On conformally flat \((\alpha, \beta)\)-metrics with relatively isotropic mean Landsberg curvature (Q2875458)

The paper under review is devoted to a particular class of Finsler spaces. Namely, given a Riemannian metric \(\alpha\) and a one-form \(\beta\) on a manifold \(M\), the authors deal with an \((\alpha,\beta)\)-metric defined as a Finsler metric on \(TM\) by \(F = \alpha \phi(\frac{\beta}{\alpha})\), where \(\phi\) is a smooth function on \(M\). The main attention is paid to the conformal properties of \((\alpha, \beta)\)-metrics. It is shown that if an \((\alpha, \beta)\)-metric \(F\) on \(M\) is a conformally flat weak Landsberg metric, i.e., \(e^\sigma F\) is a Minkowski metric for some smooth function \(\sigma\) on \(M\) and its mean Landsberg curvature is vanishing, then it must be either a Riemannian metric or a locally Minkowski metric. Moreover, it is proved that if an \((\alpha, \beta)\)-metric \(F\) on \(M\) is conformally flat, \(\phi\) is a polynomial, and \(F\) is of relatively isotropic mean Landsberg curvature, i.e., its mean Cartan torsion \(I\) and mean Landsberg curvature \(J\) satisfy \(J+cFI=0\) for some smooth function \(c\) on \(M\), then it must be either a Riemannian metric or a locally Minkowski metric [\textit{G. Chen} and \textit{X. Cheng}, Int. J. Math. 24, No. 1, Paper No. 1350003, 15 p. (2013; Zbl 1274.53032)].