On some projectively flat Finsler metrics in terms of hypergeometric functions (Q1758900)

A Finsler metric \(F=F(x,y)\) on an open subset \(\mathcal{U}\subset\mathbb{R}^n\) is said to be projectively flat if all geodesics are straight line segments in \(\mathcal{U}\). This is equivalent to the equality \(G^i=P(x,y)y^i\), where \(G^i=G^i(x,y)\) are the spray coefficients of \(F\) and \(P=P(x,y)\) is given by \(P=\frac{F_{x^k}y^k}{2F}\). Finsler metrics of the form \(F=\alpha\phi(\frac{\beta}{\alpha})\) are called \((\alpha,\beta)\)-metrics where \(\alpha:=\sqrt{a_{ij}y^iy^j}\) is a Riemannian metric, \(\beta:=b_iy^i\) is a \(1-\)form, and \(\phi=\phi(s)\) is a positive smooth function defined in a neighborhood of the origin \(s=0\). When \(\phi(s)=1+s\), we get Randers metrics. In the present paper, using hypergeometric functions, the author gives an explicit construction of a family of projectively flat \((\alpha,\beta)\)-metrics through the deformation of a special class of projectively flat Randers metrics.