On Einstein Kropina metrics

Abstract

In this paper, a characteristic condition of Einstein Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric $F=\frac{{\alpha }^2}{\beta }$ with constant Killing form $\beta$ on an n-dimensional manifold $M$, $n\ge 2$, is an Einstein metric if and only if $\alpha$ is also an Einstein metric. By using the navigation data $(h,W)$, it is proved that an n-dimensional ($n\ge 2$) Kropina metric $F=\frac{{\alpha }^2}{\beta }$ is Einstein if and only if the Riemannian metric $h$ is Einstein and $W$ is a unit Killing vector field with respect to $h$. Moreover, we show that every Einstein Kropina metric must have vanishing S-curvature, and any conformal map between Einstein Kropina metrics must be homothetic.