Some projectively flat \((\alpha, \beta)\)-metrics (Q862728)

If \(\alpha =\sqrt{\alpha_{ij}y^iy^j}\) is a Riemannian metric and \(\beta=b_iy^i\) is a \(1-\)form, a Finsler \((\alpha , \beta)-\)metric is a function \(F(\alpha , \beta)\) homogeneous of degree \(1\). This class of metrics includes Randers and Kropina metrics. In this paper the authors consider the polynomial \((\alpha , \beta)\)-metrics \(F=\alpha (1+\sum_{i=1}^4a_is^i)\), \(s=\frac{\beta}{\alpha}\) where \(a_i\) \((i=1,2,3,4)\) are constants. This is reduced to \(F=\alpha+\varepsilon \beta +\frac{2k \beta}{\alpha}-\frac{k^2 \beta ^4}{3 \alpha ^3}\). A necessary and sufficient condition for this \(F\) to be locally projectively flat is proved. Then non-trivial special solutions are found. Finally, it is shown that such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian.