On a class of weakly Einstein Finsler metrics (Q2017140)

A Finsler metric \(F\) is called an Einstein metric if its Ricci curvature is equal to a multiple of the square of the metric, i.e., \(\mathrm{Ric}=(n-1)\lambda F^2\), where \(\lambda=\lambda (x)\) is a scalar function. It is called weakly Einstein if \[ \mathrm{Ric}=(n-1)(\frac{3\theta}{F}+\lambda)F^2, \] where \(\lambda\) is a scalar function and \(\theta\) is a \(1\)-form. An \(m\)-Kropina metric is a Finsler metric of the form \(F=\alpha^{1-m}\beta^m\), \(m\neq 0, 1\), where \(\alpha\) is a Riemannian metric and \(\beta\) is a \(1\)-form. In this paper, the authors show that a weakly Einstein \(m\)-Kropina metric must be Einstein. Some interesting results on the curvature of Einstein \(m\)-Kropina metrics are also presented.