Einstein Criterion for Finsler Space With Special $(\alpha, \beta)$-Metrics

Abstract

Einstein-Finsler metrics are very useful to study geometric structure of spacetime and to build applications in theory of relativity. In this paper, we consider the special $(\alpha, \beta)$-metric $L=\mu \frac{\alpha^2}{\beta}+\nu \frac{\beta^2}{\alpha}$ and obtained the Riemann curvature. Then we obtained the necessary and sufficient condition for that $(\alpha, \beta)$-metric to be Einstein metric, when $\beta$ is a constant killing form. Finally, we proved that the above metric is Einstein if and only if it is Ricci flat.