Historical remarks

The section brings a survey of Finsler geometry and product manifolds; a Finsler metric on a manifold is a collection of Minkowski norms \( F_{x} \) in tangent space at x such that \( F_{x} \) varies smoothly in x.

In 1854, Reimann introduced the geometry based on the element of arc length

$$\begin{aligned} ds=F(x^{1},x^{2},\ldots ,x^{n},{\mathrm{{d}}}x^{1},{\mathrm{{d}}}x^{2},\ldots ,{\mathrm{{d}}}x^{n}). \end{aligned}$$

where F is positively homogeneous of degree 1 in \( dx^{i} \) [2, 3]. For more than half a century, there had been no progress until P.Finsler introduced the Riemannian–Finsler geometry (Finsler geometry for short) in his thesis in 1918. A Finsler metric \(F=F(x,y) \) is defined on tanget bundle TM, such that gives a for \( x\in M \) by Minkowski norm [4,5,6,7],

$$\begin{aligned} F\mid _{T_{x}M}=F_{x}: T_{x}M\mapsto [0,\infty ) . \end{aligned}$$

Chern and Shen are defined product metric on Finsler spaces [8] as:

(1) For Finsler manifolds \( (M_{1},F_{1}) \) and \((M_{2},F_{2}), \) \( x=(x_{1},x_{2}) \in M=M_{1}\times M_{2}\) and \(y=(y_{1},y_{2})\in T_{(x_{1},x_{2})}(M_{1}\times M_{2})\), let \( F: M=M_{1}\times M_{2} \mapsto [0,\infty ) \) are defined by

$$\begin{aligned} F(x,y):=\left\{ \begin{array}{ccc} F_{1}(x_{1},y_{1}) &{} \quad {\mathrm{{if}}} \quad &{} y=y_{1}\oplus 0\\ F_{2}(x_{2},y_{2}) & \quad {\mathrm{{if}}} \quad &{} y=0\oplus y_{2} \end{array}\right. \end{aligned}$$
(1.1)

Then F is Finsler metric on \( M_{1}\times M_{2} \), where \(T_{x}M=T_{x_{1}}M_{1}\oplus T_{x_{2}}M_{2} \).

(2) For Riemannian manifolds \( (M_{1},g_{1}) , (M_{2},g_{2}) \) let \( f:[0, \infty )\times [0,\infty )\mapsto [0,\infty ) \) be a \( c^{\infty } \) function satisfying:

\(f(\lambda s,\lambda t)=\lambda f(s,t), \forall \lambda > 0, \) and \( f(s,t)>0, \forall (s,t)\ne (0,0). \)

Then for all \( (x_{1}, x_{2}) \in M_{1} \times M_{2} \) and \( (y_{1}, y_{2}) \in TM_{1} \times TM_{2} ,\) F defined function by

$$\begin{aligned} F(x,y):=\sqrt{f\left( \left[ g_{1}(x_{1},y_{1})\right] ^{2},\left[ g_{2}(x_{2},y_{2})\right] ^{2} \right) } \end{aligned}$$

is a Finsler metric \( M=M_{1}\times M_{2}\).

(3) Kozma et al. have defined a Twisted Products Finsler Manifolds [9] as:

Let \( (M_{1},F_{1}) ,(M_{2},F_{2}) \) be two Finsler manifolds and \( M=M_{1}\times M_{2} \), then for all \( (x_{1}, x_{2}) \in M_{1} \times M_{2} \) and \( (y_{1},y_{2})\in T_{(x_{1},x_{2})}(M_{1}\times M_{2})-\{(0,0)\}\equiv (T_{x_{1}}M_{1}-\{0\})\times (T_{x_{2}}M_{2}-\{0\}) \) and \( c^{\infty } \) function \( f : M_{1} \times M_{2} \mapsto R^{+} \), the Twisted Products metric is defined by

$$\begin{aligned} F(y_{1},y_{2}):=\sqrt{F^{2}_{1}(x_{1},y_{1})+f^{2}(x_{1},x_{2})F^{2}_{2}(x_{2},y_{2})}. \end{aligned}$$

\((y_{1}, y_{2}) \in TM_{1}-\left\{ 0\right\} \)

In this paper, we are going to generalize a new product Finsler metric F on \( M=M_{1}\times M_{2}\), then we call (MF) as canonical product Finsler manifolds.

Introduction and preliminaries

We recall some definitions and fundamental results in Finsler geometry.

Definition 2.1

Let V be a n-dimensional real vector space. A Minkowski norm on V is a functional F on V, which is smooth on \(V-\lbrace 0\rbrace \) and satisfies the following conditions:

  1. (1)

    \(F(u)\ge 0, \ \forall \ u\in V;\)

  2. (2)

    \(F(\lambda u)=\lambda F(u), \ \forall \ \lambda >0, \ \forall u\in V\);

  3. (3)

    for any basis \(e_{1}, \ldots , e_{n}\) of V, write \(F(y)= F(y^{1}, \ldots , y^{n})\), \(y=y^{j}e_{j}\). Then the Hessian matrix

    $$\begin{aligned} (g_{ij}):=\left( \left[ \dfrac{1}{2}F^{2}\right] _{y^{i}y^{j}}\right) \end{aligned}$$

    is positive definite at any point of \(V-\lbrace 0\rbrace \).

The pair (VF) is called Minkowski space.

Definition 2.2

Let M be a (connected) smooth manifold. A Finsler metric on M is a function \(F: TM\rightarrow [0, +\,\infty )\) such that

  1. (1)

    F is \(C^{\infty }\) on the slit tangent bundle \(TM-\lbrace 0\rbrace \);

  2. (2)

    The restriction of F to any \(T_{p}M\), \(p\in M\) is a Minkowski norm.

The pair (MF) is called Finsler manifold or Finsler space.

Let (MF) be a Finsler space and \((x^{1}, \ldots , x^{n})\) be a local coordinate system on an open subset U of M. Then \(\left\{ \dfrac{\partial }{\partial x^{1}}, \ldots , \dfrac{\partial }{\partial x^{n}} \right\} \) form a basis for the tangent space at any point in U.

Theorem 2.3

(Euler’s) Suppose a real-valued H on \(R^{n}\) is differentiable away from the origin of \(R^{n}\). Then the following two statements are equivalent [5]

  1. (I)

    H is positively homogeneous of degree r. That is:

    $$\begin{aligned} H(\lambda y)=\lambda ^{r} H(y);\quad \forall \ \lambda ,\quad \lambda >0 \end{aligned}$$
  2. (II)

    The radial directional derivative of H is r times H. Namely,

    $$\begin{aligned} y^{i}H_{y^{i}}(y)=rH(y). \end{aligned}$$

Corollary 2.4

Let F be positively homogeneous of degree 1 on \(R^{n}\). By using Euler’s theorem, we can show that:

  1. (a)

    \(y^{i}F_{y^{i}}=F.\)

  2. (b)

    \(y^{i}F_{y^{i}y^{j}}=0.\)

  3. (c)

    \(y^{k}F_{y^{i}y^{j}y^{k}}=-F_{y^{i}y^{j}.}\)

  4. (d)

    \(y^{l}F_{y^{i}y^{j}y^{k}y^{l}}=-2F_{y^{i}y^{j}y^{k}}.\)

Corollary 2.5

Let (MF) be a Finsler manifold . Then \(F^{2}\) defined by:

\(F_{p}^{2}(y)=(F_{p}(y))^{2}\); for all \( p\in M \) and \( y\in T_{p}M\); is a positively homogeneous function of degree 2 on \( T_{p}M \).

Corollary 2.6

Let (MF) be a Finsler manifolds. Then

$$\begin{aligned} y^{i}\frac{\partial F^{2}(y)}{\partial y^{i}}=2(F(y))^{2} . \end{aligned}$$

For simplicity, we put \( F_{1}(y_{1}) ,F_{2}(y_{2})\);

as \( F_{p_{1}}(y_{1}) , F_{p_{2}}(y_{2})\), respectively. \(\left( i.e : F_{1}=F_{p_{1}} , F_{2}=F_{p_{2}} \right). \)

Proposition 2.7

Let (MF) be a Finsler manifold of dimension n. Then

$$\begin{aligned} y^{i}\dfrac{\partial ^{2}(F(y))^{2}}{\partial y^{i}\partial y^{j}}=\dfrac{\partial (F(y))^{2}}{\partial y^{j}}. \end{aligned}$$

Proof

By definition and Corollary 2.6, we have:

$$\begin{aligned}&y^{i}\dfrac{\partial ^{2}(F(y))^{2}}{\partial y^{i}\partial y^{j}}=\mathop {\sum }\limits _{i=1}^{n} y^{i}\dfrac{\partial ^{2}(F(y))^{2}}{\partial y^{i} \partial y^{j}}= \mathop {\sum }\limits _{i=1}^{n}y^{i}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{i}}\right) \\&\quad =\mathop {\sum }\limits _{{i=1}_{i\ne j}}^{n} y^{i}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{i}}\right) +y^{j}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) \\&\quad =\mathop {\sum }\limits _{{i=1}_{i\ne j}}^{n}\dfrac{\partial }{\partial y^{j}}\left( y^{i}\dfrac{\partial (F(y))^{2}}{\partial y^{i}} \right) +y^{j}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) \\&\quad =\dfrac{\partial }{\partial y^{j}}\left( \mathop {\sum }\limits _{{i=1}_{i\ne j}}^{n} y^{i}\dfrac{\partial (F(y))^{2}}{\partial y^{i}} \right) +y^{j}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) \\&\quad =\dfrac{\partial }{\partial y^{j}}\left( \mathop {\sum }\limits _{i=1}^{n} y^{i}\dfrac{\partial (F(y))^{2}}{\partial y^{i}} -y^{j} \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) +y^{j}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) \\&\quad =\dfrac{\partial }{\partial y^{j}}\left( 2(F(y))^{2} -y^{j} \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) +y^{j}\dfrac{\partial }{\partial y^{j}}\left( \dfrac{\partial (F(y))^{2}}{\partial y^{j}}\right) =\dfrac{\partial (F(y))^{2}}{\partial y^{j}}. \end{aligned}$$

\(\square \)

Minkowski structure on product of Finsler manifolds

Theorem 3.1

Let \((M_{1},F_{1})\), \((M_{2},F_{2})\) be Finsler Manifolds of dimensions n and m, respectively. Suppose that \( (p_{1},p_{2}) \in M_{1}\times M_{2}\) and \(F_{(p_{1},p_{2})}: T_{(p_{1},p_{2})}(M_{1}\times M_{2}) \rightarrow \left[ 0,+\,\infty \right) \) be a function defined by: \(F_{(p_{1},p_{2})}(y_{1},y_{2}):=(F_{1} \oplus F_{2})_{(p_{1},p_{2})}(y_{1},y_{2}):=F_{1}(y_{1})+ F_{2}(y_{2})\), \(\forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1}\times M_{2})\). Then,

  1. (I)

    \(F_{(p_{1},p_{2})}(y_{1},y_{2}) \ge 0 , \quad \forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1},M_{2})\).

  2. (II)

    \(F_{(p_{1},p_{2})}(\lambda y_{1},\lambda y_{2})=\lambda F_{(p_{1},p_{2})}(y_{1},y_{2}); \,\, \forall \lambda > 0,\,\, \forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1},M_{2})\).

Proof

  1. (I)

    Since for all \( y_{1} \in T_{p_{1}}M_{1} ; F_{1}(y_{1}) \ge 0 \) and for all \( y_{2} \in T_{p_{2}}M_{2} ; F_{2}(y_{2}) \ge 0 \), it follows that \( F_{(p_{1},p_{2})}(y_{1},y_{2}) \ge 0 .\)

  2. (II)

    By the definition of Finsler metric, we have \(\forall \lambda >0 \), \( F_{1}(\lambda y_{1})=\lambda F_{1}(y_{1}), F_{2}(\lambda y_{2})=\lambda F_{2}(y_{2}).\) It follows that \( F_{(p_{1},p_{2})}(\lambda y_{1},\lambda y_{2})=\lambda F_{(p_{1},p_{2})}(y_{1},y_{2}); \,\, \forall \lambda >0\).

\(\square \)

Let us denote by

$$\begin{aligned} y^{i}:=\left\{ \begin{array}{cc} y_{1}^{i} &{}\quad 1\le i\le n \\ y_{2}^{i-n} &{}\quad n+1 \le i \le n+m \end{array}\right. \end{aligned}$$
(3.1)

and

$$\begin{aligned} \left\{ \begin{array}{cc} y_{1}^{i}=0 &{}\quad n+1\le i\le n+m \\ y_{2}^{i}=0 &{}\quad 1 \le i \le n \end{array}.\right. \end{aligned}$$
(3.2)

Theorem 3.2

Let \((M_{1},F_{1})\), \((M_{2},F_{2})\) be Finsler Manifolds of dimensions n and m, respectively. Suppose that \( (p_{1},p_{2}) \in M_{1}\times M_{2} \) and \(F_{(p_{1},p_{2})}: T_{(p_{1},p_{2})}(M_{1}\times M_{2}) \rightarrow \left[ 0,+\,\infty \right) \) be a function defined by:

\(F_{(p_{1},p_{2})}(y_{1},y_{2}):=(F_{1}\oplus F_{2})_{(p_{1},p_{2})}(y_{1},y_{2}):=F_{1}(y_{1})+ F_{2}(y_{2})\), \(\forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1},M_{2})\). Then the Hessian matrix \(\left( g_{ij}\right) :=\left( \left[ \frac{1}{2}F^{2}\right] _{y^{i}y^{j}}\right) \,\,(y\ne 0)\) is positively definite.

Proof

By the definition of positive definiteness in linear algebra, it is sufficient to show that:

\(\forall y=(y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1}\times M_{2})-\lbrace 0\rbrace \), we have

$$\begin{aligned} \left[ y\right] ^{t}\left[ g_{ij}\right] \left[ y\right] >0. \end{aligned}$$

where \(y_{1}=(y_{1}^{1},\ldots y_{1}^{n}) \in T_{p_{1}}M_{1}-\lbrace 0\rbrace \) and \( y_{2}=(y_{2}^{1},\ldots y_{2}^{m}) \in T_{p_{2}}M_{2}-\lbrace 0\rbrace \). \( [y]^{t} \) is transpose of matrix [y]. According to the definition of F, we have

$$\begin{aligned} F^{2}(y)=[F_{1}(y_{1})+F_{2}(y_{2})]^{2}=F_{1}^{2}(y_{1})+F_{2}^{2}(y_{2})+2F_{1}(y_{1})F_{2}(y_{2}). \end{aligned}$$

We now compute the Hessian matrix of component at slit tangent space

\(T_{(p_{1},p_{2})}(M_{1}\times M_{2})-\lbrace 0 \rbrace \) that is:

$$\begin{aligned} g_{ij}:=\frac{1}{2}\dfrac{\partial ^{2}F^{2}(y) }{\partial y^{i}\partial y^{j}}=\frac{1}{2}\dfrac{\partial ^{2}}{\partial y^{i}\partial y^{j}}\left[ F_{1}^{2}(y_{1})+F_{2}^{2}(y_{2})+2F_{1}(y_{1})F_{2}(y_{2})\right] . \end{aligned}$$

For simplicity, we write \( \left( g_{ij}\right) \) as, \(\begin{pmatrix} A_{11} &{} A_{12}\\ A_{21} &{} A_{22} \end{pmatrix} \) where \( A_{11}, A_{12} , A_{21} \) and \( A_{22} \) are matrices, where.

$$\begin{aligned} (A_{11})_{ij}&= {} \frac{1}{2}\dfrac{\partial ^{2}F^{2}(y)}{\partial y_{1}^{i}\partial y_{1}^{j}} ; \quad 1\le i, \ j \le n, \\ (A_{12})_{ij}&= {} \frac{1}{2}\dfrac{\partial ^{2}F^{2}(y)}{\partial y_{1}^{i}\partial y_{2}^{j}} ; \quad 1\le i \le n , \ 1\le j\le m \\ (A_{21})_{ij}&= {} \frac{1}{2}\dfrac{\partial ^{2}F^{2}(y)}{\partial y_{2}^{i}\partial y_{1}^{j}} ; \quad 1\le i \le m , \ 1\le j\le n , \\ (A_{22})_{ij}&= {} \frac{1}{2}\dfrac{\partial ^{2}F^{2}(y)}{\partial y_{2}^{i}\partial y_{2}^{j}} ; \quad 1\le i, \ j \le m . \end{aligned}$$

By definition of F and its partial derivatives, we have:

$$\begin{aligned} (A_{11})_{ij}&= {} \frac{1}{2}\left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{i}\partial y_{1}^{j}}+2F_{2}(y_{2})\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{i}\partial y_{1}^{j}}\right) ; \quad 1\le i,j \le n, \\ (A_{12})_{ij}&= {} \frac{1}{2}\left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) ;\quad 1\le i \le n , \ 1\le j\le m, \\ (A_{21})_{ij}&= {} \frac{1}{2}\left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{j}}\times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{i}}\right) ;\quad 1\le j\le n , \ 1\le i\le m, \\ (A_{22})_{ij}&= {} \frac{1}{2}\left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{i}\partial y_{2}^{j}}+2F_{1}(y_{1})\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{i}\partial y_{2}^{j}}\right) ; \quad 1\le i, \ j \le m. \end{aligned}$$

It can be seen that:

$$\begin{aligned} \left( A_{11} \right) _{n\times n}=\frac{1}{2} \begin{bmatrix} \left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{1}\partial y_{1}^{1}}+2F_{2}(y_{2}) \dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{1}\partial y_{1}^{1}}\right)&\cdots&\left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{1}\partial y_{1}^{n}}+2F_{2}(y_{2})\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{1}\partial y_{1}^{n}}\right) \\ \left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{2}\partial y_{1}^{1}}+2F_{2}(y_{2})\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{2}\partial y_{1}^{1}}\right)&\cdots&\left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{2}\partial y_{1}^{n}}+2F_{2}(y_{2})\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{2}\partial y_{1}^{n}}\right) \\ \quad \vdots \\ \left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{n}\partial y_{1}^{1}}+2F_{2}(y_{2})\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{n}\partial y_{1}^{1}}\right)&\cdots&\left( \dfrac{\partial ^{2}F_{1}^{2}(y_{1})}{\partial y_{1}^{n}\partial y_{1}^{n}}+2F_{2}(y_{2})\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{n}\partial y_{1}^{n}}\right) \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned}&\left( A_{12} \right) _{n\times m}=\frac{1}{2} \begin{bmatrix} \left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}}\times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{1}}\right) \cdots \left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}} \times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\right) \\ \left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{2}}\times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{1}}\right) \cdots \left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{2}}\times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\right) \\ \vdots \\ \left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\times \dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\right) \cdots \left( 2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\times \dfrac{\partial F_{2}(y_{n})}{\partial y_{2}^{m}}\right) \end{bmatrix} , \\&\left( A_{21} \right) _{m\times n}=\frac{1}{2} \begin{bmatrix} \left( 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{1}}\times \dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}}\right) \cdots \left( 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{1}}\times \dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\right) \\ \left( 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{2}}\times \dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}}\right) \cdots \left( 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{2}}\times \dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\right) \\ \vdots \\ \left( 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\times \dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}}\right) \cdots \left( 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\times \dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\right) \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned} \left( A_{22} \right) _{m\times m}=\frac{1}{2} \begin{bmatrix} \left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{1}\partial y_{2}^{1}}+2F_{1}(y_{1}) \dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{1}\partial y_{2}^{1}}\right)&\cdots&\left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{1}\partial y_{2}^{m}}+2F_{1}(y_{1})\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{1}\partial y_{2}^{m}}\right) \\ \left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{2}\partial y_{2}^{1}}+2F_{1}(y_{1})\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{2}\partial y_{2}^{1}}\right)&\cdots&\left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{2}\partial y_{2}^{m}}+2F_{1}(y_{1})\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{2}\partial y_{2}^{m}}\right) \\ \quad \vdots \\ \left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{m}\partial y_{2}^{1}}+2F_{1}(y_{1})\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{m}\partial y_{2}^{1}}\right)&\cdots&\left( \dfrac{\partial ^{2}F_{2}^{2}(y_{2})}{\partial y_{2}^{m}\partial y_{2}^{m}}+2F_{1}(y_{1})\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{m}\partial y_{2}^{m}}\right) \end{bmatrix}. \end{aligned}$$

This give the Hessian matrix \((g_{ij}):=\left( \left[ \frac{1}{2}F^{2}\right] _{y^{i}y^{j}}\right) = \begin{pmatrix} A_{11} &{}\quad A_{12}\\ A_{12} &{}\quad A_{22} \end{pmatrix}\).

It is sufficient to show that: \(\forall (y_{1},y_{2})\in T_{(p_{1},p_{2})}(M_{1}\times M_{2})-\lbrace 0\rbrace \) we have:

$$\begin{aligned}&\left[ y\right] ^{t}\left[ g_{ij}\right] \left[ y\right] >0. \begin{bmatrix} y_{1}^{1},\ldots , y_{1}^{n},y_{2}^{1},\ldots ,y_{2}^{m} \end{bmatrix} \begin{bmatrix} g_{ij} \end{bmatrix} \begin{bmatrix} y_{1}^{1} \\ \,\, \vdots \\ y_{1}^{n} \\ y_{2}^{1}\\ \,\, \vdots \\ y_{2}^{m} \end{bmatrix} = \begin{bmatrix} y_{1}^{1},\ldots , y_{1}^{n},y_{2}^{1},\ldots ,y_{2}^{m} \end{bmatrix} \times \dfrac{1}{2} \\&\begin{bmatrix} \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{1}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{1}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \\ \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{2}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{2}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{2}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \\ \vdots \\ \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{n}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{n}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \\ 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{1}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{1}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{1}\partial y_{2}^{j}}\right) \\ 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{2}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{2}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{2}\partial y_{2}^{j}}\right) \\ \vdots \\ 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{m}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{m}\partial y_{2}^{j}}\right) \end{bmatrix} \end{aligned}$$
$$\begin{aligned}&= \frac{1}{2} y_{1}^{1} \begin{bmatrix} \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{1}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{1}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{1}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \end{bmatrix} \\&+ \, \frac{1}{2} y_{1}^{2} \begin{bmatrix} \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{2}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{2}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{2}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \end{bmatrix} \\&+ \\&\ \vdots \\&+\, \frac{1}{2} y_{1}^{n} \begin{bmatrix} \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{n}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{n}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{n}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \end{bmatrix} \\&+\, \frac{1}{2} y_{2}^{1} \begin{bmatrix} 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{1}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{1}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{1}\partial y_{2}^{j}}\right) \end{bmatrix} \\&+\, \frac{1}{2} y_{2}^{2} \begin{bmatrix} 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{2}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{2}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{2}\partial y_{2}^{j}}\right) \end{bmatrix} \\&+ \\&\ \vdots \\&+\, \frac{1}{2} y_{2}^{m} \begin{bmatrix} 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{m}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{m}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{m}\partial y_{2}^{j}}\right) \end{bmatrix} \\&= \frac{1}{2} y_{1}^{k} \begin{bmatrix} \left( y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{k}\partial y_{1}^{i}}\right) +2F_{2}(y_{2})\left( y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{k}\partial y_{1}^{i}}\right) +2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{k}}\left( y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}\right) \end{bmatrix} \\&+\, \frac{1}{2} y_{2}^{h} \begin{bmatrix} 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{h}}\left( y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}\right) +\left( y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{h}\partial y_{2}^{j}}\right) +2F_{1}(y_{1})\left( y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{h}\partial y_{2}^{j} }\right) \end{bmatrix}. \end{aligned}$$

In the first expression by Corollaries 2.4 (b), and 2.5, 2.6 and Proposition 2.7, it follows that

$$y_{1}^{i}\dfrac{\partial ^{2}F_{1}(y_{1})}{\partial y_{1}^{k}\partial y_{1}^{i}}=0 , \ y_{2}^{j}\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{j}}=F_{2}(y_{2}), \ y_{1}^{i}\dfrac{\partial ^{2}(F_{1}(y_{1}))^{2}}{\partial y_{1}^{k}\partial y_{1}^{i}}=\dfrac{\partial (F_{1}(y_{1}))^{2}}{\partial y_{1}^{k}}.$$

In the second expression,

$$\begin{aligned} y_{2}^{j}\dfrac{\partial ^{2}F_{2}(y_{2})}{\partial y_{2}^{h}\partial y_{2}^{j}}=0 ,\quad y_{1}^{i}\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{i}}=F_{1}(y_{1}), \quad y_{2}^{j}\dfrac{\partial ^{2}(F_{2}(y_{2}))^{2}}{\partial y_{2}^{h}\partial y_{2}^{j}}=\dfrac{\partial (F_{2}(y_{2}))^{2}}{\partial y_{2}^{h}}. \end{aligned}$$

Then we will have

$$\begin{aligned} \left[ y\right] ^{t}\left[ g_{ij}\right] \left[ y\right] = \frac{1}{2} y_{1}^{k} \begin{bmatrix} \dfrac{\partial (F_{1}(y_{1}))^{2}}{\partial y_{1}^{k}}+2\dfrac{\partial F_{1}(y_{1})}{\partial y_{1}^{k}}F_{2}(y_{2}) \end{bmatrix} + \frac{1}{2} y_{2}^{h} \begin{bmatrix} 2\dfrac{\partial F_{2}(y_{2})}{\partial y_{2}^{h}}F_{1}(y_{1})+\dfrac{\partial (F_{2}(y_{2}))^{2}}{\partial y_{2}^{h}} \end{bmatrix}. \end{aligned}$$

Therefore \(\left[ y\right] ^{t}\left[ g_{ij}\right] \left[ y\right] =\left[ F_{1}^{2}(y_{1})+2F_{1}(y_{1})F_{2}(y_{2})+F_{2}^{2}(y_{2})\right] =\left[ F_{1}(y_{1})+F_{2}(y_{2})\right] ^{2}\). It is clear that \(\forall y=(y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1}\times M_{2})-\lbrace 0\rbrace \) we will have

$$\begin{aligned} \left[ y\right] ^{t}\left[ g_{ij}\right] \left[ y\right] >0. \end{aligned}$$

And since the Hessian matrix \((g_{ij}):=\left( \left[ \frac{1}{2}F^{2}\right] _{y^{i}y^{j} }\right) ; (y\ne 0) \) is positive definite. \(\square \)

Corollary 3.3

Let \((M_{1},F_{1})\), \((M_{2},F_{2})\) be Finsler Manifolds of dimensions n and m, respectively. Suppose that \( (p_{1},p_{2}) \in M_{1}\times M_{2} \) and \( F_{(p_{1},p_{2})}: T_{(p_{1},p_{2})}(M_{1}\times M_{2}) \rightarrow \left[ 0,+\,\infty \right) \) be a function defined by:

$$\begin{aligned}&F_{(p_{1},p_{2})}(y_{1},y_{2})=(F_{1}\oplus F_{2})_{(p_{1},p_{2})}(y_{1},y_{2})=F_{1}(y_{1})+ F_{2}(y_{2}), \\&\forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1}\times M_{2}) \end{aligned}$$

Then \(F_{(p_{1},p_{2})}: T_{(p_{1},p_{2})}(M_{1}\times M_{2}) \rightarrow \left[ 0,+\,\infty \right) \) is a Minkowski norm.

Proof

The proof is result of Theorems 3.1 and 3.2. \(\square \)

Theorem 3.4

Let \((M_{1},F_{1})\), \((M_{2},F_{2}) \) be Finsler manifolds of dimensions n and m, respectively. Then the function \(F : T(M_{1}\times M_{2})\rightarrow \left[ 0,+ \infty \right) \) defined by:

$$\begin{aligned}&F_{(p_{1},p_{2})}(y_{1},y_{2}):=(F_{1}\oplus F_{2})_{(p_{1},p_{2})}(y_{1},y_{2}):=F_{1}(y_{1})+ F_{2}(y_{2}) \\&\forall (p_{1},p_{2})\in M_{1}\times M_{2}, \forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1}\times M_{2})=T_{p_{1}}M_{1}\oplus T_{p_{2}}M_{2} \end{aligned}$$

is a \(c^{\infty } \)-function on the slit‘ tangent bundle \(T(M_{1}\times M_{2}-\left\{ 0 \right\} \).

Proof

Since \(F_{1} \) and \( F_{2} \) are \( c^{\infty } \)-functions, it follows that vector function \( \theta :T_{p_{1}}M_{1}\oplus T_{p_{2}}M_{2}\rightarrow \left[ 0,+\,\infty \right) \times \left[ 0,+\,\infty \right) \) defined by: \( \theta (y_{1},y_{2})=\left( F_{1}(y_{1}),F_{2}(y_{2})\right) \) is a \( c^{\infty } \) -function, and so \( \lambda : R\times R \rightarrow R \) defined by \( \lambda (s,t)=s+t \) is a \( c^{\infty } \)-function. It follows that \( F_{(p_{1},p_{2})}=\lambda o \theta \) is a \( c^{\infty } \) -function on \( T(M_{1}\times M_{2})-\left\{ 0 \right\} \).

As the restriction of F to any \(T_{(p_{1},p_{2})}(M_{1}\times M_{2})-\left\{ 0 \right\} \) is \( c^{\infty }, \) we have F is a \( c^{\infty } \) function on the slit tangent bundle \( T(M_{1}\times M_{2})-\left\{ 0 \right\} \). \(\square \)

Finsler structure on product of Finsler manifolds

Theorem 4.1

Let \((M_{1},F_{1})\), \((M_{2},F_{2}) \) be Finsler manifolds of dimensions n and m, respectively. Then the function \(F : T(M_{1}\times M_{2})\rightarrow \left[ 0,+ \infty \right) \) defined by:

$$\begin{aligned}&\forall (p_{1},p_{2})\in M_{1}\times M_{2}, \forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1}\times M_{2})=T_{p_{1}}M_{1}\oplus T_{p_{2}}M_{2} \\&F_{(p_{1},p_{2})}(y_{1},y_{2}):=(F_{1}\oplus F_{2})_{(p_{1},p_{2})}(y_{1},y_{2}):=F_{1}(y_{1})+ F_{2}(y_{2}) \end{aligned}$$

is a Finsler metric on \( M_{1}\times M_{2}\).

Proof

By the Corollary 3.3, the restriction of F to \(T_{(p_{1},p_{2})}(M_{1}\times M_{2}), \forall (p_{1},p_{2})\in M_{1}\times M_{2}\) is s Minkowski norm, and according to the Theorem 3.4 F is \( c^{\infty } \) function on the slit tangent bundle \(T(M_{1}\times M_{2})-\left\{ 0 \right\} \). Therefore F is a Finsler metric on \( M_{1}\times M_{2}\). \(\square \)

Definition 4.2

Let \((M_{1},F_{1}) , (M_{2},F_{2}) \) be Finsler manifolds of dimensions n and m, respectively. Then Finsler metric \(F=F_{1}+F_{2}\) on \(M_{1}\times M_{2}\) is called canonical product Finsler metric.