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In the present paper, we find equations to characterize the projective changes between two important
  International Journal of Modelling, Simulation and Applications (IJMSA) Vol.1, No. 1 9            ),(  β α    Gauree Shanker and Vijeta Singh Department of Mathematics and Statistics Banasthali University, Banasthali Rajasthan-304022, India  ABSTRACT  In the present paper, we find equations to characterize the projective changes between two important ),(  β α  -metric which are 232 α  β α  β  β α   +++= F   (second approximate Matsumoto metric) and  β α  ~~~ += F   (Randers metric) and also between second approximate Matsumoto metric and  β α  ~~~ 2 = F   (Kropina metric), on a manifold with dimension 3 ≥ n  where α  and α  ~ are two Riemannian metrics,  β  and  β  ~  are two non-zero one forms. Moreover we consider this projective change when F has some special curvature properties.  M.S.C 2010 : 53B40, 53C60.  KEYWORDS : Finsler metric, ),(  β α  -metric, Kropina metric, Projective Change, Douglas metric and S-curvature. 1.   INTRODUCTION The Projective changes between two Finsler spaces have been researched and thought through by many geometers (see [4], [14], [15], [17]). It's been defined that two Finsler metrics on a smooth manifold M are considered to be Projectively equivalent in case they consists of the same geodesics as point sets and their geodesic coefficients is determined by the relation .),( ~ iii  y y xPGG  +=   where ),(  y xP   is supposed to be a scalar function on }0{ \  TM   with ),(),(  y xP y xP  λ λ   =   and the two Riemmanian metrics are considered to be Projectively equivalent on the condition that their spray coefficients are related by ,~ ik  xii  y yGG k  τ  α α   +=    International Journal of Modelling, Simulation and Applications (IJMSA) Vol.1, No. 1 10 here )(  x τ τ   =   represents a scalar function on the manifold  M  . Local coordinates in the tangent bundle TM  is denoted by ),(  ji  y x . In Finsler geometry ),(  β α  -metric is notified as a substantial and significant class of Finsler metrics. It can be depicted in the form ,),( α  β αφ   == ssF  ,where α   is the Riemannian metric,  β   represents one form and φ   denotes the positive ∞ C   function on the domain of definition. Exceptionally, when s 1 = φ  , the Finsler metric  β α  2 = F   is called Kropina metric. L. Berwald was the first one to introduce Kropina metric in connection with a two-dimensional Finsler space with rectilinear extremals and was studied upon by V. K. Kropina [7]. Whereas, Randers metric is regular Finsler metric, on the other hand Kropina metric is non-regular Finsler metric. Kropina metric is considered to be one of the significant and elementary Finsler metric with abundance of interesting and useful applications in physics, irreversible thermodynamics, dissipative mechanics and electron optics with a magnetic field ([6], [16]). Besides this, it has uses in applications related to control theory, relativistic field theory, developmental biology and evolution. Rapsack's paper [13] has provided us a very important and necessary result related with the projective change, which deals with the necessity and sufficiency of Projective change. H. Park and Y. Lee , in 1984 [11] studied and put limelight on the projective change between a Finsler space with ),(  β α  - metric and the associated Riemmanian metric. In similar way, numerous papers have been devoted on the topic ‘Projective change’. As we have more examples in its context like Projective change between Finsler spaces with ),(  β α  - metric, studied by S. Bacso and M. Matsumoto [2]. A class of Projectively at metrics with constant flag curvature has been researched upon by Z. Shen and Civi Yildrim in [15]. In 2009, N. Cui and Y. Shen [4] were the ones who did a deep study on projective change between Z. Shen square metric and a Randers metric. Recently in 2012, Jingjnong and Xinyue Cheng carried further the topic of projective changes between ),(  β α  - metric dealing with Randers metric and Kropina metric. 2.   PRELIMINARIES The geodesics of F   are defined by a system of 2nd order differential equations as follows, ,0,2 22 =      + dt dx xGdt  xd  ii  where ),(  y xF F   =  represents a Finsler metric. A Finsler metric on a manifold  M   is a ∞ C  -function ),0[:  ∞→ TM F   satisfies the following properties: 1. Regularity: F   is ∞ C   on }0{ \  TM  ;  International Journal of Modelling, Simulation and Applications (IJMSA) Vol.1, No. 1 11 2. Positive homogeneity: ),(),(  y xF  y xF   λ λ   =  for 0 > λ  ; 3. Strong convexity: The fundamental tensor ),(  y xg ij  is positive for all }0{ \ ),( TM  y x  ∈ ; where ).,(][ 21 2  y xF g  ji  y yij  =  The pair n F F  M   = ),(  is called Finsler space. F   is called the fundamental function and ij g  is called the fundamental tensor of the Finsler space n F  . ),(  y xGG ii =  are called spray coefficients of F  , given by [ ] [ ][ ] llm  xm y xili F  yF gG 22 41 −= . Let      ∂∂+−+∂∂+−∂∂∂∂== immiimmilk  ji jkli jkl  y yT nT  y yGnG y y y D D 1111~ 3 α α         ∂∂+−∂∂∂∂= immilk  j  y yT nT  y y y 11 3  (2.1) The tensor lk  jii jkl dxdxdx D D  ⊗⊗⊗∂= :  is called the Douglas tensor. A Finsler metric is called the Douglas metric if the Douglas tensor vanishes. It can be easily reviewed that the Douglas tensor is a projective invariant. In addition to this we have a vital fact which states that all Berwald metrics must be Douglas metrics. For a ),(  β α  -metric, ,),( α  β αφ   == ssF    where  jiij  y ya = α  represents a Riemannian metric and ii  y xb )( =  β   denotes a one form with 0 b <  β  . For      = α  β αφ  F   to be a regular Finsler metric ([1], [3]), the function )( s φ   has to be positive ∞ C   function on an open interval ),( 00 bb − satisfying, 0''22' ,0)()()()( bbsssbsss  ≤≤>−+−  φ φ φ  . One knows that Randers metric is regular on the other hand Kropina metric is not regular, still the relation 0)()()()( ''22' >−+− ssbsss  φ φ φ   is completely valid for 0 > s .  International Journal of Modelling, Simulation and Applications (IJMSA) Vol.1, No. 1 12 The geodesic coefficients of F   and α   are depicted in the form ),(  y xG i  and ),(  y xG i α  , respectively and the covariant derivative of  β   with respect to α   is denoted by  ji ji dxdxb  ⊗=∇ |  β  . Thus we have  jijii j jiiji j jiij br r bbsbbr   =−=+= :),(21:),(21: ||||  and put 000000 :,:,:,: llilil j j jiij sbs yss yr r  y yr r   ==== , etc. Importantly the geodesic coefficient ),(  y xG i  of F   is defined by, [11] iiiii br Qs yr QsQsGG )2()2( 0000000  +−++Θ++=  α ψ α α α  α   , (2.2) where kjik i j sas  = :  and '' φ φ φ  sQ −= , ])()[(2 )( ''22' ''''' φ φ φ φ  φ φ φφ φφ  sbss −+−+−=Θ , ])()[(2 ''22' '' φ φ φ φ ψ  sbs  −+−= , (2.3) The ),(  β α  - metrics of Douglas type have been illustrated in [8]. Further to find the desired results, firstly we calculate the douglas tensor of ),(  β α  - metrics. Since iiii br QsQsGG )2( 0000  +−++=  α ψ α  α  . Clearly, the sprays i G  and i G ~  are projective invariant providing the same Dou- glas tensor. Let iii br QsQsT  )2( 0000  +−+=  α ψ α  . (2.4) Then iii T GG  += α  ~ . From (2.3), we get ]2)[(])([2 000221' 022'000' r QssbssbQQssr sQ  yT T  mmm y m  −−−−−−+= ∂∂=  − α α ψ ψ  . (2.5)  International Journal of Modelling, Simulation and Applications (IJMSA) Vol.1, No. 1 13 Now, if the metrics F   and F  ~  consists of the same Douglas tensor, i.e. i jkli jkl  D D ~ = , , by definition of Douglas tensor and (2.5), we get 0)~(11~ 3 =      −+−−∂∂∂∂ im ym yiilk  j  yT T nT T  y y y mm  Thus we have a class of scalar functions, given by )(:  x H  H  i jk i jk   = , such that iim ym yii  H  yT T  nT T  mm 00 )~(11~ =−+−− , (2.6) where k  ji jk i  y y H  H   = : 00 , i T   and m y m T   are given by the relations (2.3) and (2.5) respectively. 3.   PROJECTIVE CHANGE BETWEEN TWO ),(  β α  - METRICS. For a Finsler space ),( F  M F  n = , the metric ),(  y xF F   =  is considered as a Finsler metric provided 0 b <  β    and their geodesic coefficients are given by (2.1) and (2.2). One can easily obtain the following: (a.) For  Second approximate Matsumoto metric   232 α  β α  β  β α   +++= F   , we have ;21321 322 ssssQ −−++=   ;)1)}(31(2831{2 12151261 2232 5432 sssbss ssss ++++−− −−−− =Θ   .)31(2831 31 232 sbss s ++−− += ψ   (3.1) (b.)  For Randers metric    β α  ~~~ += F  , we have ;1~ = Q   ;)1(2 1~ s +=Θ   .0~ = ψ   (3.2) (c.)  For Kropina metric    β α  ~~~ 2 = F  , we have ;21~ sQ  −=   ;~~ 2 bs −=Θ   .~21~ 2 b = ψ   (3.3) Now we discuss the projective change between two ),(  β α  -metrics,
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