Randers metrics with special curvature properties
In Finsler geometry, besides the flag curvature \(\mathbb K\) there exist several non-Riemannian quantities: \(\tau\) (distortion),\(\mathbb I\) (the mean Cartan torsion), \(\mathbb S\) (the S--curvature), \(\mathbb E\) (the E--curvature), \(\mathbb J\) (the mean Landsberg cuvature) cf. [\textit{Zhongmin Shen}, Differential geometry of spray and Finsler spaces. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1009.53004)]. The authors study these quantities for the Randers metrics and prove Theorem 1.1: Let \(F = \alpha + \beta \) be a Randers metric on an \(n\)- dimensional manifold \(M\). (i) For a scalar function \(c=c(x)\) on \(M\), the following are equivalent: a) \({\mathbb S}= (n+1)cF\); b)\({\mathbb E} = (1/2)(n+1)cF^{-1}h\), where \(h\) is the angular metric; (ii) For a scalar function \(c=c(x)\) on \(M\), the following are equivalent: \({\mathbb J} +cF{\mathbb I}=0;{\mathbb S} = (n+1)cF \) and \(\beta\) is closed. Then they classify the Randers metrics with constant flag curvature \({\mathbb K} = \lambda\) and \({\mathbb J}/I = -c(x)F \).
scientific article; zbMATH DE number 1894628