Modification of a Finsler space by a normalized semi-parallel vector field

LetF _n be a Finsler space with metric functionF(x, y). M. Matsumoto 6] has defined a modified Finsler spaceF _n ^* whose metric functionF ^*(x, y) is given byF ^*2 = = F² + (X_i(x)yⁱ)², whereX _i are the components of a covariant vector which is a function of coordintae only. Since a concurrent vector is a function of coordinate only, Matsumoto and Eguchi 9] have studied various properties of the modified Finsler spaceF _n ^* under the assumption thatX _i are the components of a concurrent vector field inF _n. In this paper we shall introduce the concept of semi-parallel vector field inF _n and study the properties of modified Finsler spaceF _n ^* .