关键词: Finsler度量;射影平坦, 对偶平坦

Abstract: Finsler geometry is just Riemannian geometry without quadratic restriction, and we know that the projectively flat and dually flat Finsler metrics are two of important problems in Finsler geometry. In this paper, we study a class of Finsler metrics with 3 parameters in the form $F=\alpha+\beta$, where $\alpha(x,y)=\frac{\sqrt{\kappa^2{\langle x,y\rangle}^2+\varepsilon{\mid y\mid}^2(1+\zeta{\mid x\mid}^2)}}{1+\zeta{\mid x\mid}^2}$ and
$\beta(x,y)=\frac{\kappa\langle x,y\rangle}{1+\zeta{\mid x\mid}^2}$.By using the Hamel's equations and dually flat equations, the necessary and sufficient conditions for the Finsler metrics to be projectively flat and dually flat are obtained.