On Einstein Kropina metrics

TL;DR

This paper characterizes Einstein Kropina metrics, showing they are Einstein if and only if their Riemannian components are Einstein and the associated vector fields are Killing, with implications for their curvature and conformal maps.

Contribution

It provides a complete characterization of Einstein Kropina metrics using characteristic conditions and navigation data, linking their Einstein property to Riemannian components and Killing vector fields.

Findings

A non-Riemannian Kropina metric with constant Killing form is Einstein iff its Riemannian part is Einstein.

Every Einstein Kropina metric has zero S-curvature.

Conformal maps between Einstein Kropina metrics are homothetic.

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 $M$, $n\geq 2$, 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 ($n\geq2$) 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.