4𝑓_𝛽𝐹²₂. (123)(124)

Now, let 𝐹 be a locally dually flat Finsler metric. Taking derivative with respect to 𝑣^𝛾 from (123) yields 𝑓_𝑙=0, which means that 𝑓 is a constant function on 𝑀₁. In this case, the relations (123) and (124) reduce to the following: 𝜕²𝐹²₁𝜕𝑥^𝑘𝜕𝑦^𝑙𝑦^𝑘=2𝜕𝐹²₁𝜕𝑥^𝑙,𝑓𝜕²𝐹²₂𝜕𝑢^𝛼𝜕𝑣^𝛽𝑣^𝛼+4𝑓_𝛼𝑣_𝛽𝑣^𝛼=2𝑓𝜕𝐹²₂𝜕𝑢^𝛽+4𝑓_𝛽𝐹²₂. (125)(126) By (125), we deduce that 𝐹₁ is locally dually flat.

Now, we assume that 𝐹₁ and 𝐹₂ are locally dually flat Finsler metrics. Then we have 𝜕²𝐹²₁𝜕𝑥^𝑘𝜕𝑦^𝑙𝑦^𝑘=2𝜕𝐹²₁𝜕𝑥^𝑙,𝜕²𝐹²₂𝜕𝑢^𝛼𝜕𝑣^𝛽𝑣^𝛼=2𝜕𝐹²₂𝜕𝑢^𝛽. (127) By (127), we derive that (123) and (124) are hold if and only if the following hold: 𝑓_𝑙=0,𝑓_𝛼𝑣_𝛽𝑣^𝛼=𝑓_𝛽𝐹²₂. (128) Therefore we can conclude the following.

Theorem 29. Let (𝑀₁×_𝑓𝑀₂,𝐹) be a twisted product Finsler manifold.(i)If 𝐹 is locally dually flat, then 𝐹₁ is locally dually flat, 𝑓 is a function with respect to (𝑢^𝛼) only, and 𝐹₂ satisfies in (126).(ii)If 𝐹₁ and 𝐹₂ are locally dually flat, then 𝐹 is locally dually flat if and only if 𝑓 is a function with respect (𝑢^𝛼) only and 𝐹₂ satisfies in (128).

By Theorem 29, we conclude the following.

Corollary 30. There is not exist any locally dually flat proper twisted product Finsler manifold.