On conformally doubly warped product Finsler manifold

Abstract

The aim of the present paper is to introduce the notion of conformally doubly warped product Finsler manifold (CDWPF). For such a Finsler manifold, the coefficients of Barthel connection and its curvature tensor are investigated. The coefficients of Cartan, Berwald, Hashiguchi and Chern (Rund) connections of CDWPF are calculated. Some special Finsler spaces are studied.

Introduction

The doubly warped product of Riemannian (semi-Riemannian) manifolds has been studied by many authors, for example, we refer to [1–5]. Several applications to theoretical physics can be found in the literature. For instance, in [3], Beem-Powell considered the doubly warped product for Lorentzian manifolds. Moreover, in [6, 7], Asanov studied the generalization of the Schwarzschild metric in the Finslerian setting and obtained some models of relativity theory described through the warped product of Finsler metrics. Then, Shen used a construction of warped of Riemannian metrics at the vertical bundle and obtained a Finslerian warped product metric [8]. Recently, E. Peyghan and A. Tayebi ([9]) introduced horizontal and vertical warped product Finsler manifold and they proved that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian.

In this paper, we study a more general product Finsler manifold that will be called conformally doubly warped product Finsler manifold (CDWPF), that is, the product manifold M:=M₁×M₂ endowed with the metric $F : TM^{o}_{1} \times TM^{o}_{2} \rightarrow R$ defined by

$$ F(v_{1},v_{2})=e^{\sigma(\pi_{1}(v_{1}),\pi_{2}(v_{2}))}\sqrt{f^{2}_{2}(\pi_{2}(v_{2}))F^{2}_{1}(\pi_{1}(v_{1}))+f^{2}_{1}(\pi_{1}(v_{1}))F^{2}_{2}(\pi_{2}(v_{2}))} \,, $$

where (M₁,F₁) and (M₂,F₂) are two Finsler manifolds; f₁:M₁→R⁺ and f₂:M₂→R⁺ are two smooth functions on M₁ and M₂, respectively; π₁:M₁×M₂→M₁,π₂:M₁×M₂→M₂ are the natural projection maps; and σ:M₁×M₂→R⁺ is a positively smooth function on $M_{1}\times M_{2}, TM^{o}_{1} = TM_{1}-{0}$ and $TM^{o}_{2} = TM_{2}-{0}$.

For a conformally doubly warped product Finsler manifolds (CDWPF), the coefficients of Barthel connection and its curvature tensor are studied. Moreover, the coefficients of Cartan connection for CDWPF are given. Finally, some special Finsler spaces concerning a conformally doubly warped product Finsler manifold are derived.

Finally, the obtained results in this paper generalize and retrieve some results concerning the doubly warped product Finsler manifold, warped product Finsler manifold, conformally warped product Finsler manifold, product Finsler manifold and conformally product Finsler manifold.

Notations and preliminaries

In this section, we give a brief account of the basic concepts of Finsler geometry that will be needed throughout. This means that all notations and results which appear in this section refer to [10–14].

Let M be an n-dimensional smooth manifold. Let (xⁱ) be the coordinates of any point of the base manifold M and (yⁱ) a supporting element at the same point. We mean by T_xM the tangent space at x∈M and $TM={\bigcup }_{x\in M}\,\,T_{x}M$ the tangent bundle of M. A Finsler structure on M is defined as follows:

Definition 1

A Finsler structure on a manifold M is a function [11–13]

$$F:TM\rightarrow \mathbb{R}$$

with the following properties: (a) $F\geqslant 0$ and F(x,y)=0 if and only if y=0. (b) F is C^∞ on the slit tangent bundle $\mathcal {T} M:=TM\backslash \{0\}$. (c) F(x,y) is positively homogenous of degree one in y: F(x,λy)=λF(x,y) for all y∈TM and λ>0. (d) The Hessian matrix $g_{ij}(x,y):=\frac {1}{2}\frac {\partial ^{2} F^{2}}{\partial y^{j} \partial y^{i}}$ is positive-definite at each point of $\mathcal {T} M$.

The pair (M,F) is called a Finsler space and the symmetric bilinear form g=g_ij(x,y)dxⁱ⊗dx^j is called the Finsler metric tensor of the Finsler space (M,F).

The tensor $C_{ijk}:= \frac {1}{4}\frac {\partial ^{3} F^{2}}{\partial y^{k} \partial y^{j} \partial y^{i}}$ is called the Cartan torsion. It is well known that F is called Riemannian if and only if the Cartan tensor C_ijk vanishes identically [10,13]. By Deicke’s Theorem, F is Riemannian if and only if I_i=0, where I_i:=g^jkC_ijk called the mean (contraction) Cartan torsion.

The Matsumoto torsion for a Finsler structure (M,F) with dimension n is given by

$$M_{ijk}:=C_{ijk}+\frac{1}{n+1}\{I_{i} \hbar_{jk} +I_{j} \hbar_{ki}+I_{k} \hbar_{ij}\},$$

where $\hbar _{ij}:=g_{ij}-\frac {1}{F^{2}}y_{i} \, y_{j}$ is the angular metric tensor and $y_{i}:=g_{ij}y^{j}=\frac {\partial F}{\partial y^{i}}$. A Finsler structure F is said to be C-reducible if the Matsumoto torsion vanishes identically [15].

If a Finsler manifold (M,F(x,y)) is given, then the components of the associated canonical spray G^h and the components of the associated nonlinear connection (Barthel connection) $G^{h}_{i}$ are defined respectively by

$$G^{i} (x,y):= \frac{1}{4}g^{ih}\{\frac{\partial^{2} F^{2}}{\partial y^{h} \partial x^{j}}\, y^{j}-\frac{\partial F^{2}}{\partial x^{h}}\}(x,y), \, \, G^{h}_{i}:= \frac{\partial G^{h} }{\partial y^{i}}.$$

Also, the Berwald curvature tensor is defined by

$$B^{h}_{ijk}:=\frac{\partial^{3} G^{h}}{\partial y^{k} \partial y^{j} \partial y^{i}}.$$

A Finsler structure F is called Berwaldian if the Berwald curvature tensor $B^{h}_{ijk}$ vanishes identically.

Finally, we know that there exist at least four linear Finsler connections associated with a Finsler structure F and they have the same nonlinear connection $G^{h}_{i}$ namely, the Cartan connection $C\Gamma \equiv (\ \Gamma ^{h}_{ij}, G^{h}_{i}, C^{h}_{ij})$, the Berwald connection $B\Gamma \equiv (\ G^{h}_{ij}, G^{h}_{i},0)$, the Hashiguchi connection $H\Gamma \equiv (\ G^{h}_{ij}, G^{h}_{i}, C^{h}_{ij})$, and the Chern (Rund) connection $R\Gamma \equiv (\ \Gamma ^{h}_{ij}, G^{h}_{i},0)$, where $C^{h}_{ij}:= \frac {1}{2}\,g^{h\ell }\, \frac {\partial g_{\ell j}}{\partial y^{i}}, G^{h}_{ij}:= \frac {\partial G^{h}_{i}}{\partial y^{j}}$ and $\Gamma ^{k}_{ij}:=\frac {1}{2}\,g^{kh}\{\frac {\delta g_{hj}}{\delta x^{i}}+\frac {\delta g_{hi}}{\delta x^{i}}-\frac {\delta g_{ij}}{\delta x^{h}}\};$$\frac {\delta }{\delta x^{k}}:=\frac {\partial }{\partial x^{k}}-G^{m}_{k} \frac {\partial }{\partial y^{m}}$ being the horizontal basis adapted to the Barthel connection $G^{i}_{j}$.

Barthel connection for CDWPF

In this section, we investigate the coefficients of canonical spray for the conformally doubly warped product Finsler manifold (CDWPF). Moreover, the coefficients of the Barthel connection and its curvature tensor for CDWPF are obtained.

First, we begin with the following definition.

Definition 2

Let (M₁,F₁) and (M₂,F₂) be two Finsler manifolds with dimM₁=n₁ and dimM₂=n₂ and f₁:M₁→R⁺ and f₂:M₂→R⁺ be two smooth functions. Let π₁:M₁×M₂→M₁,π₂:M₁×M₂→M₂ be the natural projection maps and σ:M₁×M₂→R⁺ be positively smooth function on M₁×M₂. The product manifold M:=M₁×M₂ endowed with the metric $F : TM^{o}_{1} \times TM^{o}_{2} \rightarrow R$ defined by

$$ F(v_{1},v_{2})=e^{\sigma(\pi_{1}(v_{1}),\pi_{2}(v_{2}))}\sqrt{f^{2}_{2}(\pi_{2}(v_{2}))F^{2}_{1}(\pi_{1}(v_{1}))+f^{2}_{1}(\pi_{1}(v_{1}))F^{2}_{2}(\pi_{2}(v_{2}))} \,, $$

(1)

where $TM^{o}_{1} = TM_{1}-{0}$ and $TM^{o}_{2} = TM_{2}-{0}$, called the conformally doubly warped product Finsler manifolds (CDWPF) of the manifolds M₁ and M₂, and denoted by $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$. In this case, σ will be called the conformally factor and f₁ and f₂ will be called the warping functions.

Specially, if either f₁=1 or f₂=1, but not both, and σis not constant function, then $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ will be called conformally warped product Finsler manifolds (CWPF) of the manifolds M₁ and M₂. If both f₁=1,f₂=1, and σ is not constant function, then $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ will be called a conformally product Finsler manifold (CPF). If neither f₁ nor f₂ is constant and σ=0, then $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ will be called a doubly warped product Finsler manifold (DWPF).

Now, let (M₁,F₁) and (M₂,F₂) be two Finsler manifolds with dimensions n₁ and n₂, respectively. Hence, the two functions

$$\begin{array}{@{}rcl@{}} g_{ij}(x,y) &:=& \frac{\partial^{2} F^{2}_{1} }{\partial y^{i} \partial y^{j}} {\qquad\quad\qquad} g_{\alpha\beta}(u,v) := \frac{\partial^{2} F^{2}_{2} }{\partial v^{\alpha} \partial v^{\beta}} \end{array} $$

(2)

define Finsler metrics on M₁ and M₂, respectively. Let $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ be a conformally doubly warped Finsler manifold (CDWPF) and let x∈M and y∈T_xM, where x=(x,u),y=(y,v),M:=M₁×M₂ and T_xM=T_xM₁⊕T_uM₂.

Consequently, from Eqs. (1) and (2), the conformally doubly warped Finsler metric and its inverse are given by

$$\begin{array}{@{}rcl@{}} \mathbf{g}_{ab}(\mathrm{x},\mathrm{y}) &:=& \frac{\partial^{2} F^{2} (\mathrm{x},\mathrm{y}) }{\partial \mathrm{y}^{a} \partial \mathrm{y}^{b}}=\left(\begin{array}{ccc} e^{2\sigma(x,u)}f^{2}_{2} g_{ij}& 0 \\ 0 &e^{2\sigma(x,u)}f^{2}_{1} g_{\alpha\beta} \\ \end{array} \right) \end{array} $$

(3)

$$\begin{array}{@{}rcl@{}} \mathbf{g}^{ab}(\mathrm{x},\mathrm{y}) &=&\left(\begin{array}{ccc} e^{-2\sigma(x,u)}\frac{1}{f^{2}_{2}} g^{ij}& 0 \\ 0 &e^{-2\sigma(x,u)}\frac{1}{f^{2}_{1}} g^{\alpha\beta} \\ \end{array} \right), \end{array} $$

(4)

where $\mathrm {y}^{a} := (y^{i}, v^{\alpha }), \; \mathrm {y}^{b}:= (y^{j}, v^{\beta }),\; \mathbf {g}_{ij}= e^{2\sigma (x,u)}f^{2}_{2} g_{ij}, \;\mathbf {g}_{\alpha \beta }= e^{2\sigma (x,u)}f^{2}_{1} g_{\alpha \beta }, \mathbf {g}_{i\beta } = \mathbf {g}_{\alpha j} =~0 ; \:i, j,...\in \{1,..., n_{1}\},\, \alpha, \beta,...\in \{1,..., n_{2}\}$ and a,b,...∈{1,...,n₁+n₂}.

Proposition 1

The coefficients of conformally doubly warped canonical spray for CDWPF are given by

$$\mathbb{G}^{a}(x,u,y,v)=(\mathbb{G}^{i}(x,u,y,v),\mathbb{G}^{\alpha}(x,u,y,v)),$$

where

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{i}(x,u,y,v) &=& G^{i}(x,y)+ \frac{1}{4}g^{ih}\{2(\frac{\partial\sigma}{\partial x^{j}}\,y^{j}+\frac{\partial\sigma}{\partial u^{\alpha}}\,v^{\alpha})\frac{\partial F^{2}_{1}}{\partial y^{h}} -\frac{1}{f^{2}_{2}}\frac{\partial\sigma}{\partial x^{h}}(f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2}) \\ && +\frac{1}{f^{2}_{2}}(\frac{\partial f^{2}_{2}}{\partial u^{\alpha}} \frac{\partial F^{2}_{1}}{\partial y^{h}}\, v^{\alpha}-\frac{\partial f^{2}_{1}}{\partial x^{h}}F^{2}_{2})\} \end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{\alpha}(x,u,y,v) &=& G^{\alpha}(u,v)+ \frac{1}{4}g^{\alpha\gamma}\{2(\frac{\partial\sigma}{\partial u^{\beta}}\,v^{\beta}+\frac{\partial\sigma}{\partial x^{j}}\,y^{j})\frac{\partial F^{2}_{2}}{\partial v^{\alpha}} -\frac{1}{f^{2}_{1}}\frac{\partial\sigma}{\partial u^{\gamma}}(f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2}) \\ && +\frac{1}{f^{2}_{1}}(\frac{\partial f^{2}_{1}}{\partial x^{j}} \frac{\partial F^{2}_{2}}{\partial v^{\beta}}\, y^{j}-\frac{\partial f^{2}_{2}}{\partial u^{\gamma}}F^{2}_{1})\}\,. \end{array} $$

(6)

Proof

We know that the coefficients of canonical spray for (M₁,F₁),(M₂,F₂), and $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ are defined respectively by

$$\begin{array}{@{}rcl@{}} G^{i} (x,y)&=& \frac{1}{4}g^{ih}\{\frac{\partial^{2} F^{2}_{1}}{\partial y^{h} \partial x^{j}}\, y^{j}-\frac{\partial F^{2}_{1}}{\partial x^{h}}\}(x,y) \end{array} $$

(7)

$$\begin{array}{@{}rcl@{}} G^{\alpha}(u,v) &=& \frac{1}{4}g^{\alpha\gamma}\{\frac{\partial^{2} F^{2}_{2}}{\partial v^{\gamma} \partial u^{\beta}}\, v^{\beta}-\frac{\partial F^{2}_{2}}{\partial u^{\gamma}}\}(u,v) \end{array} $$

(8)

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{a} (\mathrm{x},\mathrm{y}) &=& \frac{1}{4}\mathbf{g}^{ab}\{\frac{\partial^{2} F^{2}}{\partial \mathrm{y}^{b} \partial \mathrm{x}^{c}}\, \mathrm{y}^{c}-\frac{\partial F^{2}}{\partial \mathrm{x}^{b}}\}(\mathrm{x},\mathrm{y}) \,. \end{array} $$

(9)

Setting a=i into (9) and noting the fact that g^iβ=0, we get

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{i} (\mathrm{x},\mathrm{y}) &=& \frac{1}{4}\mathbf{g}^{ib}\{\frac{\partial^{2} F^{2}}{\partial \mathrm{y}^{b} \partial \mathrm{x}^{c}}\, \mathrm{y}^{c}-\frac{\partial F^{2}}{\partial \mathrm{x}^{b}}\}(\mathrm{x},\mathrm{y}) \\ &=&\frac{1}{4}\mathbf{g}^{ih}\{\frac{\partial^{2} F^{2}}{\partial {y}^{h} \partial {x}^{j}}\, {y}^{j} + \frac{\partial^{2} F^{2}}{\partial {y}^{h} \partial {u}^{\alpha}}\, {v}^{\alpha} -\frac{\partial F^{2}}{\partial {x}^{h}}\} \,. \end{array} $$

(10)

On the other hand, from (1) one can show that

$$\begin{array}{@{}rcl@{}} \frac{\partial F^{2}}{\partial {x}^{j}} &=& e^{2\sigma} \left\{2\frac{\partial\sigma}{\partial x^{j}}(f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})+ f^{2}_{2} \frac{\partial F^{2}_{1}}{\partial x^{j}}+\frac{\partial f^{2}_{1}}{\partial x^{j}} F^{2}_{2}\right\} \\ && \\ \frac{\partial F^{2}}{\partial {u}^{\alpha}} &=& e^{2\sigma}\left\{2\frac{\partial\sigma}{\partial {u}^{\alpha}}(f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})+ f^{2}_{1} \frac{\partial F^{2}_{2}}{\partial {u}^{\alpha}}+\frac{\partial f^{2}_{2}}{\partial {u}^{\alpha}} F^{2}_{1} \right\} \\ && \\ \frac{\partial^{2} F^{2}}{\partial y^{h} \partial {x}^{j}} &=& e^{2\sigma} f^{2}_{2} \left\{2\frac{\partial\sigma}{\partial x^{j}} \frac{\partial^{2} F^{2}_{1}}{\partial y^{h}}+ \frac{\partial^{2} F^{2}_{1}}{\partial y^{h} \partial x^{j}} \right\} \\ && \\ \frac{\partial^{2} F^{2}}{\partial y^{h} \partial {u}^{\alpha}} &=& e^{2\sigma} \left\{2f^{2}_{2}\frac{\partial\sigma}{\partial u^{\alpha}} \frac{\partial^{2} F^{2}_{1}}{\partial y^{h}}+ \frac{\partial f^{2}_{2}}{\partial {u}^{\alpha}}\frac{\partial F^{2}_{1}}{\partial y^{h}} \right\} \,. \end{array} $$

Hence, Relation (5) follows by substituting the above relations into (10), taking into account (3), (4) and (7).

Similarly, by putting b=α into (9), using Eq. (4), g^αj=0 and after some calculations, one can deduce Relation (6). This completes the proof. □

Proposition 2

The coefficients of conformally doubly warped product Barthel connection for CDWPF are given by

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{a}_{b}(\mathrm{x},\mathrm{y}) &:=& \frac{\partial \mathbb{G}^{a}(\mathrm{x},\mathrm{y})}{\partial \mathrm{y}^{b}}=\left(\begin{array}{ccc} \mathbb{G}^{i}_{j}(x,u,y,v)& \quad \mathbb{G}^{\alpha}_{j}(x,u,y,v) \\ \mathbb{G}^{i}_{\beta}(x,u,y,v) & \quad \mathbb{G}^{\alpha}_{\beta}(x,u,y,v) \\ \end{array} \right) \,, \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{i}_{j}(x,u,y,v) &:=& \frac{\partial \mathbb{G}^{i}}{\partial y^{j}}= G^{i}_{j}-\frac{1}{4f^{2}_{2}}\frac{\partial g^{ih}}{\partial y^{j}}\frac{\partial\sigma}{\partial x^{h}}\left(f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2}\right)+ \left(\frac{\partial\sigma}{\partial x^{r}}\,y^{r} + \frac{\partial\sigma}{\partial u^{\alpha}}\,v^{\alpha}\right)\delta^{i}_{j} \\ &&+ \frac{\partial\sigma}{\partial x^{j}}\,y^{i}-\frac{1}{4}\, g^{ih}\frac{\partial\sigma}{\partial x^{h}}\frac{\partial F^{2}_{1}}{\partial y^{j}} -\frac{1}{4 f^{2}_{2}}\frac{\partial g^{ih}}{\partial y^{j}}\frac{\partial f^{2}_{1}}{\partial x^{h}}\,F^{2}_{2}+\frac{1}{2f^{2}_{2}}\frac{\partial f^{2}_{2}}{\partial u^{\alpha}}\,v^{\alpha} \delta^{i}_{j} \,,\\ && \\ \mathbb{G}^{i}_{\beta}(x,u,y,v) &:=& \frac{\partial \mathbb{G}^{i}}{\partial v^{\beta}}= \frac{1}{4}\,g^{ih}\! \left\{2\frac{\partial\sigma}{\partial u^{\beta}} \frac{\partial F^{2}_{1}}{\partial y^{h}}-\frac{1}{f^{2}_{2}}f^{2}_{1} \frac{\partial\sigma}{\partial x^{h}} \frac{\partial F^{2}_{2}}{\partial v^{\beta}}+ \frac{1}{f^{2}_{2}}\left(\!\frac{\partial f^{2}_{2}}{\partial u^{\beta}} \frac{\partial F^{2}_{1}}{\partial y^{h}}- \frac{\partial f^{2}_{1}}{\partial x^{h}} \frac{\partial F^{2}_{2}}{\partial v^{\beta}}\!\right)\!\right\},\\ && \\ \mathbb{G}^{\alpha}_{j}(x,u,y,v) &:=& \frac{\partial \mathbb{G}^{\alpha}}{\partial y^{j}}= \frac{1}{4}\,g^{\alpha\gamma}\! \left\{\!2\frac{\partial\sigma}{\partial x^{j}} \frac{\partial F^{2}_{2}}{\partial v^{\gamma}}-\frac{1}{f^{2}_{1}}f^{2}_{2} \frac{\partial\sigma}{\partial u^{\gamma}} \frac{\partial F^{2}_{1}}{\partial y^{j}}+ \frac{1}{f^{2}_{1}}\left(\!\frac{\partial f^{2}_{1}}{\partial x^{j}} \frac{\partial F^{2}_{2}}{\partial v^{\gamma}}- \frac{\partial f^{2}_{2}}{\partial u^{\gamma}} \frac{\partial F^{2}_{1}}{\partial y^{j}}\!\right)\!\right\}\!,\\ && \\ \mathbb{G}^{\alpha}_{\beta}(x,u,y,v) &:=& \frac{\partial \mathbb{G}^{\alpha}}{\partial v^{\beta}}= G^{\alpha}_{\beta}-\frac{1}{4f^{2}_{1}}\frac{\partial g^{\alpha\gamma}}{\partial v^{\beta}}\frac{\partial\sigma}{\partial u^{\gamma}}(f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})+\left(\frac{\partial\sigma}{\partial x^{r}}\,y^{r}+\frac{\partial\sigma}{\partial u^{\alpha}}\,v^{\alpha}\right)\delta^{\alpha}_{\beta} \\ &&+ \frac{\partial\sigma}{\partial u^{\beta}}\,v^{\alpha}-\frac{1}{4}\, g^{\alpha\gamma}\frac{\partial\sigma}{\partial u^{\gamma}}\frac{\partial F^{2}_{2}}{\partial v^{\beta}} -\frac{1}{4 f^{2}_{1}}\frac{\partial g^{\alpha\gamma}}{\partial v^{\beta}}\frac{\partial f^{2}_{2}}{\partial u^{\gamma}}\,F^{2}_{1}+\frac{1}{2f^{2}_{1}}\frac{\partial f^{2}_{1}}{\partial x^{r}}\,y^{r} \delta^{\alpha}_{\beta}. \end{array} $$

Proof

The proof follows from Proposition 1 and taking into account the fact that $\frac {\partial g^{ih}}{\partial y^{j}}\frac {\partial F^{2}_{1}}{\partial y^{h}}=0 \left (\frac {\partial g^{\alpha \gamma }}{\partial v^{\beta }}\frac {\partial F^{2}_{2}}{\partial v^{\gamma }}=0\right)$. □

Corollary 1

In view of the above proposition and [9], the basis of the vertical and the horizontal distributions VTM^o and HTM^o; M:=M₁×M₂ for the CDWPF $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ are given respectively by

$$\begin{array}{@{}rcl@{}} \frac{\partial}{\partial \mathrm{y}^{a}} &=& \frac{\partial}{\partial y^{i}} \, \delta^{i}_{a} + \frac{\partial}{\partial v^{\alpha}} \, \delta^{\alpha}_{a} \\ \frac{\delta^{d}}{\delta^{d} \mathrm{x}^{a}} &=& \frac{\delta^{d}}{\delta^{d} x^{i}}\, \delta^{i}_{a} + \frac{\delta^{d}}{\delta^{d} u^{\alpha}} \, \delta^{\alpha}_{a} \,, \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \frac{\delta^{d}}{\delta^{d} x^{j}}&:=& \frac{\partial}{\partial x^{j}}-\mathbb{G}^{r}_{j} \frac{\partial}{\partial y^{j}}-\mathbb{G}^{\beta}_{j} \frac{\partial}{\partial v^{\beta}} = \frac{\delta}{\delta x^{j}}- \mathbb{M}^{r}_{j} \frac{\partial}{\partial y^{r}} -\mathbb{G}^{\beta}_{j} \frac{\partial}{\partial v^{\beta}} \,,\\ && \\ \frac{\delta^{d}}{\delta^{d} u^{\alpha}}&:=& \frac{\partial}{\partial u^{\alpha}}-\mathbb{G}^{r}_{\alpha} \frac{\partial}{\partial y^{r}}- \mathbb{G}^{\mu}_{\alpha} \frac{\partial}{\partial v^{\mu}} = \frac{\delta}{\delta u^{\alpha}}- \mathbb{M}^{\mu}_{\alpha} \frac{\partial}{\partial v^{\mu}} -\mathbb{G}^{r}_{\alpha} \frac{\partial}{\partial y^{r}},\\ \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \frac{\delta}{\delta x^{j}}&:=&\frac{\partial}{\partial x^{j}}-{G}^{r}_{j} \frac{\partial}{\partial y^{j}} \,, \, \, \, \frac{\delta}{\delta u^{\alpha}}:=\frac{\partial}{\partial u^{\alpha}}-{G}^{\mu}_{\alpha} \frac{\partial}{\partial v^{\mu}} \,, \\ \mathbb{M}^{r}_{j}&:=&\frac{1}{2f^{2}_{2}} \frac{\partial f^{2}_{2}}{\partial u^{\alpha}}\, v^{\alpha} \, \delta^{r}_{j} -\frac{1}{4f^{2}_{2}} \frac{\partial g^{r}h}{\partial y^{j}} \frac{\partial f^{2}_{1}}{\partial x^{h}} \, F^{2}_{2} -\frac{1}{4f^{2}_{2}} \frac{\partial g^{r}h}{\partial y^{j}} \frac{\partial \sigma}{\partial x^{h}} (f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}) +\frac{\partial \sigma}{\partial x^{j}} y^{r}\\ && + (\frac{\partial \sigma}{\partial x^{t}} y^{t}+\frac{\partial \sigma}{\partial u^{\alpha}} v^{\alpha})\,\delta^{r}_{j} -\frac{1}{4} \, g^{rh}\frac{\partial \sigma}{\partial x^{h}}\frac{\partial F^{2}_{1}}{\partial y^{j}} \,,\\ \mathbb{M}^{\mu}_{\alpha}&:=&\frac{1}{2f^{2}_{1}} \frac{\partial f^{2}_{1}}{\partial x^{i}}\, y^{i} \, \delta^{\mu}_{\alpha} -\frac{1}{4f^{2}_{1}} \frac{\partial g^{\mu\lambda}}{\partial v^{\alpha}} \frac{\partial f^{2}_{2}}{\partial u^{\lambda}} \, F^{2}_{1} -\frac{1}{4f^{2}_{1}} \frac{\partial g^{\mu\lambda}}{\partial v^{\alpha}} \frac{\partial \sigma}{\partial u^{\lambda}} (f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}) +\frac{\partial \sigma}{\partial x^{j}} y^{r}\\ && + \left(\frac{\partial \sigma}{\partial x^{t}}\, y^{t}+\frac{\partial \sigma}{\partial u^{\lambda}} \, v^{\lambda}\right)\,\delta^{\mu}_{\alpha} -\frac{1}{4} \, g^{\mu\lambda} \frac{\partial \sigma}{\partial u^{\lambda}}\frac{\partial F^{2}_{2}}{\partial v^{\alpha}} \,, \end{array} $$

$\mathbb {G}^{r}_{\alpha }, \mathbb {G}^{\beta }_{j}$ are given by Proposition 2.

Proposition 3

The coefficients of the conformally doubly warped product Barthel curvature tensor for CDWPF are given by

$$\mathbb{R}^{c}_{ab}(x,u,y,v):=\frac{\delta^{d} \mathbb{G}^{c}_{a}}{\delta^{d} \mathrm{x}^{b}}-\frac{\delta^{d} \mathbb{G}^{c}_{b}}{\delta^{d} \mathrm{x}^{a}} = (\mathbb{R}^{k}_{ij}, \mathbb{R}^{k}_{i\beta}, \mathbb{R}^{k}_{\alpha j}, \mathbb{R}^{k}_{\alpha\beta},\mathbb{R}^{\gamma}_{ij}, \mathbb{R}^{\gamma}_{i\beta}, \mathbb{R}^{\gamma}_{\alpha j}, \mathbb{R}^{\gamma}_{\alpha\beta}), $$

where

$$\begin{array}{@{}rcl@{}} \mathbb{R}^{k}_{ij}&:=& \frac{\delta^{d} \mathbb{G}^{k}_{i}}{\delta^{d} x^{j}}-\frac{\delta^{d} \mathbb{G}^{k}_{j}}{\delta^{d} x^{i}} =R^{k}_{ij}+\mathfrak{U}_{ij}\left\{\frac{\delta \mathbb{M}^{k}_{i}}{\delta x^{j}} - \mathbb{M}^{r}_{j} G^{k}_{ir} -\mathbb{M}^{r}_{j} \frac{\partial \mathbb{M}^{k}_{i}}{\partial y^{r}} -\mathbb{G}^{\mu}_{j} \frac{\partial \mathbb{M}^{k}_{i}}{\partial v^{\mu}} \right\} \\ && \\ \mathbb{R}^{k}_{i\beta}&:=& \frac{\delta^{d} \mathbb{G}^{k}_{i}}{\delta^{d} u^{\beta}}-\frac{\delta^{d} \mathbb{G}^{k}_{\beta}}{\delta^{d} x^{i}} \,, \quad \mathbb{R}^{k}_{\alpha j}:= \frac{\delta^{d} \mathbb{G}^{k}_{\alpha}}{\delta^{d} x^{j}}-\frac{\delta^{d} \mathbb{G}^{k}_{j}}{\delta^{d} u^{\alpha}} \,, \\ && \\ \mathbb{R}^{k}_{\alpha\beta}&:=&\frac{\delta^{d} \mathbb{G}^{k}_{\alpha}}{\delta^{d} u^{\beta}}-\frac{\delta^{d} \mathbb{G}^{k}_{\beta}}{\delta^{d} u^{\alpha}} \,, \quad \mathbb{R}^{\gamma}_{ij}:= \frac{\delta^{d} \mathbb{G}^{\gamma}_{i}}{\delta^{d} x^{j}}-\frac{\delta^{d} \mathbb{G}^{\gamma}_{j}}{\delta^{d} x^{i}} \,, \end{array} $$

$$\begin{array}{@{}rcl@{}} \mathbb{R}^{\gamma}_{i\beta}&:=&\frac{\delta^{d} \mathbb{G}^{\gamma}_{i}}{\delta^{d} u^{\beta}}-\frac{\delta^{d} \mathbb{G}^{\gamma}_{\beta}}{\delta^{d} x^{i}} \,,\quad \mathbb{R}^{\gamma}_{\alpha j}:= \frac{\delta^{d} \mathbb{G}^{\gamma}_{\alpha}}{\delta^{d} x^{j}}-\frac{\delta^{d} \mathbb{G}^{\gamma}_{j}}{\delta^{d} u^{\alpha}} \,, \\ && \\ \mathbb{R}^{\gamma}_{\alpha\beta}&:=& \frac{\delta^{d} \mathbb{G}^{\gamma}_{\alpha}}{\delta^{d} u^{\beta}}-\frac{\delta^{d} \mathbb{G}^{\gamma}_{\beta}}{\delta^{d} u^{\alpha}} =R^{\gamma}_{\alpha\beta}+\mathfrak{U}_{\alpha\beta}\left\{\frac{\delta \mathbb{M}^{\gamma}_{\alpha}}{\delta u^{\beta}} - \mathbb{M}^{\mu}_{\beta} G^{\gamma}_{\alpha\mu} -\mathbb{M}^{\mu}_{\beta} \frac{\partial \mathbb{M}^{\gamma}_{\alpha}}{\partial v^{\mu}} -\mathbb{G}^{r}_{\beta} \frac{\partial \mathbb{M}^{\gamma}_{\alpha}}{\partial y^{r}} \right\}; \end{array} $$

$\mathfrak {U}_{i,j}\{A_{ij}\}:=A_{ij}-A_{ji}$ and $G^{k}_{ij}:=\frac {\partial G^{k}_{i}}{\partial y^{j}}, G^{\gamma }_{\alpha \beta }:=\frac {\partial G^{\gamma }_{\alpha }}{\partial v^{\beta }}, (\mathbb {G}^{i}_{j}, \mathbb {G}^{\alpha }_{j}, \mathbb {G}^{i}_{\beta }, \mathbb {G}^{\alpha }_{\beta })$ are the coefficients of conformally doubly warped product Barthel connection given by Proposition 2.

In view of the above proposition, we have

Corollary 2

If the conformally doubly warped product Finsler manifold $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ is horizontally integrable, then (M₁,F₁) and (M₂,F₂) are horizontally integrable if and only if the following conditions satisfy

$$\begin{array}{@{}rcl@{}} \mathfrak{U}_{ij}\left\{\frac{\delta \mathbb{M}^{k}_{i}}{\delta x^{j}} - \mathbb{M}^{r}_{j} G^{k}_{ir} -\mathbb{M}^{r}_{j} \frac{\partial \mathbb{M}^{k}_{i}}{\partial y^{r}} -\mathbb{G}^{\mu}_{j} \frac{\partial \mathbb{M}^{k}_{i}}{\partial v^{\mu}} \right\}&=&0 \,, \\ \mathfrak{U}_{\alpha\beta}\left\{\frac{\delta \mathbb{M}^{\gamma}_{\alpha}}{\delta u^{\beta}} - \mathbb{M}^{\mu}_{\beta} G^{\gamma}_{\alpha\mu} -\mathbb{M}^{\mu}_{\beta} \frac{\partial \mathbb{M}^{\gamma}_{\alpha}}{\partial v^{\mu}} -\mathbb{G}^{r}_{\beta} \frac{\partial \mathbb{M}^{\gamma}_{\alpha}}{\partial y^{r}} \right\}&=&0. \end{array} $$

Berwald connection for CDWPF

Here, the coefficients of the conformally doubly warped product Berwald connection for CDWPF are studied and investigated.

Proposition 4

The coefficients $\mathbb {G}^{c}_{ab}(x,u,y,v)$ of the conformally doubly warped product Berwald connection for CDWPF are given by

$$\mathbb{G}^{c}_{ab}(\mathrm{x},\mathrm{y}):=\frac{\partial \mathbb{G}^{c}_{a} (\mathrm{x},\mathrm{y})}{\partial \mathrm{y}^{b}} = (\mathbb{G}^{k}_{ij}, \mathbb{G}^{k}_{i\beta}, \mathbb{G}^{k}_{\alpha j}, \mathbb{G}^{k}_{\alpha\beta},\mathbb{G}^{\gamma}_{ij}, \mathbb{G}^{\gamma}_{i\beta}, \mathbb{G}^{\gamma}_{\alpha j}, \mathbb{G}^{\gamma}_{\alpha\beta}), $$

where

$$\begin{array}{@{}rcl@{}} \mathbb{G}^{k}_{ij}&:=& \frac{\partial \mathbb{G}^{k}_{i}}{\partial y^{j}}=G^{k}_{ij}-\frac{1}{4f^{2}_{2}} \frac{\partial^{2} g^{kh}}{\partial y^{j} \partial y^{i}} \frac{\partial f^{2}_{1}}{\partial x^{h}} F^{2}_{2} -\frac{1}{4f^{2}_{2}} \frac{\partial^{2} g^{kh}}{\partial y^{j} \partial y^{i}} \frac{\partial \sigma}{\partial x^{h}} (f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})\\ &&-\frac{1}{4} \frac{\partial g^{kh}}{ \partial y^{i}} \frac{\partial \sigma}{\partial x^{h}} \frac{\partial F^{2}_{1}}{ \partial y^{j}} -\frac{1}{4} \frac{\partial g^{kh}}{ \partial y^{j}} \frac{\partial \sigma}{\partial x^{h}} \frac{\partial F^{2}_{1}}{ \partial y^{i}} + \frac{\partial \sigma}{\partial x^{i}}\, \delta^{k}_{j} + \frac{\partial \sigma}{\partial x^{j}}\, \delta^{k}_{i} -\frac{1}{2}\, g^{kh} \, g_{ij} \frac{\partial \sigma}{\partial x^{h}} \\ && =\mathbb{G}^{k}_{ji} \,, \\ && \\ \mathbb{G}^{k}_{i \beta}&:=& \frac{\partial \mathbb{G}^{k}_{i}}{\partial v^{\beta}}= \frac{1}{2f^{2}_{2}} \frac{\partial f^{2}_{2}}{\partial u^{\beta}}\, \delta^{k}_{i} -\frac{1}{4 f^{2}_{2}} \frac{\partial g^{kh}}{ \partial y^{i}} \frac{\partial f^{2}_{1}}{\partial x^{h}} \frac{\partial F^{2}_{2}}{ \partial v^{\beta}} -\frac{f^{2}_{1}}{4 f^{2}_{2}} \frac{\partial g^{kh}}{ \partial y^{i}} \frac{\partial \sigma}{\partial x^{h}} \frac{\partial F^{2}_{2}}{ \partial v^{\beta}} +\frac{\partial \sigma}{\partial u^{\beta}}\, \delta^{k}_{i} =\mathbb{G}^{k}_{\beta i} \,, \\ && \\ \mathbb{G}^{k}_{\alpha \beta}&:=& \frac{\partial \mathbb{G}^{k}_{\alpha}}{\partial v^{\beta}}= -\frac{1}{2 f^{2}_{2}} g^{kh}\, g_{\alpha\beta}\left\{f^{2}_{1} \frac{\partial \sigma}{\partial x^{h}} +\frac{\partial f^{2}_{1}}{\partial x^{h}} \right\}= \mathbb{G}^{k}_{\beta \alpha} \,, \\ && \\ \mathbb{G}^{\gamma}_{i j}&:=& \frac{\partial \mathbb{G}^{\gamma}_{i}}{\partial y^{j}}= -\frac{1}{2 f^{2}_{1}} g^{\gamma \alpha} \, g_{ij}\left\{f^{2}_{2} \frac{\partial \sigma}{\partial u^{\alpha}} +\frac{\partial f^{2}_{2}}{\partial u^{\alpha}} \right\}=\mathbb{G}^{\gamma}_{j i} \,, \\ && \\ \mathbb{G}^{\gamma}_{i \beta}&:=& \frac{\partial \mathbb{G}^{\gamma}_{i}}{\partial v^{\beta}}= \frac{1}{2f^{2}_{1}} \frac{\partial f^{2}_{1}}{\partial x^{i}}\, \delta^{\gamma}_{\beta} -\frac{1}{4 f^{2}_{1}} \frac{\partial g^{\gamma\alpha}}{ \partial v^{\beta}} \frac{\partial f^{2}_{2}}{\partial u^{\alpha}} \frac{\partial F^{2}_{1}}{ \partial y^{i}} -\frac{f^{2}_{2}}{4 f^{2}_{1}} \frac{\partial g^{\gamma\alpha}}{ \partial v^{\beta}} \frac{\partial \sigma}{\partial u^{\alpha}} \frac{\partial F^{2}_{1}}{ \partial y^{i}} +\frac{\partial \sigma}{\partial x^{i}}\, \delta^{\gamma}_{\beta} =\mathbb{G}^{\gamma}_{\beta i} \,, \\ && \\ \mathbb{G}^{\gamma}_{\alpha\beta}&:=& \frac{\partial \mathbb{G}^{\gamma}_{\alpha}}{\partial v^{\beta}}=G^{\gamma}_{\alpha\beta}-\frac{1}{4f^{2}_{1}} \frac{\partial^{2} g^{\gamma\lambda}}{\partial v^{\beta} \partial v^{\alpha}} \frac{\partial f^{2}_{2}}{\partial u^{\lambda}} F^{2}_{1} -\frac{1}{4f^{2}_{1}} \frac{\partial^{2} g^{\gamma\lambda}}{ \partial v^{\beta} \partial v^{\alpha}} \frac{\partial \sigma}{\partial u^{\lambda}} (f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})\\ && -\frac{1}{4} \frac{\partial g^{\gamma\lambda}}{ \partial v^{\alpha}} \frac{\partial \sigma}{\partial u^{\lambda}} \frac{\partial F^{2}_{2}}{ \partial v^{\beta}}-\frac{1}{4} \frac{\partial g^{\gamma\lambda}}{ \partial v^{\beta}} \frac{\partial \sigma}{\partial u^{\lambda}} \frac{\partial F^{2}_{2}}{ \partial v^{\alpha}} + \frac{\partial \sigma}{\partial u^{\alpha}}\, \delta^{\gamma}_{\beta} + \frac{\partial \sigma}{\partial u^{\beta}}\, \delta^{\gamma}_{\alpha} -\frac{1}{2}\, g^{\gamma\lambda} \, g_{\alpha\beta} \frac{\partial \sigma}{\partial u^{\lambda}} \\ && =\mathbb{G}^{\gamma}_{\beta\alpha} \,, \end{array} $$

and $(\mathbb {G}^{i}_{j}, \mathbb {G}^{\alpha }_{j}, \mathbb {G}^{i}_{\beta }, \mathbb {G}^{\alpha }_{\beta })$ are the coefficients of conformally doubly warped product Barthel connection given by Proposition 2.

According to Propositions 2 and 4, we have

Theorem 1

The conformally doubly warped product Berwald connection for CDWPF is given by

$$B\Gamma^{d} \equiv (\mathbb{G}^{c}_{ab}(x,u,y,v), \mathbb{G}^{a}_{b}(x,u,y,v), 0), $$

where $\mathbb {G}^{a}_{b}$ and $\mathbb {G}^{c}_{ab}$ are respectively given by Propositions 2 and 4.

Cartan connection for CDWPF

As in the preceding section, the coefficients of the conformally doubly warped product Cartan connection for CDWPF are obtained and studied.

Proposition 5

The coefficients $\overline {\Gamma }^{c}_{ab}(x,u,y,v)$ of the conformally doubly warped product Cartan connection for CDWPF are given by

$${} \overline{\Gamma}^{c}_{ab}(x,u,y,v):=\!\frac{1}{2}\, \mathbf{g}^{ce}\left\{\!\frac{\delta^{d} \mathbf{g}_{ea}}{\delta^{d} \mathrm{x}^{b}}+ \frac{\delta^{d} \mathbf{g}_{eb}}{\delta^{d} \mathrm{x}^{a}}-\frac{\delta^{d} \mathbf{g}_{ab}}{\delta^{d} \mathrm{x}^{e}}\!\right\} \,=\,(\overline{\Gamma}^{k}_{ij}, \overline{\Gamma}^{k}_{i\beta}, \overline{\Gamma}^{k}_{\alpha j}, \overline{\Gamma}^{k}_{\alpha\beta},\overline{\Gamma}^{\gamma}_{ij}, \overline{\Gamma}^{\gamma}_{i\beta}, \overline{\Gamma}^{\gamma}_{\alpha j}, \overline{\Gamma}^{\gamma}_{\alpha\beta}\!), $$

where

$$\begin{array}{@{}rcl@{}} \overline{\Gamma}^{k}_{ij}&=& \Gamma^{k}_{ij}+\frac{\partial \sigma}{\partial x^{j}} \, \delta^{k}_{i} +\frac{\partial \sigma}{\partial x^{i}} \, \delta^{k}_{j}-\frac{\partial \sigma}{\partial x^{h}} g^{kh} g_{ij} -\frac{1}{2}g^{kh}\{\mathbb{M}^{r}_{j}\frac{\partial g_{hi}}{\partial y^{r}} +\mathbb{M}^{r}_{i}\frac{\partial g_{hj}}{\partial y^{r}} -\mathbb{M}^{r}_{h}\frac{\partial g_{ij}}{\partial y^{r}}\} \,, \\ && \\ \overline{\Gamma}^{k}_{i \beta}&=& \frac{1}{2 f^{2}_{2}}\,g^{kh}\{\frac{\partial f^{2}_{2}}{\partial u^{\beta}}\, g_{hi}-f^{2}_{2} \mathbb{G}^{r}_{\beta} \frac{\partial g_{hi}}{\partial y^{r}}+2f^{2}_{2}\frac{\partial \sigma}{\partial u^{\beta}}\,g_{hi} \}=\overline{\Gamma}^{k}_{\beta i} \,, \\ && \\ \overline{\Gamma}^{k}_{\alpha\beta}&=& -\frac{1}{2 f^{2}_{2}}\,g^{kh}\{\frac{\partial f^{2}_{1}}{\partial x^{h}}\, g_{\alpha\beta}- f^{2}_{1} \mathbb{G}^{\lambda}_{h} \frac{\partial g_{\alpha\beta}}{\partial v^{\lambda}}+2f^{2}_{1}\frac{\partial \sigma}{\partial x^{h}}\,g_{\alpha\beta}\} \,, \\ && \\ \overline{\Gamma}^{\gamma}_{ij}&=& -\frac{1}{2 f^{2}_{1}}\,g^{\gamma\lambda}\{\frac{\partial f^{2}_{2}}{\partial u^{\lambda}}\, g_{ij}- f^{2}_{2} \mathbb{G}^{r}_{\lambda} \frac{\partial g_{ij}}{\partial y^{r}}+2f^{2}_{2}\frac{\partial \sigma}{\partial u^{\lambda}}\,g_{ij}\} \,, \\ && \\ \overline{\Gamma}^{\gamma}_{i \beta}&=& \frac{1}{2 f^{2}_{1}}\,g^{\gamma\lambda}\{\frac{\partial f^{2}_{1}}{\partial x^{i}}\, g_{\lambda\beta} -f^{2}_{1} \mathbb{G}^{\alpha}_{i} \frac{\partial g_{\lambda\beta}}{\partial v^{\alpha}}+2f^{2}_{1}\frac{\partial \sigma}{\partial x^{i}}\,g_{\lambda\beta} \}=\overline{\Gamma}^{\gamma}_{\beta i} \,, \\ && \\ \overline{\Gamma}^{\gamma}_{\alpha\beta}&=& \Gamma^{\gamma}_{\alpha\beta}+\!\frac{\partial \sigma}{\partial u^{\alpha}} \, \delta^{\gamma}_{\beta} +\!\frac{\partial \sigma}{\partial u^{\beta}} \, \delta^{\gamma}_{\alpha}-\frac{\partial \sigma}{\partial u^{\lambda}} g^{\gamma\lambda} g_{\alpha\beta} -\frac{1}{2}g^{\gamma\lambda}\left\{\!\mathbb{M}^{\mu}_{\alpha}\frac{\partial g_{\lambda\beta}}{\partial v^{\mu}} +\mathbb{M}^{\mu}_{\beta}\frac{\partial g_{\lambda\alpha}}{\partial v^{\mu}} -\mathbb{M}^{\mu}_{\lambda}\frac{\partial g_{\alpha\beta}}{\partial v^{\mu}}\!\right\}\!, \end{array} $$

and $\Gamma ^{k}_{ij}:=\frac {1}{2}\,g^{kh}\left \{\frac {\delta g_{hj}}{\delta x^{i}}+\frac {\delta g_{hi}}{\delta x^{i}}-\frac {\delta g_{ij}}{\delta x^{h}}\right \}, \,\, \Gamma ^{\gamma }_{\alpha \beta }:=\frac {1}{2}\,g^{\gamma \lambda }\left \{\frac {\delta g_{\lambda \beta }}{\delta u^{\alpha }}+ \frac {\delta g_{\lambda \alpha }}{\delta u^{\beta }}-\frac {\delta g_{\alpha \beta }}{\delta u^{\lambda }}\right \}, \, \mathbb {M}^{r}_{j}, \mathbb {M}^{\mu }_{\alpha }$ are defined by Corollary 1.

Proof

The proof follows from the definition of $\overline {\Gamma }^{c}_{ab}(x,u,y,v)$ taking into account Relations (3) and (4) and Corollary 1. □

Proposition 6

The coefficients $\overline {C}^{c}_{ab}(x,u,y,v)$ of the conformally doubly warped product Cartan tensor field for CDWPF are given by

$$\begin{array}{@{}rcl@{}} \overline{C}^{c}_{ab}(x,u,y,v)&:=&\frac{1}{2}\, \mathbf{g}^{ce} \frac{\partial \mathbf{g}_{ab}}{\partial \mathrm{y}^{e}} \\ &=&\left(\overline{C}^{k}_{ij}, \overline{C}^{k}_{i\beta}, \overline{C}^{k}_{\alpha j}, \overline{C}^{k}_{\alpha\beta},\overline{C}^{\gamma}_{ij}, \overline{C}^{\gamma}_{i\beta}, \overline{C}^{\gamma}_{\alpha j}, \overline{C}^{\gamma}_{\alpha\beta}\right)\\ &=&\left({C}^{k}_{ij}, 0,0,0,0,0,0,{C}^{\gamma}_{\alpha\beta}\right), \end{array} $$

where ${C}^{k}_{ij}(x,y):=\frac {1}{2}\, {g}^{kh} \frac {\partial {g}_{ij}}{\partial {y}^{h}}$ and ${C}^{\gamma }_{\alpha \beta }(u,v):=\frac {1}{2}\, {g}^{\gamma \lambda } \frac {\partial {g}_{\alpha \beta }}{\partial {v}^{\lambda }}$ are the coefficients of Cartan tensor fields of (M₁,F₁) and (M₂,F₂), respectively.

Summing up, we have

Theorem 2

The conformally doubly warped product Cartan connection for CDWPF is given by

$${C\Gamma}^{d} \equiv (\overline{\Gamma}^{c}_{ab}(x,u,y,v), \mathbb{G}^{a}_{b}(x,u,y,v), \overline{C}^{c}_{ab}(x,u,y,v)),$$

where $\overline {\Gamma }^{c}_{ab}(x,u,y,v)$ and $\overline {C}^{c}_{ab}(x,u,y,v)$ are respectively given by Proposition 5 and 6, also $\mathbb {G}^{a}_{b}$ is given by Proposition 2.

Corollary 3

For the conformally doubly warped product Finsler manifold CDWPF, we have (a) the conformally doubly warped product Rund connection RΓ^d is given by

$${R\Gamma}^{d} \equiv (\overline{\Gamma}^{c}_{ab}(x,u,y,v), \mathbb{G}^{a}_{b}(x,u,y,v), 0).$$

(b) the conformally doubly warped product Hasiguchi connection HΓ^d is given by

$${H\Gamma}^{d} \equiv (\mathbb{G}^{c}_{ab}(x,u,y,v), \mathbb{G}^{a}_{b}(x,u,y,v), \overline{C}^{c}_{ab}(x,u,y,v)),$$

where $\mathbb {G}^{a}_{b}(x,u,y,v), \mathbb {G}^{c}_{ab}(x,u,y,v), \overline {\Gamma }^{c}_{ab}(x,u,y,v)$, and $\overline {C}^{c}_{ab}(x,u,y,v)$ are respectively given by Propositions 2, 4, 5 and 6.

Some special Finsler spaces

In this section, some special Finsler spaces such as Riemannian, C-reducible, and Berwaldian spaces are studied for CDWPF.

First, we begin with the following two lemmas which are useful for this section.

Lemma 1

The coefficients $\overline {I}_{a}(x,u,y,v)$ of the conformally doubly warped product contraction Cartan torsion tensor for CDWPF are given by

$$\overline{I}_{a}(x,u,y,v):= \mathbf{g}^{bc} \overline{C}_{abc}(x,u,y,v)=(I_{i}(x,y), 0,0,0,0,0,0,I_{\alpha}(u,v)),$$

where $\overline {C}_{abc}(x,u,y,v):=\mathbf {g}_{dc}\overline {C}^{d}_{ab}, I_{i}(x,y):=g^{jk} {C}_{ijk}(x,y)$ and I_α(u,v):=g^βγC_αβγ(u,v).

Proof

The proof follows from the definition of $\overline {C}_{abc}(x,u,y,v)$ together with (3), (4) and the fact that $\overline {C}_{abc}:=\mathbf {g}_{dc}\overline {C}^{d}_{ab} =(e^{2\sigma } f^{2}_{2} C_{ijk}(x,y), 0,0,0,0,0,0,e^{2\sigma } f^{2}_{1}C_{\alpha \beta \gamma }(u,v)).$ □

Lemma 2

The coefficients $\mathbf {\hbar }_{ab}(x,u,y,v)$ of the conformally doubly warped product angular metric tensor for CDWPF are given by

$$\mathbf{\hbar}_{ab}(x,u,y,v):= \mathbf{g}_{ab}-\frac{1}{F^{2}} \mathrm{y}_{a} \mathrm{y}_{b} =(\mathbf{\hbar}_{ij}, \mathbf{\hbar}_{i\beta}, \mathbf{\hbar}_{\alpha j}, \mathbf{\hbar}_{\alpha\beta}), $$

where

$$\begin{array}{@{}rcl@{}} \mathbf{\hbar}_{ij}(x,u,y,v)&:=& \mathbf{g}_{ij}-\frac{1}{F^{2}} \mathrm{y}_{i} \mathrm{y}_{j} =e^{2\sigma} f^{2}_{2} (g_{ij}-\frac{f^{2}_{2}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}}\,y_{i}y_{j}). \\ && \\ \mathbf{\hbar}_{i\beta}(x,u,y,v)&:=& \mathbf{g}_{i\beta}-\frac{1}{F^{2}} \mathrm{y}_{i} \mathrm{y}_{\beta} =-\frac{e^{2\sigma} f^{2}_{1}f^{2}_{2}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}}\,y_{i}v_{\beta}.\\ && \\ \mathbf{\hbar}_{\alpha j}(x,u,y,v)&:=& \mathbf{g}_{\alpha j}-\frac{1}{F^{2}} \mathrm{y}_{j} \mathrm{y}_{\alpha} =-\frac{e^{2\sigma} f^{2}_{1}f^{2}_{2}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}}\,y_{j} v_{\alpha}.\\ && \\ \mathbf{\hbar}_{\alpha\beta}(x,u,y,v)&:=& \mathbf{g}_{\alpha\beta}-\frac{1}{F^{2}} \mathrm{y}_{\alpha} \mathrm{y}_{\beta} =e^{2\sigma} f^{2}_{1} (g_{\alpha\beta}-\frac{f^{2}_{1}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}}\,v_{\alpha} v_{\beta}). \end{array} $$

Proof

The proof follows from the definition of $\mathbf {\hbar }_{ab}(x,u,y,v)$ together with (3) and the fact that $\mathrm {y}_{a}:=\mathbf {g}_{ab} \mathrm {y}^{b}=(e^{2\sigma } f^{2}_{2}\,y_{i}, e^{2\sigma } f^{2}_{2}\,v_{\alpha })$. □

In view of Lemma 1, we have

Theorem 3

The conformally doubly warped product Finsler manifold $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ is Riemannian if and only if (M₁,F₁) and (M₂,F₂) are Riemannian manifolds.

A doubly warped product Finsler manifold CDWPF is C-reducible if the associated Matsumoto conformally doubly warped product tensor field $\mathbb {M}_{abc}(x,u,y,v)$ vanishes identically.

Theorem 4

Every C-reducible conformally doubly warped product Finsler manifold $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ is Riemannian.

Proof

The Matsumoto conformally doubly warped product tensor field $\mathbb {M}_{abc}(x,u,y,v)$ is defined by

$$ \mathbb{M}_{abc}(x,u,y,v):=\overline{C}_{abc}-\frac{1}{n+1}\{ \overline{I}_{a} \mathbf{\hbar}_{bc}+ \overline{I}_{b} \mathbf{\hbar}_{ca}+\overline{I}_{c} \mathbf{\hbar}_{ab}\}. $$

(11)

Hence, using Lemmas 1 and 2, the component $\mathbb {M}_{\alpha jk}(x,u,y,v)$ has the form

$$\begin{array}{@{}rcl@{}} \mathbb{M}_{\alpha jk}(x,u,y,v) &=& \frac{1}{n+1} \frac{e^{2\sigma} f^{2}_{1}f^{2}_{2}v_{\alpha}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}}(I_{j}\, y_{k}+ I_{k}\, y_{j}) - \frac{e^{2\sigma} f^{2}_{2} I_{\alpha}} {(n+1)} (g_{jk}-\frac{f^{2}_{2}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}} \,y_{j} \,y_{k}). \end{array} $$

Consequently, one can show that

$$\begin{array}{@{}rcl@{}} \mathbb{M}_{\alpha jk}(x,u,y,v) \,y_{j} \,y_{k}&=& - \frac{e^{2\sigma} f^{2}_{2} F^{2}_{1} I_{\alpha}} {(n+1)} (1-\frac{f^{2}_{2} F^{2}_{1}}{f^{2}_{2}F^{2}_{1}+f^{2}_{1} F^{2}_{2}}). \end{array} $$

Now, if $\mathbb {M}_{\alpha jk}(x,u,y,v)$ vanishes, then I_α vanishes. This means that (M₂,F₂) is Riemannian.

Similarly, if $\mathbb {M}_{i \alpha \beta }(x,u,y,v)=0$, then I_i=0. Hence, (M₁,F₁) is also Riemannian. Therefore, using Theorem 3, the result follows. □

In view of Proposition 4, we have

Proposition 7

The coefficients $\mathbb {B}^{d}_{abc}(x,u,y,v)$ of the conformally doubly warped product Berwald curvature tensor for CDWPF are given by

$$\mathbb{B}^{d}_{abc}(x,u,y,v):= \frac{\partial \mathbb{G}^{d}_{ab}}{\partial \mathrm{y}^{c}} =(\mathbb{B}^{k}_{ijl}, \mathbb{B}^{k}_{i\beta l},\mathbb{B}^{k}_{\alpha\beta l},\mathbb{B}^{k}_{\alpha\beta\lambda}, \mathbb{B}^{\gamma}_{ijl}, \mathbb{B}^{\gamma}_{i\beta l},\mathbb{B}^{\gamma}_{\alpha\beta l},\mathbb{B}^{\gamma}_{\alpha\beta\lambda}), $$

where

$$\begin{array}{@{}rcl@{}} \mathbb{B}^{k}_{ijl}&:=& \frac{\partial \mathbb{G}^{k}_{ij}}{\partial y^{l}}=B^{k}_{ijl} -\frac{1}{4f^{2}_{2}} \frac{\partial^{3} g^{kh}}{\partial y^{l} \partial y^{j} \partial y^{i}} \frac{\partial f^{2}_{1}}{\partial x^{h}} F^{2}_{2} -\frac{1}{4f^{2}_{2}} \frac{\partial^{3} g^{kh}}{\partial y^{l} \partial y^{j} \partial y^{i}} \frac{\partial \sigma}{\partial x^{h}} (f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})\\ && -\frac{1}{4} \frac{\partial^{2} g^{kh}}{\partial y^{j} \partial y^{i}} \frac{\partial F^{2}_{1}}{\partial y^{l}} \frac{\partial \sigma}{\partial x^{h}} -\frac{1}{4} \frac{\partial^{2} g^{kh}}{ \partial y^{l} \partial y^{i}} \frac{\partial \sigma}{\partial x^{h}} \frac{\partial F^{2}_{1}}{ \partial y^{j}} -\frac{1}{2} \frac{\partial g^{kh}}{ \partial y^{i}} \frac{\partial \sigma}{\partial x^{h}} \, g_{jl} -\frac{1}{4} \frac{\partial^{2} g^{kh}}{\partial y^{l} \partial y^{j}} \frac{\partial \sigma}{\partial x^{h}} \frac{\partial F^{2}_{1}}{ \partial y^{i}}\\ &&-\frac{1}{2} \frac{\partial g^{kh}}{ \partial y^{j}} \frac{\partial \sigma}{\partial x^{h}} \, g_{il} -\frac{1}{2}\, \frac{\partial g^{kh}}{\partial y^{l}} \frac{\partial \sigma}{\partial x^{h}} \, g_{ij} - g^{kh} \frac{\partial \sigma}{\partial x^{h}}\, C_{ijl} \,, \\ && \\ \mathbb{B}^{k}_{i \beta l}&:=& \frac{\partial \mathbb{G}^{k}_{i\beta}}{\partial y^{l}}= -\frac{1}{4 f^{2}_{2}} \frac{\partial^{2} g^{kh}}{ \partial y^{l} \partial y^{i}} \frac{\partial F^{2}_{2}}{ \partial v^{\beta}} \{ \frac{\partial f^{2}_{1}}{\partial x^{h}} +f^{2}_{1} \frac{\partial \sigma}{\partial x^{h}}\} \,, \\ && \\ \mathbb{B}^{k}_{\alpha \beta l}&:=& \frac{\partial \mathbb{G}^{k}_{\alpha \beta}}{\partial y^{l}}= -\frac{1}{2 f^{2}_{2}} \frac{\partial g^{kh}}{\partial y^{l}} \, g_{\alpha\beta} \{ \frac{\partial f^{2}_{1}}{\partial x^{h}}+f^{2}_{1} \frac{\partial \sigma}{\partial x^{h}}\} \,, \end{array} $$

$$\begin{array}{@{}rcl@{}} \mathbb{B}^{k}_{\alpha \beta \lambda}&:=& \frac{\partial \mathbb{G}^{k}_{\alpha \beta}}{\partial v^{\lambda}}= -\frac{1}{f^{2}_{2}} \, g^{kh}C_{\alpha\beta\lambda} \{\frac{\partial f^{2}_{1}}{\partial x^{h}} +f^{2}_{1} \frac{\partial \sigma}{\partial x^{h}}\} \,, \\ && \\ \mathbb{B}^{\gamma}_{i j l}&:=& \frac{\partial \mathbb{G}^{\gamma}_{i j}}{\partial y^{l}}= -\frac{1}{f^{2}_{1}}\, g^{\gamma \mu} C_{ijl}\{\frac{\partial f^{2}_{2}}{\partial u^{\mu}}+f^{2}_{2} \frac{\partial \sigma}{\partial u^{\mu}} \} \,, \\ && \\ \mathbb{B}^{\gamma}_{i \beta l}&:=& \frac{\partial \mathbb{G}^{\gamma}_{i \beta}}{\partial y^{l}}= -\frac{1}{2 f^{2}_{1}} \frac{\partial g^{\gamma\mu}}{ \partial v^{\beta}} \, g_{il} \{ \frac{\partial f^{2}_{2}}{\partial u^{\mu}} +f^{2}_{2} \frac{\partial \sigma}{\partial u^{\mu}}\} \,, \\ && \\ \mathbb{B}^{\gamma}_{\alpha\beta l}&:=& \frac{\partial \mathbb{G}^{\gamma}_{\alpha\beta}}{\partial y^{l}}= -\frac{1}{4f^{2}_{1}} \frac{\partial^{2} g^{\gamma\mu}}{\partial v^{\beta} \partial v^{\alpha}} \frac{\partial F^{2}_{1}}{\partial y^{l}}\{ \frac{\partial f^{2}_{2}}{\partial u^{\mu}} + f^{2}_{2} \frac{\partial \sigma}{\partial u^{\mu}}\} \,, \\ && \\ \mathbb{B}^{\gamma}_{\alpha\beta\lambda}&:=& \frac{\partial \mathbb{G}^{\gamma}_{\alpha\beta}}{\partial v^{\lambda}} =B^{\gamma}_{\alpha\beta\lambda}-\frac{1}{4f^{2}_{1}} \frac{\partial^{3} g^{\gamma\mu}}{\partial v^{\lambda} \partial v^{\beta} \partial v^{\alpha}} \frac{\partial f^{2}_{2}}{\partial u^{\mu}} F^{2}_{1} -\frac{1}{4f^{2}_{1}} \frac{\partial^{3} g^{\gamma\mu}}{\partial v^{\lambda} \partial v^{\beta} \partial v^{\alpha}} \frac{\partial \sigma}{\partial u^{\mu}} (f^{2}_{2} F^{2}_{1}+f^{2}_{1} F^{2}_{2})\\ && -\frac{1}{4} \frac{\partial^{2} g^{\gamma\mu}}{\partial v^{\beta} \partial v^{\alpha}} \frac{\partial F^{2}_{2}}{\partial v^{\lambda}} \frac{\partial \sigma}{\partial u^{\mu}} -\frac{1}{4} \frac{\partial^{2} g^{\gamma\mu}}{\partial v^{\lambda} \partial v^{\alpha}} \frac{\partial \sigma}{\partial u^{\mu}} \frac{\partial F^{2}_{2}}{ \partial v^{\beta}} -\frac{1}{2} \frac{\partial g^{\gamma\mu}}{ \partial v^{\alpha}} \frac{\partial \sigma}{\partial u^{\mu}} \, g_{\beta\lambda} - g^{\gamma\mu} \frac{\partial \sigma}{\partial u^{\mu}} \, C_{\alpha\beta\lambda} \\ &&-\frac{1}{2} \frac{\partial g^{\gamma\mu}}{ \partial v^{\beta}} \frac{\partial \sigma}{\partial u^{\mu}} \, g_{\alpha\lambda} -\frac{1}{2}\, \frac{\partial g^{\gamma\mu}}{\partial v^{\lambda}} \frac{\partial \sigma}{\partial u^{\mu}} \,g_{\alpha\beta} -\frac{1}{4} \frac{\partial^{2} g^{\gamma\mu}}{\partial v^{\lambda} \partial v^{\beta}} \frac{\partial \sigma}{\partial u^{\mu}} \frac{\partial F^{2}_{2}}{ \partial v^{\alpha}} \,, \end{array} $$

and $\mathbb {G}^{k}_{ij}, \mathbb {G}^{k}_{i\beta }, \mathbb {G}^{k}_{\alpha \beta },\mathbb {G}^{\gamma }_{ij}, \mathbb {G}^{\gamma }_{i\beta }, \mathbb {G}^{\gamma }_{\alpha \beta }$ are the coefficients of conformally doubly warped product Berwald connection given by Proposition 4.

Definition 3

A conformally doubly warped product Finsler manifold $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ satisfying the following conditions: $\frac {\partial f^{2}_{1}}{\partial x^{h}} +f^{2}_{1} \frac {\partial \sigma }{\partial x^{h}} \neq 0$ and $\frac {\partial f^{2}_{2}}{\partial u^{\mu }} +f^{2}_{2} \frac {\partial \sigma }{\partial u^{\mu }} \neq 0$ is called a conditionally conformally doubly warped product Finsler manifold.

Theorem 5

Every conditionally conformally doubly warped product Finsler manifold $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ with vanishing Berwald curvature is Riemannian.

Definition 4

A conformally doubly warped product Finsler manifold $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ with constant conformal factor σ is called a homothety doubly warped product Finsler manifold.

A Finsler manifold is called Berwald if its hv-Berwald curvature tensor vanishes identically.

Theorem 6

Let $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ be a homothety doubly warped product Finsler manifold and f₁ is constant on M₁(f₂ is constant on M₂). Then, $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ is Berwaldian if and only if (M₁,F₁) is Riemannian, (M₂,F₂) is Berwaldian and $\frac {\partial g^{\alpha \gamma }}{ \partial v^{\lambda }} \frac {\partial f^{2}_{2}}{\partial u^{\alpha }}=0 ((M_{2}, F_{2})$ is Riemannian, (M₁,F₁) is Berwaldian and $\frac {\partial g^{ij}}{ \partial y^{l}} \frac {\partial f^{2}_{1}}{\partial u^{i}}=0)$

Proof

Suppose that $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ is a homothety doubly warped product Finsler manifold and f₁ is constant on M₁. Then, from Proposition 7, one can show that

$$\begin{array}{@{}rcl@{}} \mathbb{B}^{\gamma}_{i j l}&=& -\frac{1}{f^{2}_{1}}\, g^{\gamma \mu} C_{ijl}\frac{\partial f^{2}_{2}}{\partial u^{\mu}} \,, \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} \mathbb{B}^{\gamma}_{i \beta l}&=& -\frac{1}{2 f^{2}_{1}} \frac{\partial g^{\gamma\mu}}{ \partial v^{\beta}} \, g_{il} \frac{\partial f^{2}_{2}}{\partial u^{\mu}} \,, \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} \mathbb{B}^{\gamma}_{\alpha\beta\lambda}&=&B^{\gamma}_{\alpha\beta\lambda}-\frac{1}{4f^{2}_{1}} \frac{\partial^{3} g^{\gamma\mu}}{\partial v^{\lambda} \partial v^{\beta} \partial v^{\alpha}} \frac{\partial f^{2}_{2}}{\partial u^{\mu}} F^{2}_{1}. \end{array} $$

(14)

Now, if $(~_{f_{2}}M_{1} \times ~_{f_{1}} M_{2},F)$ is Berwaldian, then using (12), we conclude that C_ijk vanishes, and hence (M₁,F₁) is Riemanian. On the other hand, from (13), we get $\frac {\partial g^{\gamma \mu }}{ \partial v^{\beta }} \frac {\partial f^{2}_{2}}{\partial u^{\mu }}=0$. Consequently, $\frac {\partial ^{3} g^{\gamma \mu }}{\partial v^{\lambda } \partial v^{\beta } \partial v^{\alpha }} \frac {\partial f^{2}_{2}}{\partial u^{\mu }}=0$. From which together with (14), we get $B^{\gamma }_{\alpha \beta \lambda }=0$. This means that (M₂,F₂) is Berwaldian. The converse is proved by the same manner. This completes the proof. □

Concluding remarks

• In this paper, we obtained some results concerning the conformally doubly warped product Finsler manifold CDWPF; namely, we got formulas for the following:

  • Canonical spray, Barthel connection and its curvature tensor (Propositions 1, 2, and 3) are calculated.

  • Berwald and Cartan connections (Theorems 1 and 2) are computed.

  • Some special Finsler spaces such as Riemannian, C-reducible and Berwald spaces (Theorems 3, 4 and 6) are studied.

• The above results can be obtained for the conformally warped product Finsler manifold CWPF by setting f₁=1.

• The same results can be achieved for the doubly warped product Finsler manifold DWPF by setting σ=0 which was investigated by Peyghan and Tayebi [9].

• The mentioned results above can be obtained for the warped product Finsler manifold WPF by setting σ=0,f₁=1, and f₂=1.