Einstein Finsler Metrics and Killing Vector Fields on Riemannian Manifolds

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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 S³ with Ric = 2 F², Ric=0 and Ric=- 2 F², 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.