Einstein Finsler Metrics and Killing Vector Fields on Riemannian Manifolds

Abstract

In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with $\mathrm{Ric}=2F^2$, $\mathrm{Ric}=0$ and $\mathrm{Ric}=-2F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.