Generalized Berwald spaces with \((\alpha, \beta)\)-metrics (Q297973)

The paper is devoted to the study of \((\alpha,\beta)\)-metrics in Finsler geometry. The authors study generalized Berwald manifold with \((\alpha,\beta)\)-metrics. The sign property notion for \((\alpha,\beta)\)-metrics is introduced and the authors study a class of \((\alpha,\beta)\)-metrics with sign property from the view point of generalized Berwald manifolds. In Section 3 they prove that the Finsler manifold \((M,F)\) is a generalized Berwald manifold if and only if there exists a covariant derivative on the manifold \(M\), which is compatible with \(\alpha\) and \(\beta\). The paper contains 7 sections. In the last section, the authors analyze Finsler functions on Lie groups using the theory developed in the first sections of this paper. In conclusion, the paper is well written and contains new and interesting results in Finsler geometry.