On Einstein–Kropina metrics

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=α2β with constant Killing form β on an n-dimensional manifold M  , n⩾2n⩾2, is an Einstein metric if and only if α   is also an Einstein metric. By using the navigation data (h,W)(h,W), it is proved that an n  -dimensional (n⩾2n⩾2) Kropina metric F=α2β 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.