Communications of the
Korean Mathematical Society CKMS

The $(\alpha,\beta)$-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-form $\beta$; it has been sometimes treated in theoretical physics. The condition for a Finsler space with an $(\alpha,\beta)$-metric $L(\alpha,\beta)$ to be projectively flat was given by Matsumoto [11]. The present paper is devoted to studying the condition for a Finsler space with $L=\alpha^{1-r(x)}\beta^{r(x)}$ or $L=\alpha+\beta^2/\alpha$ to be projectively flat on the basis of Matsumoto's results.

Keywords: Berwald connection, special Kropina metric, Finsler space, projectively flat

MSC numbers: 53B40