On the Ricci curvature of a Randers metric of isotropic \(S\)-curvature
An \(n\)-dimensional compact manifold \(M\) is considered with a Randers metric \(F=\alpha+\beta\) and with constant \(c\) isotropic \(S\)-curvature i.e. having the form \(\mathcal S(x,y)=(n+1)c F(x,y)\). A number of Finsler manifolds of isotropic \(\mathcal S\)-curvature were given by the first author and \textit{C. Yang} [Differ. Geom. Appl. 24, No.~2, 119--129 (2006; Zbl 1094.53014)]. First an integral inequality involving the Riemannian metric of \(\alpha\), the vector field \(V\) of \(\beta\) and \(\text{Ric}(x,V)\) is proved. This leads to the main theorem saying: If \(\text{Ric}< -(n+1)c^2\), then \(F=\alpha\), i.e. \(F\) is a Riemannian metric. Also some other related theorems and corollaries are given. \textit{Z. Shen} [Math. Z. 249, No. 3, 625--639 (2005; Zbl 1076.53096)] proved that every negatively curved closed Finsler manifold with constant \(\mathcal S\)-curvature is Riemannian. In the present paper the authors weaken Shen's condition on the flag curvature, and request the Finsler metric to be of Randers type.
scientific article; zbMATH DE number 5323432
scientific article