The Characterizations on a Class of Weakly Weighted Einstein–Finsler Metrics

Abstract

In this paper, we mainly introduce and study the weakly weighted Einstein–Finsler metrics. First, we show that weakly weighted Einstein–Kropina metrics must be of isotropic S-curvature with respect to the Busemann–Hausdorff volume form under a certain condition about the weight constants. Then we characterize weakly weighted Einstein–Kropina metrics completely via their navigation expressions or via $\alpha$ and $\beta$, respectively. In particular, when $\nu \ne 0$ (or $\nu =\kappa = 0$, respectively) and S-curvature with respect to the Busemann–Hausdorff volume form is isotropic, we prove that a Kropina metric determined by navigation data (h, W) is a weakly weighted Einstein metric if and only if the Riemann metric h is a weighted Einstein–Riemann metric.