In this paper, we study left invariant ða; bÞ-metrics on four-dimensional real Lie groups equipped with left invariant Einstein Riemannian metrics. We classify all left invariant ða; bÞ-metrics of Berwald type induced by a left invariant Einstein Riemannian metric and a left invariant vector ﬁeld and show that all of them are locally Minkowskian. All left invariant Randers metrics of Douglas type, and all Einstein Kropina metrics induced by a left invariant Riemannian metric and a left invariant vector ﬁeld, are classiﬁed. Finally, the ﬂag curvatures of these spaces are investigated and in a special case the geodesics are computed. Keywords Flag curvature Lie group (a, b)-metric Douglas spaces Mathematics Subject Classiﬁcation 53C30 53C60 22E60 1 Introduction In this article we study left invariant ða; bÞ-metrics of Berwald type, left invariant Randers metrics of Douglas Among Finsler metrics, ða; bÞ-metrics are very interesting, type and the classiﬁcation of left invariant Einstein Kropina since they are computable and have many applications. For metrics on four-dimensional Lie groups equipped with left example, Randers metric, which is a special type of ða; bÞ- invariant Einstein Riemannian metrics. metrics, has been introduced because of its applications in Here,
Iranian Journal of Science and Technology, Transactions A: Science – Springer Journals
Published: Jun 4, 2018
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