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

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 a and 0, respectively. In particular, when v ? 0 (or v = K = 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.

Automated summary

In this paper, the authors mainly focus on the study of weakly weighted Einstein-Finsler metrics. They first prove that weakly weighted Einstein-Kropina metrics must have isotropic S-curvature with respect to the Busemann-Hausdorff volume form under certain conditions about the weight constants. Then they characterize weakly weighted Einstein-Kropina metrics completely using their navigation expressions or the values of a and 0. Particularly, when v ? 0 (or v = K = 0), and the S-curvature with respect to the Busemann-Hausdorff volume form is isotropic, they show 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.