Generalization of Finsler metrics on the product of Finsler manifolds

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-{0} and satisfies the following conditions:

• F(u)≥0,∀u∈V;

• F(λu)=λF(u),∀λ>0,∀u∈V0, \ \forall u\in V$$\end{document}]]> ;

• for any basis e1,…,en of V , write F(y)=F(y1,…,yn) , y=yjej . Then the Hessian matrix

  (gij):=12F2yiyj
  is positive definite at any point of V-{0} .

The pair ( V ,  F ) is called Minkowski space.

Definition 2.2

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

• F is C∞ on the slit tangent bundle TM-{0} ;

• The restriction of F to any TpM , p∈M is a Minkowski norm.

The pair ( M ,  F ) is called Finsler manifold or Finsler space.

Let ( M ,  F ) be a Finsler space and (x1,…,xn) be a local coordinate system on an open subset U of M . Then ∂∂x1,…,∂∂xn form a basis for the tangent space at any point in U .

Theorem 2.3

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

• H is positively homogeneous of degree r . That is:

  H(λy)=λrH(y);∀λ,λ>00 \end{aligned}$$\end{document}]]>

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

  yiHyi(y)=rH(y).

Corollary 2.4

Let F be positively homogeneous of degree 1 on Rn . By using Euler’s theorem, we can show that:

• yiFyi=F.

• yiFyiyj=0.

• ykFyiyjyk=-Fyiyj.

• ylFyiyjykyl=-2Fyiyjyk.

Corollary 2.5

Let ( M ,  F ) be a Finsler manifold . Then F2 defined by:

Fp2(y)=(Fp(y))2 ; for all p∈M and y∈TpM ; is a positively homogeneous function of degree 2 on TpM .

Corollary 2.6

Let ( M ,  F ) be a Finsler manifolds. Then

as Fp1(y1),Fp2(y2) , respectively. i.e:F1=Fp1,F2=Fp2.

Proposition 2.7

Let ( M ,  F ) be a Finsler manifold of dimension n. Then

Proof

By definition and Corollary ^2.6 , we have:

Theorem 3.1

Let (M1,F1) , (M2,F2) be Finsler Manifolds of dimensions n and m, respectively. Suppose that (p1,p2)∈M1×M2 and F(p1,p2):T(p1,p2)(M1×M2)→0,+∞ be a function defined by: F(p1,p2)(y1,y2):=(F1⊕F2)(p1,p2)(y1,y2):=F1(y1)+F2(y2) , ∀(y1,y2)∈T(p1,p2)(M1×M2) . Then,

• F(p1,p2)(y1,y2)≥0,∀(y1,y2)∈T(p1,p2)(M1,M2) .

• F(p1,p2)(λy1,λy2)=λF(p1,p2)(y1,y2);∀λ>0,∀(y1,y2)∈T(p1,p2)(M1,M2) 0,\,\, \forall (y_{1},y_{2}) \in T_{(p_{1},p_{2})}(M_{1},M_{2})$$\end{document}]]> .

Proof

• Since for all y1∈Tp1M1;F1(y1)≥0 and for all y2∈Tp2M2;F2(y2)≥0 , it follows that F(p1,p2)(y1,y2)≥0.

• By the definition of Finsler metric, we have ∀λ>00 $$\end{document}]]> , F1(λy1)=λF1(y1),F2(λy2)=λF2(y2). It follows that F(p1,p2)(λy1,λy2)=λF(p1,p2)(y1,y2);∀λ>00$$\end{document}]]> .

Let us denote by

Theorem 3.2

Let (M1,F1) , (M2,F2) be Finsler Manifolds of dimensions n and m, respectively. Suppose that (p1,p2)∈M1×M2 and F(p1,p2):T(p1,p2)(M1×M2)→0,+∞ be a function defined by:

F(p1,p2)(y1,y2):=(F1⊕F2)(p1,p2)(y1,y2):=F1(y1)+F2(y2) , ∀(y1,y2)∈T(p1,p2)(M1,M2) . Then the Hessian matrix gij:=12F2yiyj(y≠0) is positively definite.

Proof

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

∀y=(y1,y2)∈T(p1,p2)(M1×M2)-{0} , we have

T(p1,p2)(M1×M2)-{0} that is:

It is sufficient to show that: ∀(y1,y2)∈T(p1,p2)(M1×M2)-{0} we have: