Anisotropic conformal change of conic pseudo-Finsler surfaces, I
S. G. Elgendi Nabil L. Youssef A. A. Kotb Ebtsam H. Taha

 
 
 


Anisotropic conformal change of conic pseudo-Finsler surfaces, I
    S. G. Elgendi^1,  Nabil L. Youssef^2,    A. A. Kotb^3 and Ebtsam H. Taha^2 
    
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 ^1 Department of Mathematics, Faculty of Science,
 Islamic University of Madinah,  Madinah, Kingdom of Saudia Arabia








^2Department of Mathematics, Faculty of Science,

Cairo University, Giza, Egypt










^3Department of Mathematics, Faculty of Science,

Damietta University, Damietta , Egypt







salah.ali@fsc.bu.edu.eg, salahelgendi@yahoo.com

nlyoussef@sci.cu.edu.eg, nlyoussef2003@yahoo.fr

alikotb@du.edu.eg,  kotbmath2010@yahoo.com

ebtsam.taha@sci.cu.edu.eg, ebtsam.h.taha@hotmail.com





 






The present work is devoted to  investigate  anisotropic conformal transformation of  conic pseudo-Finsler surfaces (M,F), that is, F(x,y)⟼F(x,y)=e^ϕ(x,y)F(x,y), where the function ϕ(x,y) depends on both position x and direction y, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If F is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface (M,F) to be transformed to a (conic) pseudo-Finsler surface (M,F) under the transformation F=e^ϕ(x,y)F. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form.  This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed  Finsler function  F.  In contrast to isotropic conformal transformation, we have a non-homothetic  conformal factor ϕ(x,y) that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change.  Furthermore, we investigate a sufficient condition for F to be dually flat or/and  projectively flat.  Finally, we study some special cases of the conformal factor ϕ(x,y). Various examples are provided whenever the situation needs.


Keywords:    conic pseudo-Finsler surface; modified Berwald frame; anisotropic conformal change;  projectively flat; dually flat

MSC 2020:  53B40, 53C60





§ INTRODUCTION

 Conformal transformations have been investigated in different frameworks, since they have some important geometric features <cit.> and successful applications such as those in physics, biology and ecology <cit.>. 
 In the conformal transformation theory, it is useful to study the geometric properties and  geometric objects that are preserved under  conformal transformations. In Riemannian  geometry, (isotropic) conformal transformation is an angle-preserving. In pseudo Riemannian metrics with signature (Lorentzian metrics),  it preserves causality. In parallelizable manifolds, it preserves both angles and causality as each parallelization structure induces a pseudo-Riemannian (or Lorentzian) metric, cf. <cit.>.
In Finsler geometry, (isotropic) conformal transformation is angle-preserving <cit.> and leaves the geodesic spray  invariant if the conformal factor is a homothety <cit.>. Further, each Randers space (M, F= α + β) has a globally defined nonholonomic frame which is called the Holland frame <cit.>. This frame is an isotropic conformally invariant in the sense that the transformation F(x,y)⟼F(x,y)=e^ϕ(x)F(x,y) leaves the frame elements fixed <cit.>.    

Given two pseudo-Finsler metrics F(x,y) and F(x,y),  the anisotropic conformal transformation F(x,y)⟼F(x,y)=e^ϕ(x,y)F(x,y) preserves the lightcone <cit.> and lightlike geodesics and their focal points <cit.>.  
The anisotropic conformal change in Finsler Geometry is not only valuable in applications, but also it has a great impact in the study of Finsler geometry. For example, in contrast to isotropic conformal change, the anisotropic conformal change of a pseudo-Finsler metric is not necessarily a pseudo-Finsler metric (see Theorem <ref>). In addition,  the anisotropic conformal change can send a psudo-Finsler metric to a pseudo-Riemannian one and vice-versa. 
 
Since conic pseudo-Finsler surfaces  are  used widely in applications, especially in physics, we study here the anisotropic conformal transformation F(x,y)⟼F(x,y)=e^ϕ(x,y)F(x,y) of a conic pseudo-Finsler metric F(x,y).  First, under the anisotropic conformal change, we compute the components g_ij of the metric tensor of F. Actually, the expression of g_ij contains the second derivative of the function ϕ with respect to directional arguments. Generally, this term  represents a severe obstacle to compute the components g^ij of the inverse metric tensor.
Keeping in mind that the inverse metric tensor is the door to find the geodesic spray, which enables to study the geometry of the transformed space (M,F), this motivates us to consider the two-dimensional case to make use of the modified Berwald frame.  Consequently, we become able to derive some important geometric objects associated with the transformed metric F, for example,  the transformed  Cartan  tensor,  main scalar, geodesic spray and  Barthel connection.  Further, we find out some invariant objects under anisotropic conformal transformations (see Theorem <ref>).

The present paper is organized in the following  manner.  
In 1,  we recall some basic facts on the geometry of sprays and Finsler manifolds. 
In 2, we first show that the fact that  two Finsler metrics are anisotropically conformally related is not equivalent to the proportionality of their associated  metric tensors, contrary to what has been mentioned in  <cit.>. Then, we investigate the anisotropic conformal transformations of  pseudo-Finsler surfaces. 
In 3, we characterize the anisotropic conformal transformations and  study their action on some important Finslerian geometric objects. Some anisotropic conformal invariant geometric properties are found out. 
In 4,  we study the case when the anisotropic conformal transformation is a projective transformation.  Moreover, we obtain sufficient conditions for F to be projectively flat and dually flat.   
Finally,  in 5, we consider two important special cases: the factor ϕ is a function of position only and the factor ϕ is a function of direction only. Interesting results in both cases are obtained. 

It should be noted that, various examples have been provided whenever the situation needs. The Maple’s code of these examples is presented in the Appendix at the end of the paper.






§ NOTATION AND PRELIMINARIES

Let M be an n-dimensional smooth  manifold and π : TM ⟶ M the canonical projection of the tangent bundle TM onto M. Let TM_0:=TM∖(0) be the slit tangent bundle, (0) being the null section, and (x^i, y^i)   the corresponding  coordinate system TM.  
  The natural almost-tangent structure J of T M is the vector 1-form   defined  by J = ∂/∂ y^i⊗ dx^i. The vertical vector field
=y^i∂/∂ y^i on TM is called  the
Liouville vector field. The set of smooth functions on TM_0 is denoted by C^∞ (TM_0).

A spray on M is a vector field S on T M such that JS = and
[, S] = S. Locally, it is  given by <cit.>

    S = y^i ∂/∂ x^i - 2G^i (x,y)  ∂/∂ y^i,

where the spray coefficients G^i(x,y) are positively 2-homogeneous
functions in  y (or simply h(2)-functions). A nonlinear connection is defined by an n-dimensional distribution H(TM_0) on TM_0  which is supplementary to the vertical distribution V(TM_0). This means that for all u ∈ TM_0, we have

    T_u(TM_0) = V_u(TM_0) ⊕ H_u(TM_0) .

The local basis of V_u(TM_0) and H_u(TM_0) are given, respectively by,  ∂̇_i:=∂∂ y^i   and δ_i:=δδ x^i=∂∂ x^i-G^j_i(x,y)∂∂ y^j. The coefficients of  Cartan nonlinear (or Barthel) connection  and Berwald connection are defined by  G^j_i(x,y):=∂ G^j∂ y^i and  G^h_ij(x,y):=∂ G^h_j∂ y^i, respectively.



In the following we set the definition of a (conic pseudo-) Finsler manifold which will be used throughout the paper.

A conic sub-bundle of TM is a non-empty open subset 𝒜⊂ TM_0 such that  π(𝒜) = M and 
λ v∈𝒜  ∀ v∈𝒜 and  ∀λ∈ℝ^+.


  A conic pseudo-Finsler  metric on M is a function
  F:𝒜⟶ℝ
  which satisfies the following conditions:
  
    
  (i) F is smooth on 𝒜, 
    
  (ii) F(x,y) is positively homogeneous of degree one in y: F(x,λ y)=λ F(x,y) for all (x,y)∈𝒜  and λ∈ℝ^+,
    
  (iii) The Finsler metric tensor g_ij (x,y)=1/2∂̇_i∂̇_jF^2 (x,y)
    is non-degenerate at each point of 𝒜.
  
The pair (M, F) is called a conic pseudo-Finsler manifold.



The Finsler metric F is said to be:

∙ a conic Finsler metric if the  metric tensor g_ij in (iii) is positive definite, or equivalently, F is non-negative, and in this case (M, F) is  a conic Finsler manifold.

∙ a pseudo-Finsler metric if 𝒜 is replaced by TM, and in this case (M, F) is  a pseudo-Finsler manifold.

∙ a Finsler metric if 𝒜 is replaced by TM and g_ij in (iii) is positive definite,  and in this case (M, F) is  a  Finsler manifold.
  
 
Each pseudo-Finsler metric F induces a spray S on M, for which the spray coefficients are given by G^i = 1/4 g^ij(y^k ∂̇_j∂_kF^2-∂_jF^2) <cit.>, this spray is called the  geodesic spray of F.  






§ ANISOTROPIC CONFORMAL TRANSFORMATION

In this section, we introduce the anisotropic conformal transformation of a conic pseudo-Finsler metric F and investigate its elementary properties. 
  
Conformality is classically defined as follows.
 
<cit.>
Let F and F be two Finsler metric functions defined on a smooth manifold M. The two Finsler metric tensors g_ij and  g_ij resulting from F and F, respectively, are called conformal if there exists a factor of proportionality Ψ(x,y) between the metric tensors:  

    g_ij(x,y)=Ψ(x,y) g_ij(x,y),
 
where Ψ(x,y) is a positive smooth function of position x and direction y.


     Suppose g_ij and g_ij are conformal, then (<ref>) holds. 
  As g_ijy^iy^j=F^2, we get
  
    F(x,y)=√(Ψ(x,y)) F(x,y).

  
  This means that if g_ij(x,y) and g_ij(x,y) are proportional, then F(x,y) and F(x,y) are proportional, or, in other words, (<ref>) implies  (<ref>). The converse is not true in general <cit.>: proportional Finsler metric functions may yield non proportional Finsler metric tensors. Nevertheless, we have
   

  Let two Finsler metric functions F and F on M be proportional, i.e., F(x,y)=Ψ(x,y) F(x,y). A necessary and sufficient condition for the associated Finsler metric tensors g_ij and g_ij to be proportional is that Ψ(x,y) is a functions of position x only.


 Let F and F be proportional, that is,
  
    F(x,y)=Ψ(x,y) F(x,y).
   
  By squaring and differentiating with respect to y^i and y^j, we get
  
    g_ij  =Ψ^2 g_ij+{(∂̇_iΨ)
      (∂̇_jΨ)+Ψ(∂̇_i∂̇_jΨ)}F^2 + Ψ{(∂̇_jΨ)∂̇_iF^2
        +(∂̇_iΨ)∂̇_jF^2}.

  Consequently, the metric tensors g_ij and g_ij are proportional if and only if
  
    {(∂̇_iΨ)
      (∂̇_jΨ)+Ψ(∂̇_i∂̇_jΨ)}F^2 + Ψ{(∂̇_jΨ)∂̇_iF^2
        +(∂̇_iΨ)∂̇_jF^2}=0.

   Contracting both sides of the above equation by y^j and  using the homogeneity properties of F  and Ψ, we get
   
    Ψ(∂̇_iΨ)F^2=0,

  from which ∂̇_iΨ(x,y)=0 and Ψ≠0 is a function of x only.



(a) The above discussion shows that the proportionality of F and F does not imply automatically that the conformal factor Ψ is independent of direction, contrary to what has been mentioned by Y. Shen and Z. Shen in <cit.>.


(b) On the other hand, it was proved by Knebelman <cit.> that the proportionality of g_ij and g_ij does imply automatically that the conformal factor Ψ is independent of direction, as has been mentioned by H. Rund in <cit.>.  


We begin our investigation from the proportionality of F and F, regardless of the proportionality of g_ij and g_ij and regardless of the independence of the conformal factor of direction. In fact, we are particularly interested in the dependence of the conformal factor on both position and direction, motivated by the current applications in physics and other branches of science, and by the high potentiality of near future applications. 

Now, let us set our definition of conformality.

  Let (M, F) be a conic pseudo-Finsler  manifold.  The anisotropic conformal transformation of  F is defined by
  
    F⟼F(x,y)=e^ϕ(x,y)F(x,y),

    where  ϕ(x,y) is  an h(0) smooth function on 𝒜. In this case, we say that F and F are anisotropically conformally related, or F is anisotropically conformal to F.
    
If the conformal factor ϕ(x,y) is independent of direction,  (<ref>) is called isotropic conformal, or, simply, conformal transformation.     
 

 One of the benefits of the anisotropic conformal transformation is that, unlike conformal transformation <cit.>, the anisotropic conformal transformation can transform a Riemannian metric to a Finslerian one. 


  Let (M,F) be a conic pseudo-Finsler space, then under the anisotropic conforaml transformation (<ref>), we have

    g_ij=  e^2ϕ[ g_ij+2 F^2  ∂̇_̇i̇ϕ ∂̇_̇j̇ϕ + 2F ( ℓ_j ∂̇_̇i̇ϕ+    ℓ_i ∂̇_̇j̇ϕ)+F^2 ∂̇_i∂̇_jϕ].



  Since F^2=e^2ϕ(x,y)F^2, we get ∂̇_̇i̇F^2=e^2ϕ (2   F^2 ∂̇_̇i̇ϕ+∂̇_̇i̇F^2).
 Thereby,

 g_ij    = 1/2 ∂̇_̇i̇ ∂̇_̇j̇F^2    =e^2ϕ[2 F^2(∂̇_̇i̇ϕ)(∂̇_̇j̇ϕ) + 2F (ℓ_j  ∂̇_̇i̇ϕ+ℓ_i ∂̇_̇j̇ϕ)+F^2 ∂̇_i∂̇_jϕ+1/2 ∂̇_̇i̇∂̇_̇j̇F^2]

         =e^2ϕ[ g_ij+2 F^2(∂̇_̇i̇ϕ)(∂̇_̇j̇ϕ) + 2F (ℓ_j  ∂̇_̇i̇ϕ+ℓ_i ∂̇_̇j̇ϕ)+F^2 ∂̇_i∂̇_jϕ],

      where   ℓ_i=∂̇_iF.



    




The existence of the term F^2∂̇_i∂̇_jϕ in (<ref>)  makes calculating the inverse metric g^ij, in a tensorial form, very complicated in  general dimension. However, g^ij can be found out in the two-dimensional case thanks to the existence of Berwald frame in such a case.
  
 
Henceforward, we work in  a two-dimensional conic pseudo-Finsler space equipped with a modified Berwald frame (ℓ_i,m_i) <cit.>.  The components g_ij of the  metric tensor are given by
  
    g_ij=ℓ_iℓ_j+ε m_im_j,

  where  ε=±1 is called the signature of F. Further,   the angular metric coefficients h_ij can be expressed as 
  
    h_ij=ε m_im_j.

Also, we have 

    g^ij =ℓ^iℓ^j+ε m^im^j,

where ℓ^i=y^i/F. The two vector fields ℓ = (ℓ^1,ℓ^2) and m=(m^1 ,m^2) have been chosen in such a way that they satisfy 
    g(ℓ, ℓ)=1,   g(ℓ, m)=0,   g(m, m)=ε.
  Moreover, the determinant 𝔤 of the matrix (g_ij) is given by

    𝔤 :=(g_ij)=ε(ℓ_1m_2-ℓ_2m_1)^2.

 To calculate m^i and    m_i, we give the following lemma.
 
 
 The covector m_i is given by
 
    m_1=-√(ε𝔤) ℓ^2,    m_2=√(ε𝔤) ℓ^1.

 Moreover, the   vector m^i is given by
 
    m^1=-1/√(ε𝔤) ℓ_2,    m^2=1/√(ε𝔤) ℓ_1.

 
 
 By (<ref>), we have
 
    ℓ_1m_2-ℓ_2m_1=√(ε𝔤).

 Also,  the property g(ℓ ,m)=0  gives
 
    ℓ^1m_1+ℓ^2m_2=0.

Then, the result follows by solving the above  two equations for m_1 and m_2.  The proof of the formulae of m^i can be calculated in a similar manner. 
 
 
Thus, the components C_ijk of the Cartan tensor are given by <cit.> 

    F C_ijk=ℐ m_im_jm_k,

where ℐ is the  main scalar of the surface (M,F).



From now on, when we say that (M,F) is a conic pseudo-Finsler surface, this means that (M,F) is a conic pseudo-Finsler surface equipped with the modified Berwald frame.
<cit.>
  Let (M,F) be a conic pseudo-Finsler surface. Then we have the following:
  
    
  (i) ℓ^im_i=ℓ_im^i=0,
     
  (ii) m^im_i=ε,ℓ^iℓ_i=1,
     
  (iii) δ_i^j=ℓ_iℓ^j+ε m_im^j,
    
  (iv) F∂̇ ̇_̇j̇ℓ_i=ε m_im_j=h_ij,     F∂̇ ̇_̇j̇ℓ^i=ε m^im_j,
    
  (v) F∂̇ ̇_̇j̇m_i=-(ℓ_i-εℐ m_i)m_j,    F∂̇ ̇_̇j̇m^i=-(ℓ^i+εℐ m^i)m_j,
     
  (vi) For a smooth function f on 𝒜, we have F∂̇ ̇_̇i̇f= f_;1 ℓ_i + f_;2 m_i,  where f_;1=F(∂̇ ̇_̇i̇f)  ℓ^i and f_;2=ε F (∂̇ ̇_̇i̇f) m^i. In particular, if f is h(r), then  f_;1=rf,
    
  (vii) F∂̇ ̇_̇i̇∂_kf= (∂_kf)_;1 ℓ_i + (∂_kf)_;2 m_i, where ∂_k=∂/∂ x^k,
    
  (viii) δ _if= f_,1 ℓ_i + f_,2 m_i, where f_,1= (δ_if)ℓ^i and  f_;2=ε (δ_if)m^i.



Using the modified  Berwald frame, we can write the formulae of the objects ∂̇_iϕ and  ∂̇_i∂̇_jϕ in terms of the  frame elements.  Consequently,  plugging these objects into  (<ref>),  we are able to  find the formula of components g^ij of the inverse metric of F.  More precisely, we have:


   Let (M,F) be a  conic pseudo-Finsler surface. The term  ∂̇_̇i̇∂̇_̇j̇ϕ given in (<ref>) can be expressed in the form 
   
    F^2∂̇_i∂̇_jϕ =-ϕ_;2 (ℓ_im_j+ℓ_jm_i)+( ϕ_;2;2+ε ℐ ϕ_;2) m_jm_i.



From Lemma <ref> (vi) ,  
F∂̇_iϕ=ϕ_;1 ℓ_i+ϕ_;2 m_i. Since ϕ is h(0), then ϕ_;1=0. That is,

    F∂̇_iϕ=ϕ_;2 m_i.

Multiplying both sides of (<ref>) by F, then differentiating with respect to y^j and use Lemma <ref> (v)  and (vi), we get

    F^2∂̇_i∂̇_jϕ=-ϕ_;2 (ℓ_im_j+ℓ_im_j)+( ϕ_;2;2+ε ℐ ϕ_;2) m_jm_i.



    




  Let (M,F) be a conic pseudo-Finsler surface, then under the anisotropic conformal transformation (<ref>),    the   metric tensors of F and F are related by
  
    g_ij    =e^2ϕ[ g_ij+ϕ_;2(ℓ_im_j+ℓ_jm_i)+σ m_im_j]
        = e^2ϕ[ ℓ_iℓ_j+ϕ_;2(ℓ_im_j+ℓ_jm_i)+(σ+ε) m_im_j],

where σ=ϕ_;2;2+εℐϕ_;2+2 (ϕ_;2)^2.


  It follows from (<ref>), (<ref>), (<ref>) and (<ref>).


It should be noted that the anisotropic conformal transformation of a conic pseudo-Finsler surface (M,F) does not yield in general a conic pseudo-Finsler surface, as shown in Example <ref> below.
The following result gives a necessary and sufficient condition for (M,F) to be a conic pseudo-Finsler surface. 
 

Let (M,F) be a conic pseudo-Finsler surface and  (<ref>) be an anisotropic conformal transformation of F.    Then, (M,F) is a conic pseudo-Finsler surface  if and only if 
    σ-(ϕ_;2)^2+ε=F^2[∂̇_i∂̇_jϕ
     +(∂̇_iϕ)(∂̇_jϕ)] m^im^j+ε≠0.



 Since both  F(x,y) and  e^ϕ(x,y) are smooth on 𝒜,  then  F(x,y) is smooth on 𝒜. Moreover,   F(x,y) is h(1). Now, from (<ref>), we get

  
    (g_ij)=e^4ϕℓ_1ℓ_1+2ϕ_;2 ℓ_1m_1+(σ+ε)  m_1m_1   ℓ_1ℓ_2+ϕ_;2 (ℓ_1m_2+ ℓ_2m_1) +(σ+ε) m_1m_2
    ℓ_1ℓ_2+ϕ_;2(ℓ_1m_2+ ℓ_2m_1) +(σ+ε) m_1m_2   ℓ_2ℓ_2+2ϕ_;2 ℓ_2m_2+(σ+ε)  m_2m_2.
 
That is, by using (<ref>), we have

    (g_ij)      =   e^4ϕ[(σ+ε) ℓ_1ℓ_1 m_2m_2+(σ+ε) ℓ_2ℓ_2 m_1m_1-2(σ+ε) ℓ_1m_1ℓ_2m_2
               -(ϕ_;2)^2ℓ_1ℓ_1 m_2m_2-(ϕ_;2)^2ℓ_2ℓ_2 m_1m_1+2(ϕ_;2)^2ℓ_1m_1ℓ_2m_2]
        =    e^4ϕ[σ+ε-(ϕ_;2)^2](ℓ_1m_2-ℓ_2m_1)^2
       =   ε e^4ϕ[σ-(ϕ_;2)^2+ε](g_ij).

 Therefore, the matrix  (g_ij) is non-degenerate if and only if

    σ-(ϕ_;2)^2+ε=F^2[∂̇_i∂̇_jϕ +(∂̇_iϕ)(∂̇_jϕ)] m^im^j+ε≠0.

 

    

 


This example gives a conic pseudo-Finsler surface F whose anisotropic conformal transformation F=e^ϕ F is not a conic pseudo-Finsler surface for a certain smooth h(0) function  ϕ. Let F(x,y)=√(|y|^2-(|x|^2|y|^2-⟨ x, y⟩^2))1-|x|^2 be the Klein metric on unit ball B^n⊂ℝ^n. Define

    F(x,y)= e^ϕ (x,y) F(x,y)=⟨ x, y⟩1-|x|^2,

where ϕ (x,y)=ln(⟨ x, y⟩√(|y|^2-(|x|^2|y|^2-⟨ x, y⟩^2))).

It is clear that F is linear function in y, i.e.,   (g_ij)=0. That is,  σ+ε-(ϕ_;2)^2=0. Hence, F is not a conic pseudo-Finsler metric, by Theorem <ref>.



Let  (M,F) be a  conic pseudo-Finsler surface and (<ref>) be an anisotropic conformal transformation of F. Then, we have

    g^ij    =e^-2ϕ[ερ g^ij+σρℓ^iℓ^j
    -εϕ_;2 ρ(ℓ^im^j+ℓ^jm^i)]
        =e^-2ϕ[g^ij+(ϕ_;2)^2ρℓ^iℓ^j
    -εϕ_;2 ρ(ℓ^im^j+ℓ^jm^i)+(ρ-ε) m^im^j],

where
ρ:=1σ+ε-(ϕ_;2)^2.


   In a conic pseudo-Finsler surface (M,F), a tensor of type (2,0) can be written as T^ij=A_1ℓ^iℓ^j+A_2ℓ^im^j+A_3m^iℓ^j+A_4m^im^j, where A_1,  A_2,  A_3 and   A_4 are smooth function on 𝒜.
 Consequently,  the inverse of the metric tensor g_ij, given by (<ref>), can be written as
 
    g^ij=e^-2ϕ[A_1ℓ^iℓ^j+A_2ℓ^im^j+A_3m^iℓ^j+A_4m^im^j].

To determine the functions A_1,  A_2,  A_3 and   A_4, we calculate
 
    g^irg_rj =δ_j^i=[A_1ℓ^iℓ^r+A_2ℓ^im^r+A_3m^iℓ^r+A_4m^im^r][ ℓ_rℓ_j+ϕ_;2(ℓ_rm_j+ℓ_jm_r)+(σ+ε) m_rm_j].

Using Lemma <ref> ((i)-(iii)), we get

    0=      (A_1+εϕ_;2 A_2-1)ℓ_jℓ^i+(A_3+εϕ_;2  A_4)ℓ_jm^i+(ϕ_;2 A_1+ε A_2(σ+ε))m_jℓ^i
         +(ϕ_;2  A_3+ε A_4(σ+ε)-ε) m_jm^i.

 Contracting (<ref>) by ℓ^iℓ_j, m^im_j, ℓ^im_j and  m^iℓ_j, respectively, we get the following equations:

    A_1+εϕ A_2   =1,   
        A_3+εϕ_;2  A_4  =0,   ϕ_;2 A_1+ε(σ+ε) A_2   =0,   ϕ_;2  A_3+ε(σ+ε) A_4  =ε.

 Solving the system  (<ref>) yields
 
    A_1=σ +ε/σ +ε-(ϕ_;2)^2, A_2=A_3= -εϕ_;2/σ +ε-(ϕ_;2)^2,A_4=1/σ +ε-(ϕ_;2)^2.

Substituting  A_1,  A_2,  A_3 and   A_4 into (<ref>), noting that ρ: =1/σ +ε-(ϕ_;2)^2, we obtain

    g^ij=e^-2ϕ[ρ(σ+ε)ℓ^iℓ^j
    -εϕ_;2 ρ(ℓ^im^j+ℓ^jm^i)+ρ m^im^j].

By (<ref>),  we get

    g^ij=e^-2ϕ[ερ g^ij+σρℓ^iℓ^j
    -εϕ_;2 ρ(ℓ^im^j+ℓ^jm^i)].



    



Let us end this section by listing some properties of the functions ϕ,  ρ, σ and ℐ for subsequent use. 


As  ϕ_;2, σ, ρ and ℐ are h(0)-functions, the following relations hold:
  
    
  (a)σ=ϕ_;2;2+εℐϕ_;2+2(ϕ_;2)^2,  ρ=1σ+ε-(ϕ_;2)^2=1ε+ϕ_;2;2+εℐϕ_;2+(ϕ_;2)^2.
    
  (b) F∂̇_iϕ_;2=ϕ_;2;2 m_i,  F∂̇_iσ=σ_;2m_i,    F∂̇_iρ=ρ_;2 m_i,   F∂̇_iℐ=ℐ_;2 m_i.  
  







§ TRANSFORMATION OF FUNDAMENTAL GEOMETRIC OBJECTS 


 From now on, we consider the anisotropic conformal transformation of  conic pseudo-Finsler metric F given by
 
    F(x,y)=e^ϕ(x,y)F(x,y),   with   σ+ε-(ϕ_;2)^2≠0.

 
We calculate the anisotropic conformal change (<ref>) of some  important geometric objects. Namely, we find the transformation of modified Berwald frame, angular metric, Cartan tensor, main scalar, geodesic spray, Barthel connection and  Berwald connection.


  Under the anisotropic conformal transformation (<ref>), we obtain
  
    
  (i) ℓ_i=e^ϕ[ℓ_i+ϕ_;2  m_i],     ℓ^i=e^-ϕℓ^i,
    
  (ii)h _ij=e^2ϕ[h_ij+(σ-(ϕ_;2)^2) m_im_j],
    
  (iii) m_i=e^ϕ√(ε/ρ) m_i,     m^j=e^-ϕ√(ερ)[m^j-εϕ_;2 ℓ^j],
    
  (iv) m_im^i=ε,     ℓ_iℓ^i=1,     ℓ^im_i=ℓ_im^i=0.
  


(i) Since F=e^ϕF, by (<ref>), we get

    ∂̇_̇i̇F=ℓ_i=e^ϕF∂̇_̇i̇ϕ +e^ϕ ∂̇_̇i̇F=e^ϕ[ℓ_i+F∂̇_̇i̇ϕ]=e^ϕ[ℓ_i+ϕ_;2m_i].

Then from (<ref>), we obtain

     ℓ^j  :=g^ij ℓ_i   =e^-ϕ[ρ(σ+ε)ℓ^iℓ^j-εϕ_;2 ρ(ℓ^im^j+ℓ^jm^i)+ρm^im^j](ℓ_i+ϕ_;2  m_i) 

         =e^-ϕρ ℓ^j(σ+ε-(ϕ_;2)^2)=e^-ϕ ℓ^j.


(ii) Differentiating (<ref>) with respect to y^j, we get

    ∂̇_̇i̇∂̇_̇j̇F= e^ϕ[F(∂̇_̇i̇ϕ)(∂̇_̇j̇ϕ)+F∂̇_̇i̇∂̇_̇j̇ϕ+(∂̇_̇i̇F)∂̇_̇j̇ϕ+(∂̇_̇j̇F)∂̇_̇i̇ϕ+∂̇_̇i̇∂̇_̇j̇F].

From (<ref>) and (<ref>), we obtain 

    h_ij  =F ∂̇_̇i̇∂̇_̇j̇F =e^2ϕ[h_ij+(σ-(ϕ_;2)^2)m_im_j].


(iii) From  (ii), we have
 εm_im_j =e^2ϕ[ σ+ε-(ϕ_;2)^2] m_im_j =  e^2ϕ/ρ  m_im_j.

 Hence,  m_i=√(ε/ρ) e^ϕ m_i.
 But m^i=g^ijm_j, then from (<ref>), we have
    
    m^j   = √(ερ) e^-ϕ [m^j-εϕ_;2 ℓ^j].

    
(iv) It follows from (i) and (iii).
  

    

 





In view of Proposition <ref> (iv), we have the following corollary.

Let (M,F) be a conic pseudo-Finsler surface. The anisotropic conformal transformation  of a modified Berwald frame is a modified Berwald frame.


Now, we find the anisotropic conformal transformation of some  non-Riemannian quantities.
 
Under the anisotropic conformal transformation (<ref>), we  get
  
    
  (i) F C_ijk = e^2ϕF C_ijk+e^2ϕ[εℐσ+1/2σ_;2+ϕ_;2(σ+2ε)]m_im_jm_k

  =e^2ϕ[ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε)]m_im_jm_k,
    
  (ii) ℐ=(ερ)^3/2[ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε)],
    
  (iii)FC_jk^i=ερ[F C_jk^i+{εℐσ+1/2σ_;2+ϕ_;2(σ+2ε)} m^i-εϕ_;2{ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε)}ℓ^i]m_jm_k.
      
    

 
(i) Differentiating (<ref>) with respect to y^k, we have 
  
    2C_ijk =   ∂̇_̇k̇(g_ij)

    =   2e^2ϕ∂̇_kϕ[ℓ_iℓ_j+ϕ_;2(ℓ_im_j+ℓ_jm_i)+(σ+ε)m_im_j]

        +e^2ϕ∂̇_k[g_ij+ϕ_;2 (ℓ_im_j+ℓ_jm_i)+σm_im_j]. 
     
    Multiplying both sides of the above equation by F and using Lemma <ref> (iv) and (v), we have
       
    2FC_ijk     =e^2ϕ [2ϕ_;2ℓ_iℓ_jm_k+2(ϕ_;2)^2 (ℓ_im_jm_k+ℓ_jm_im_k)+2ϕ_;2(σ+ε)m_im_jm_k] 

          +2Fe^2ϕC_ijk+e^2ϕ[ϕ_;2;2(ℓ_im_jm_k+ℓ_jm_im_k)+ϕ_;2(εm_im_jm_k-ℓ_iℓ_jm_k

           +εℐℓ_im_jm_k+εm_im_jm_k -ℓ_iℓ_jm_k+εℐℓ_jm_im_k)+σ_;2m_im_jm_k 

          +σ(-ℓ_im_jm_k+εℐm_im_jm_k-ℓ_jm_im_k+εℐm_im_jm_k)].
  
  By Remark <ref>, we get

    FC_ijk =F e^2ϕC_ijk+e^2ϕ(εℐσ+1/2σ_;2+ϕ_;2(σ+2ε))m_im_jm_k.


(ii)  Since we have F C_ijk=ℐ m_i m_j m_k, then by (i)
and Proposition <ref> (iii), we obtain

    ℐe^3ϕ(  ε/ρ) ^3/2m_im_jm_k=e^3ϕ[ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε)]m_im_jm_k.

That is, 

    ℐ=(ερ)^3/2[ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε)].


(iii) By using (<ref>) and  (i), we can write
    
    FC_jk^i  =   g^ihC_hjk =[ερ g^ih+σρℓ^iℓ^h
    -εϕ_;2ρ(ℓ^im^h+ℓ^hm^i)] 
                  [F C_ijk+(εℐσ+1/2σ_;2 +ϕ_;2(σ+2ε))m_im_jm_k]
    
         =   ερ[F C_jk^i+(εℐσ+1/2σ_;2+ϕ_;2(σ+2ε))m^im_jm_k
         -εϕ_;2 (ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε))ℓ^im_jm_k].

  	

    

 




(a) Substituting  C^i_jk= g^ihC_hjk=ℐ/Fm^im_jm_k in Proposition <ref>  (iii), we get
  
    FC^i_jk=ρ(ℐ(1+εσ)+1/2σ_;2+ϕ_;2(σ+2ε))(ε m^im_jm_k-ϕ_;2ℓ^im_jm_k).
 
  
(b) From Lemma <ref> (v)  and Proposition <ref>, we can get another equivalent formula of the main scalar ℐ of F   as follows
 
    ℐ=√(ερ) [ℐ+2εϕ_;2-ερ_;2/2ρ].

The equivalence of Proposition <ref> (ii) and (<ref>) can be proven easily.

(c) From (<ref>), ℐ=√(ερ)ℐ if and only if  4ρϕ_;2=ρ_;2. Hence, a necessary and sufficient condition for the property of being Riemannian space to be preserved under anisotropic conformal change is that   
    4ρϕ_;2=ρ_;2.



In the following we find the change of the geodesic spray under the anisotropic conformal transformation (<ref>)  and  consequently we find  the transformation of the nonlinear connection and Berwald connection.
 

Under the anisotropic conformal transformation (<ref>),  the geodesic spray coefficients G^i  and G^i of F and F, respectively, are related by

    G^i= G^i+Q  m^i+P ℓ^i,

where

    P=   1/4[-ϕ_;2 A+2 F^2(∂_kϕ)ℓ^k]),     Q=1/4[ε A+2ε F(∂_kF)m^k-2F(∂_kF)_;2ℓ^k],
    
    A=   {2Fϕ_;2 ρ (F∂_kϕ+∂_kF)+2Fρ(F∂_kϕ+∂_kF)_;2}ℓ^k-2ε Fρ (F∂_kϕ+∂_kF)m^k.



Since F^2=e^2ϕF^2, we get
∂_kF^2 =2Fe^2ϕ[F ∂_kϕ+∂_kF].  Thus,

    ∂̇_j∂_kF^2=e^2ϕ[4ϕ_;2( F∂_kϕ+∂_kF)m_j+4F(∂_kϕ)ℓ_j+2F(∂_kϕ)_;2 m_j
      +∂̇_j∂_kF^2].

Consequently,

  y^k∂̇_j∂_kF^2-∂_jF^2     =e^2ϕ[y^k∂̇_j∂_kF^2-∂_jF^2+(4ϕ_;2(  F^2∂_kϕ+F∂_kF)+2F^2(∂_kϕ)_;2)m_jℓ^k
  

        +4F^2(∂_kϕ)ℓ_jℓ^k-2F^2∂_jϕ].

Since ∂_kF^2 is h(2),  Eqn. (<ref>)  can be written as

  y^k∂̇_j∂_kF^2-∂_jF^2      =e^2ϕ[{4ϕ_;2(F^2∂_kϕ+F∂_kF)+2F(F∂_kϕ+∂_kF)_;2}m_jℓ^k
  

        +4F(F∂_kϕ+∂_kF)ℓ_jℓ^k -2F(F∂_jϕ+∂_jF)].

From (<ref>), (<ref>) and (<ref>), we get

G^i  =   1/4g^ij(y^k ∂̇_j∂_kF^2-∂_jF^2) 

    =    G^i+1/4[{-2F(ϕ_;2)^2ρ(F∂_kϕ+∂_kF)-2Fϕ_;2ρ(F∂_kϕ+∂_kF)_;2+2F^2∂_kϕ}ℓ^k
  

    +2εFρϕ_;2(F∂_kϕ+∂_kF)m^k]ℓ^i
+1/4[{2εFϕ_;2ρ(F∂_kϕ+∂_kF)+2εFρ(F∂_kϕ+∂_kF)_;2
  

    -2F(∂_kF)_;2}ℓ^k+{-2Fρ(F∂_kϕ+∂_kF)+2εF∂_kF}m^k]m^i.

Hence,  the result follows.



Using δ_kϕ=∂_k ϕ-G_k^i∂̇_iϕ and δ_kF^2=0, we obtain the following technical equalities:

    .[                                                                                (a)                                                  F^2ℓ^k ∂_kϕ=F^2ϕ_,1+2G^kϕ_;2 m_k,;                                                                                (b)                                            F m^k ∂_kϕ=ε F ϕ_,2+G^i_k ϕ_;2 m^k m_i,;                                                                                (c)                                                             ℓ^k ∂_k F^2=4 G^k ℓ_k,;                                                                                (d)                                                        m^k∂_k F^2=2FG_k^i ℓ_i m^k,;                                                                                (e) F^2ℓ^k(∂_kϕ)_;2=F^2 ϕ_,1;2+ε F G^k_i ϕ_;2 m_km^i+2G^k ϕ_;2;2 m_k+2εϕ_;2 ℐ  G^k m_k;                                                                                                                                               -2G^k ϕ_;2 ℓ_k-F^2 ϕ_,2;                                                                                (f)                                 ℓ^k(∂_k F^2)_;2=2ε F  G^k_i ℓ_k  m^i+4ε  G^k  m_k. ]}

Subsituting (<ref>) into the expressions of P and Q given by (<ref>), we obtain the following short and compact expressions:

    2Q   =ερ F^2(ϕ_;2 ϕ_,1+ϕ_,1;2-2ϕ_,2),
    
    2P   =-ρ F^2ϕ_;2(ϕ_;2 ϕ_,1+ϕ_,1;2-2ϕ_,2)+F^2ϕ_,1.

Moreover, from (<ref>) and (<ref>), we get

    2ε ϕ_;2 Q+2P=F^2ϕ_,1.


Hence, we get the following equivalent form of Proposition <ref>.

Under the anisotropic conformal transformation (<ref>),   the geodesic spray coefficients G^i  and G^i of F and F, respectively,  are related by

    G^i=G^i+Qm^i+(1/2 F^2ϕ_,1-εϕ_;2 Q)ℓ^i,

where Q is given by (<ref>).




(a) As a direct consequence of  (<ref>), the geodesic spray S associated with F has the form
  
    S=S-2(P ℓ^i+Q  m^i) ∂̇_̇i̇.

   
(b) From (<ref>), we get G^i=G^i if and only if Q=0 and ϕ_,1=0.

   
  

Under the anisotropic conformal transformation (<ref>), the coefficients of the Barthel connections  G^i_j  and G^i_j associated with F and F, respectively, are related by

    FG_j^i =FG_j^i+{ 2Pℓ^iℓ_j+(P_;2-Q)ℓ^im_j+2Qℓ_jm^i+(ε P+Q_;2-εℐQ)m^im_j}.



The proof is obtained directly by differentiating (<ref>)  with respect to y^j and using  Lemma <ref>.



Under the anisotropic conformal transformation (<ref>),   the coefficients of the Berwald connection  G^i_jk  and G^i_jk associated with F and F, respectively, are related by

    F^2 G^i_jk =   F^2 G^i_jk+(2Pℓ^i+2Qm^i)ℓ_jℓ_k+{(P_;2-Q)ℓ^i+(ε P+ Q_;2-εℐ Q)m^i}(ℓ_jm_k+ℓ_km_j)
       +{(ε P+P_;2;2-2Q_;2+εℐP_;2)ℓ^i+(2ε P_;2
      +ε Q+ Q_;2;2-εℐ_;2Q-εℐ Q_;2)m^i}m_jm_k.



It follows from (<ref>) by   differentiating G^i_j with respect to y^k and using  Lemma <ref>. 


 Geometric properties or geometric objects which are preserved under the anisotropic conformal transformation are said to be anisotropic conformal invariants. In general, the geodesic spray is not invariant under the anisotropic conformal change. 
 In the (isotropic) conformal transformation  F(x,y) = e^ϕ(x) F(x,y), the homothetic transformation is the only case that leads to unchanged geodesic spray  <cit.>. For the anisotropic conformal transformation (<ref>), the geodesic spray is not invariant in general. Nevertheless, we have 
 

  Let (M,F) be a conic pseudo-Finsler surface.  Under the anisotropic conformal transformation (<ref>),  the following assertions are equivalent:
 
   
  (i) S=S, that is, P=Q=0, where P and Q are given by (<ref>) and (<ref>), respectively.
   
  (ii) δ_iϕ=0, or equivalently, d_hϕ=0.
 



(i)⟹(ii):  F=e^ϕF implies
 
    δ_iF=δ_i(e^ϕF)=e^ϕF δ_iϕ +e^ϕ δ_iF=Fδ_iϕ +e^ϕ δ_iF.

 It is known that F is horizontally constant, that is,  δ_iF=0. By assumption S=S and so δ_i=δ_i, then 
    δ_iF=δ_iF=0.

  Thereby, (<ref>) gives Fδ_iϕ =0. Hence, δ_iϕ=0.


  (ii)⟹(i):
Suppose δ_iϕ=0. Thus, Lemma <ref> (vii) gives 

    ϕ_,1ℓ_i+ϕ_,2m_i=0.

Contracting both sides of (<ref>) by ℓ^i and m^i, respectively, we obtain  ϕ_,1=ϕ_,2=0. Then, P=0 and Q=0  by (<ref>) and (<ref>). From (<ref>),  G^i=G^i and so S=S.  



Let (M,F) be a conic pseudo-Finsler surface.  Under the anisotropic conformal transformation (<ref>),  if δ_iϕ=0, then the Barthel connection G^i_j and Berwald connection G^i_jk are invariant.

 

Let (M,F) be a conic pseudo-Finsler surface.  Under the anisotropic conformal transformation (<ref>), the following assertions are equivalent:


  (i)G^i=G^i.         (ii) G^i_j=G^i_j.         (iii) G^i_jk=G^i_jk.



(i)⟹(ii) and (ii)⟹(iii) follow directly by  differentiating G^i with respect to y^j and G^i_j with respect to y^k, respectively.

(iii)⟹(i): From (<ref>), G^i_jk=G^i_jk implies that

    0   =(2Pℓ^i+2Qm^i)ℓ_jℓ_k+{(P_;2-Q)ℓ^i+(ε P+ Q_;2-εℐ Q)m^i}(ℓ_jm_k+ℓ_km_j)+{(ε P
       +P_;2;2-2Q_;2+εℐP_;2)ℓ^i+(2ε P_;2
      +ε Q+ Q_;2;2-εℐ_;2Q-εℐ Q_;2)m^i}m_jm_k.

Contracting (<ref>) by ℓ_iℓ^jℓ^k and m_iℓ^jℓ^k, respectively, we get Q=P=0.  Hence, by Theorem <ref>,  the coefficients of the geodesic spray  are invariant. 
  

The following example provides a non-trivial anisotropic conformal transformation which leaves the geodesic spray invariant.  A Maple's code of the detailed calculations is found in the Appendix at the end of the paper.
 


Let M=𝔹^2⊂ℝ^2, x ∈ M,  y∈ T_x𝔹^2≅ℝ^2,  a=(a_1,a_2)∈ℝ^2 and a is constant vector with |a|<1. Let

    z^i=(1+⟨ a,x⟩)y^i-⟨ a,y ⟩ x^i/⟨ a,y ⟩.

Define the Finsler metric F by

    F=⟨ a,y⟩√((z^1)^2+(z^2)^2)/(1+⟨ a,x⟩)^2.

The  geodesic spray coefficients are given by

    G^i=-⟨ a,y ⟩/1+⟨ a,x ⟩y^i.

Now, let  F= e^ϕ F=⟨ a,y⟩√((z^1)^2+(z^2)^2)(1+⟨ a,x⟩)^2 exp( √((z^1)^2+(z^2)^2)),
where ϕ=√((z^1)^2+(z^2)^2). 

One can easily check that F satisfies Theorem <ref> (which means that F is a pseudo-Finsler metric) and δ_iϕ=0. Furthermore,

    G^i=G^i=-⟨ a,y ⟩/1+⟨ a,x ⟩ y^i.







§ ANISOTROPIC CONFORMAL AND PROJECTIVE CHANGES

<cit.>
Two sprays S and Ŝon  M are projectively  equivalent (or related) if there exists an h(1)-function 𝒫: TM ⟶ℝ such that
  Ŝ=S-2𝒫 𝒞. The function 𝒫 is called the projective factor.  Locally, Ĝ^i=G^i+𝒫 y^i.

The association S⟼Ŝ=S-2𝒫 𝒞 or G⟼Ĝ^i=G^i+𝒫 y^i is said to be projective transformation (or projective change).


  Under the anisotropic conformal transformation (<ref>), the following assertions are equivalent:


  (i)the geodesic sprays S and S are projectively equivalent,

  (ii)ϕ_;2ϕ_,1+ϕ_,1;2=2ϕ_,2,

  (iii)G^i=G^i+1/2 Fϕ_,1y^i.
  


 From (<ref>) and (<ref>), the two geodesic sprays S and S̅ are projectively equivalent if and only if G^i=G^i+𝒫 y^i where  𝒫=PF and Q=0. This is equivalent to
  ϕ_;2ϕ_,1+ϕ_,1;2-2ϕ_,2=0, by  (<ref>). 
In this case (<ref>) takes the form
2P=F^2ϕ_,1 and hence  the geodesic spray of F is given by

    G^i=G^i+1/2 Fϕ_,1 y^i.



    



In view of Theorem <ref>, we have

Under the anisotropic conformal transformation (<ref>) with ϕ_,1=0, the coefficients of the geodesic spray are invariant if and only if the two geodesic sprays S and S̅ are projectively equivalent. 





<cit.>
A conic pseudo-Finsler metric F=F(x,y)  on an open subset U ⊂ℝ^n is said to be projectively flat  if any one of the following equivalent conditions is satisfied:


  (i)the geodesics of F  are straight line segments in U,

  (ii)y^j∂_j∂̇_i F=∂_iF or equivalently ∂̇_i(y^j∂_j F)=2∂_iF,

  (iii)the geodesic spray coefficients G^i are of the  form G^i=𝒫 y^i, with the  projective factor 𝒫 is given by 𝒫=y^k∂_kF2F.
  


 
Under the anisotropic conformal transformation (<ref>), we have the following:


  (i)a necessary condition for F to be  projectively flat is that  Q+ε G^k m_k=0.

  (ii)a sufficient condition for F to be  projectively flat is that  F ∂_jϕ+∂_jF=0. 
 



Since F=e^ϕF, we get

    ∂_jF=e^ϕ[F ∂_jϕ+∂_jF]

  and  
y^j∂_jF=e^ϕ[F^2 (∂_jϕ)ℓ^j+F(∂_jF)ℓ^j].
Thereby,

    ∂̇_i(y^j∂_jF)   =e^ϕ[{ϕ_;2F ∂_jϕ+ϕ_;2∂_jF+F (∂_jϕ)_;2+(∂_jF)_;2}ℓ^jm_i+{2F ∂_jϕ+2∂_jF}ℓ_iℓ^j
          +{ε F ∂_jϕ+ε∂_jF}m^jm_i].

From (<ref>) and (<ref>), we get

    ∂̇_i(y^j∂_jF)-2∂_iF=   e^ϕ[{ϕ_;2(F ∂_jϕ+∂_jF)+(F ∂_jϕ+∂_jF)_;2}ℓ^jm_i+2(F ∂_jϕ+∂_jF)ℓ_iℓ^j
       +ε (F ∂_jϕ+∂_jF)m^jm_i-2(F∂_iϕ+∂_iF)].


(i) Let F be projectively flat, i.e., ∂̇_i(y^j∂_jF)-2∂_iF=0. 
Then, the RHS of (<ref>) vanishes. Contracing the later by m^i and using (<ref>), we obtain 
    2/Fρ(Q+ε G^k m_k)=0,
 which implies that Q+ε G^k m_k=0.

(ii) The proof is obtained directly from (<ref>).



    
 



(a) <cit.> Since  0=δ_i F=∂_i F-G^r_iℓ_r, we get

    2G^iℓ_i=y^i∂_iF  and    2G^r m_r=ε F^2 𝔐/h,

where h:=√(ε𝔤) and 𝔐:=∂̇_2∂_1F-∂̇_1∂_2F.
From (<ref>), we get

    2G^i =y^r(∂_r F)ℓ^i+ F^2 𝔐/hm^i.


(b) From (a), the conic pseudo-Finsler surface is projectively flat if and only if G^i m_i=0, which is equivalent to 𝔐=∂̇_2∂_1F-∂̇_1∂_2F=0 (Hammel equation in two-dimensional spaces) and the geodesic spray coefficients have the form G^i =y^r(∂_r F)/2Fy^i. 
      



Let F be a projectively flat  conic pseudo-Finsler metric and let F=e^ϕF be the anisotropic conformal transformation  (<ref>). Then, the Finsler metric  F is projectively flat if and only if  ϕ_;2ϕ_,1+ϕ_,1;2-2ϕ_,2=0. 


By Theorem <ref>,  we have ϕ_;2ϕ_,1+ϕ_,1;2-2ϕ_,2=0  is equivalent to G^i=G^i+1/2 Fϕ_,1y^i. Since F is projectively flat, then we get

    G^i=y^k∂_kF+F^2ϕ_,12F y^i.

 By using (a)  of (<ref>), we obtain

    y^k∂_kF/2F =y^k∂_kF+y^kF ∂_kϕ/2F=y^k∂_kF+F^2ϕ_,1+2G^kϕ_;2 m_k/2F=y^k∂_kF+F^2 ϕ_,1/2F,

since G^k m_k=0, by Remark <ref>.  
  From (<ref>) and (<ref>), we get
G^i =𝒫 y^i,
with     𝒫=y^k∂_kF2F.
 This is equivalent to the projective flatness of the metric F.



Let (M,F) be a conic pseudo-Finsler surface and let F=e^ϕF be the anisotropic conformal transformation  (<ref>) with ϕ_;2ϕ_,1+ϕ_,1;2-2ϕ_,2=0. Assume that F is projectively flat. Then, the Finsler metric F is projectively flat if and only if either G^km_k=0 or the conformal factor ϕ is a function of x only.   


From Theorem <ref>,  we have ϕ_;2ϕ_,1+ϕ_,1;2-2ϕ_,2=0  is equivalent to G^i=G^i+1/2 Fϕ_,1y^i. Since F is projectively flat,  we obtain

    G^i=G^i-1/2 Fϕ_,1 y^i=y^k∂_kF+y^k F∂_kϕ-F^2 ϕ_,1/2F y^i.

 By using (a)  of (<ref>), we get
 
    G^i=y^k∂_kF+2G^k ϕ_;2 m_k/2F y^i.

From (<ref>), F is projectively flat if and only if G^k ϕ_;2m_k=0, which is equivalent to  either G^km_k=0 or the conformal factor ϕ is a function of x only.  


<cit.>
A conic pseudo-Finsler metric F=F(x,y)  on an open subset U ⊂ℝ^n is said to be dually flat  if it satisfies the following equations:

    y^j∂̇_i∂_j F^2=2∂_iF^2.
 



Under the anisotropic conformal transformation (<ref>), the following hold:


  (i)a necessary condition for F to be dually flat is that 
    2/Fρ( Q+ε  G^k m_k)+2εϕ_;2/FG^k m_k-G^i_kℓ_i m^k-ϕ_;2G^i_k m_im^k+ε F(ϕ_;2ϕ_,1-ϕ_,2)=0.


  (ii) a sufficient condition for F to be dually flat is that F ∂_jϕ+∂_jF=0. 
 


As F^2=e^2ϕF^2, then ∂_jF^2=2e^2ϕ[F^2∂_jϕ+F∂_jF].  Consequently, one can show that 

    y^j∂̇_i(∂_jF^2)-2∂_iF^2=   2Fe^2ϕ[{2ϕ_;2(F ∂_jϕ+∂_jF)+(F ∂_jϕ+∂_jF)_;2}ℓ^jm_i+2(F ∂_jϕ+∂_jF)ℓ_iℓ^j
       -2(F ∂_iϕ+∂_iF)].

(i) Let F be dually flat, that is, y^j∂̇_i(∂_jF^2)-2∂_iF^2=0. Then, the RHS of (<ref>) vanishes. Contracting the later by m^i and using (<ref>) and (<ref>), we get

    2/Fρ( Q+ε  G^k m_k)+2εϕ_;2/FG^k m_k-G^i_kℓ_i m^k-ϕ_;2G^i_k m_im^k+ε F(ϕ_;2ϕ_,1-ϕ_,2)=0.

(ii) The proof is obtained directly From (<ref>). 



    



The following example shows that the condition F ∂_jϕ+∂_jF=0 is not a necessary condition for neither projective flatness nor dual flatness of F (A Maple's code of the detailed calculations is found in the Appendix). On the other hand, the same condition F ∂_jϕ+∂_jF=0 is a sufficient condition for both projective flatness and dual flatness of F


Let F be the Klein metric on the unit ball  B^n⊂ℝ^n,
F=√(|y|^2-(|x|^2|y|^2-⟨ x, y⟩^2))1-|x|^2. We get

    F= e^ϕ F=√(|y|^2-(|x|^2|y|^2-⟨ x, y⟩^2))/1-|x|^2+⟨ x, y⟩/1-|x|^2,

where 
    ϕ=ln(1+⟨ x, y⟩/√(|y|^2-(|x|^2|y|^2-⟨ x, y⟩^2))).

F is both projectively flat and dually flat <cit.>, but F ∂_jϕ+∂_jF≠0.







§ SPECIAL CASES FOR THE ANISOTROPIC CONFORMAL FACTOR  


Based on the importance of the conformal factor ϕ(x,y) in studying the geometric objects associated with F,  we  focus our attention on studying some special cases of ϕ. We consider two cases: the function ϕ is a function of y only and the function ϕ is a function of x only. In the later case the  anisotropic conformal transformation reduces to the (isotropic) conformal transformation <cit.>.


  Let (M,F) be a conic pseudo-finsler surface and consider the anisotropic conformal transformation (<ref>) with ϕ a function of y only. Assume that F is projectively flat. Then, F is projectively flat if and only if ϕ_,2=0.


  Since ϕ is a function of y only, we get

    δ_iϕ=ϕ_,1ℓ_i+ϕ_,2m_i=-ϕ_;2/FG^r_im_r.

Multiple both sides of (<ref>) by ℓ^i, we get

    F^2ϕ_,1=-2ϕ_;2G^rm_r.

As F is projectively flat, from Remark <ref> and (<ref>), we have 

    ϕ_,1=0.

By Theorem <ref> and (<ref>), F is projectively flat  if and only if ϕ_,2=0.   
  


Let (M,F) be a conic pseudo-finsler surface and consider the anisotropic conformal transformation (<ref>) with ϕ a function of y only. Assume that F is projectively flat. Then, a necessary condition for F to be dually flat is that 

    ε F(ϕ_;2ϕ_,1-ϕ_,1)-G^i_kℓ_i m^k=0.



Since F is projectively flat, then by Theorem <ref>, we have 

    Q+ε G^km_k=0.
 
Also, ϕ is  function of y only, from (a) and (b) of (<ref>), we get

    2εϕ_;2/FG^k m_k-ϕ_;2G^i_k m_im^k+ε F(ϕ_,1-ϕ_,2)=0.

From (<ref>), (<ref>) and Theorem <ref> if  F is  dually flat, then ε F(ϕ_;2ϕ_,1-ϕ_,1)-G^i_kℓ_i m^k=0.    



  Let (M, F)  be a conic pseudo-Finsler surface and F=e^ϕF be the anisotropic conformal transformation (<ref>). If ϕ is function of x only, then

    σ=0,      ρ=ε.



  Since ϕ is independent of y, then ϕ_;2=0 and σ=ϕ_;2;2+εℐϕ_;2+2(ϕ_;2)^2=0.  Also we have ρ=1σ+ε-(ϕ_;2)^2=1ε=ε.




 Let (M,F) be a conic pseudo-Finsler surface and F(x,y)=e^ϕ(x,y)F(x,y). Then, g_ij  =  e^2ϕ g_ij if and only if ϕ is either a function of x only or is  a constant (homothetic).


Let ϕ be a function of x only and F=e^ϕF. It is obvious that g_ij  =  e^2ϕ g_ij (with no further condition). Conversely, let  g_ij  =  e^2ϕ g_ij. Then, by (<ref>), we get 
    e^2ϕ[ϕ_;2(ℓ_im_j+ℓ_jm_i)+σ m_im_j]=0.

 Contracting  both sides of the previous equation by ℓ^im^j, we obtain
ϕ_;2=0.
Hence, by (<ref>), ϕ is either a function of x only or is  a constant.


  Let F=e^ϕ F be an anisotropic conformal transformation. If ϕ is a function of x only, then
 
   
  (i) g_ij=e^2ϕg_ij,    g^ij=e^-2ϕg^ij,
   
  (ii) ℓ^i=e^-ϕℓ^i,    ℓ_i=e^ϕℓ_i,
   
  (iii)m_i=e^ϕm_i,     m^i=e^-ϕm^i,    h _ij=e^2ϕh_ij,
   
  (iv) C _ijk=e^2ϕC_ijk,    C^i _jk=C^i_jk,    ℐ=ℐ,
   
  (v)G^i=  G^i+1/2F^2(ϕ_,1ℓ^i-ϕ_,2m^i).
 

 The proof of
(i)-(iv) follow  by substituting  (<ref>)  into   Propositions <ref> and   <ref>.

   (v) If ϕ is a function of x only, then by (<ref>), we have
   
    P=1/2F^2ℓ^i∂_iϕ,      Q=-1/2ε F^2m^i∂_iϕ.

By  (1) and (2) of (<ref>) and (<ref>), where ϕ_;2=0, we get

    G^i=  G^i+1/2F^2(ϕ_,1ℓ^i-ϕ_,2m^i).

   

    



In  view of  Proposition <ref>, under the anisotropic conformal transformation with ϕ is a function of x only,  we conclude the following


  (a) S and S coincide if and only if  ϕ is homothetic.

  (b)From (v) and (<ref>), we get 

    G^i=  1/2F[(ℓ^r∂_rF+Fϕ_,1)ℓ^i+F(𝔐/h-ϕ_,2)m^i].
 

  (c)From (b) the geodesic spray of F is flat if and only if ℓ^r∂_rF+Fϕ_,1=0 and 𝔐/h=ϕ_,2.

  



§ CONCLUDING REMARKS

We end the present paper with the following comments and remarks.
In this paper, we have  investigated the notion of anisotropic conformal transformation of conic pseudo-Finsler metrics in two-dimensional manifolds using the associated modified Berwald frame. The following points are to be singled out: 



  * The anisotropic conformal change of a pseudo-Finsler metric F(x, y) does not necessarily yield a pseudo-Finsler metric F(x,y).  Consequently, we find out the necessary and sufficient condition for F(x,y) = e^ϕ(x,y)F(x, y) to be a pseudo-Finsler metric. This is a crucial result which ensures that two (conic) pseudo-Finsler surfaces may or may not be anisotropically conformally related, contrary to the pure" Finslerian case where any two Finsler metrics are anisotropically conformally related ( F=(F/F)F ). This justifies our choice of the object of study and certifies that our study is nontrivial.


  * We study some geometric objects associated with the transformed metric such as, the Berwald frame, metric tensor, Cartan tensor, inverse metric tensor, main scalar, geodesic spray, Barthel connection and Berwald connection.


  * It has been proved that the geodesic spray is an anisotropic conformal invariant if and only if the conformal factor ϕ(x,y) has the property that δ_iϕ=0 or equivalently is d_h-closed. On the other hand, it is well-known that in the (isotropic) conformal case, the geodesic spray is an (isotropic) conformal invariant if and only if the conformal factor ϕ(x) is homothetic <cit.>. As the condition of being d_h-closed is more general than the condition of being homothetic, our result includes as a special case the (isotropic) conformal result. On the other hand, as a consequence of our result, the anisotropic conformal invariance of any one of the geodesic spray, Cartan nonlinear connection or Berwald connection implies the aniotropic conformal invariance of the other ones.


  * Under the anisotropic conformal transformation F= e^ϕF, for two projectively related conic pseudo-Finsler surfaces, the pojective flatness property is preserved. Moreover, the condition F∂_jϕ+∂_jF=0 is a sufficient condition for both projective flatness and dual flatness of F, but is neither a necessary condition for projective flatness nor dual flatness of F.


  * Two interesting special cases are considered: (i) when the anisotropic conformal factor depends on position only, (ii) when the anisotropic conformal factor depends on direction only. In the first case our anisotropic conformal transformation reduces to the well-known (isotropic) conformal transformation which was initiated in <cit.> and in the second case some interesting results are obtained.


This work will be continued in a forthcoming paper:  Anisotropic conformal transformation of conic pseudo-Finsler surface, II, where further geometric properties along with special Finsler spaces will be investigated.

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[pages=1-4]Maple-code.pdf

