On Einstein square metrics. (English) Zbl 1349.53038
Finsler metrics of isotropic Ricci curvature are called Einstein (Finsler) metrics. A square metric is a Finsler metric defined by \(F=\frac{{(\alpha+\beta)}^2}{\alpha}\), where \(\alpha\) is a Riemannian metric and \(\beta\) is a 1-form. Non-Randers-type \((\alpha,\beta)\)-metrics of constant flag curvature are essentially square metrics. The main purpose of the present paper is to characterize Einstein square metrics \(F=\frac{{(\alpha+\beta)}^2}{\alpha}\) and to determine their structure. Two explicit expressions for a square metric are given. A classification of Einstein square metrics is achieved up to a classification of Einstein Riemannian metrics. It should be noted that a crucial and simple result, proved by the authors, which is used throughout the paper is the following: If the Finsler metric \(F=\frac{{(\alpha+\beta)}^2}{\alpha}\) is an Einstein metric, then \(\beta\) is closed.
Reviewer: Nabil L. Youssef (Giza)
MSC:
53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |