Projectively Ricci-flat general \((\alpha, \beta)\)-metrics (Q6580085)

The notion of general \((\alpha, \beta)\)-metric was introduced in [\textit{C. Yu} and \textit{H. Zhu}, Differ. Geom. Appl. 29, No. 2, 244--254 (2011; Zbl 1215.53029)]. This is a Finsler metric \(F\) of the form \(F=\alpha\phi(b^2,\beta/\alpha),\) where \(\alpha\) is a Riemannian metric, \(\beta\) is a 1-form, \(b=||\beta_x||_\alpha\) and \(\phi(b^2,s)\) is a positive smooth function.\N\NIn the present paper, the author gives some necessary and sufficient conditions as a general \((\alpha, \beta)\)-metric \(F\), where \(\alpha\) and \(\beta\) satisfy some a priori conditions, to be projectively Ricci-flat, that is \(\mathrm{PRic}_{({G},dV)}=0\), where \({G}\) is the induced spray of \(F\) and \(dV\) is a volume form on the manifold \(M^n\).