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∈TxM, where x=(x,u),y=(y,v),M:=M1×M2 and TxM=TxM1⊕TuM2.
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 giβ=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), gαj=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 VTMo and HTMo; 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
$$\begin{array}{@{}rcl@{}} \mathfrak{U}_{ij}\left\{\frac{\delta \mathbb{M}^{k}_{i}}{\delta x^{j}} - \mathbb{M}^{r}_{j} G^{k}_{ir} -\mathbb{M}^{r}_{j} \frac{\partial \mathbb{M}^{k}_{i}}{\partial y^{r}} -\mathbb{G}^{\mu}_{j} \frac{\partial \mathbb{M}^{k}_{i}}{\partial v^{\mu}} \right\}&=&0 \,, \\ \mathfrak{U}_{\alpha\beta}\left\{\frac{\delta \mathbb{M}^{\gamma}_{\alpha}}{\delta u^{\beta}} - \mathbb{M}^{\mu}_{\beta} G^{\gamma}_{\alpha\mu} -\mathbb{M}^{\mu}_{\beta} \frac{\partial \mathbb{M}^{\gamma}_{\alpha}}{\partial v^{\mu}} -\mathbb{G}^{r}_{\beta} \frac{\partial \mathbb{M}^{\gamma}_{\alpha}}{\partial y^{r}} \right\}&=&0. \end{array} $$