IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 6 (Sep. - Oct. 2013), PP 83-89 www.iosrjournals.org www.iosrjournals.org 83 | Page Projectively Flat Finsler Space With Special (, )-Metrics P.D. Mishra & M. K. Mishra Indian Space Research Organization, INDIA. D.D.U.Gorakhpur University, Gorakhpur I. Introduction. In 1990 M. Matsumoto has considered the projectively flatness of Finsler spaces with (, )-metric [5]. In particular, the Randers space, the Kropina space and the special generalized Kropina space are considered in detail. Here is Riemannian metric and is a differential one-form. In the present chapter we consider the projective flatness of Finsler space with special (, )-metrics. In particular, Matsumoto metric 2 /( ), special generalized Matsumoto metric 2 /( ) and the metric + ( 2 /) are considered in detail. II. Projective flatness of (, )-metric. Consider a Finsler space with (, )-metric L(, ), where L is fundamental function positively homogeneous of degree one in and , j i ij y y ) x ( a is Riemannian metric and = b i (x) y i is one- form. Firstly, we are concerned with the associated Riemannian space with metric and define (1.1) (a) 2r i j = b i ; j + b j ; i = s s ij j i i j b 2 b b , (b) 2s i j = b i ; j b j ; i = j i i j b b Which are symmetric and skew symmetric tensors of order 2. Here (;) denote the covariant differentiation with respect to Riemannian Christoffel symbols ) x ( i jk . Further, we define (1.2) s i j = a i r s r j , b i = a i r b r , s i = b r s r i , b 2 = a r s b r b s . Here a i j are conjugate metric tensor of a i j . Next, we consider the Berwald connection B= ) 0 , G , G ( i j i jk of the Finsler space with the (, )- metric L(, ). As, is well known, we have i j k i jk G G , i j i j G G , F F y G g G 2 j r j r i ij j , F = L 2 /2 where g i j denote the conjugate metric tensor of metric g i j (x, y) of the Finsler space. If we put (1.3) i 00 i i G 2 B 2 , where the subscript 0 denote the contraction by y i i.e. k j i jk i 00 y y , then the equation (1.1) of [5] gives (1.4) i i i 0 i i b y 2 r C L L s L L y E B 00 , where E and C satisfy