Some projectively flat (α, β)-metrics

Abstract

In this paper, we study a class of Finsler metrics in the form $F = \alpha + \varepsilon \beta + 2k\tfrac{{\beta ^2 }}{\alpha } - \tfrac{{k^2 \beta ^4 }}{{3\alpha ^3 }}$, where $\alpha = \sqrt {\alpha _{ij} y^i y^j }$ is a Riemannian metric, β = b _i y ⁱ is a 1-form, and ε and k ≠ 0 are constants. We obtain a sufficient and necessary condition for F to be locally projectively flat and give the non-trivial special solutions. Moreover, it is proved that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.