A new quantity in Riemann-Finsler geometry (Q2902686)

The authors introduce a new Finslerian non-Riemannian quantity \(\widehat{C}\) which is referred to as \(\widehat{C}\)-curvature. The main results of the paper are the following.NEWLINENEWLINE {1.} Let \((M,F)\) be an \(n(\geq 3)\)-dimensional Finsler manifold of scalar curvature with flag curvature \(K(x,y)\). Then \(K\) is weakly isotropic if and only if the \(\widehat{C}\)-curvature vanishes (Theorem 1.1).NEWLINENEWLINE {2.} A new proof of \textit{B. Najafi}, \textit{Z. Shen} and \textit{A. Tayebi}'s theorem [Geom. Dedicata 131, No. 1, 87--97 (2008; Zbl 1147.53020)] which is simpler than the original one (Theorem 1.2).NEWLINENEWLINE {3.} For a Finsler manifold with dimension \(n=3\), the \(\widehat{C}\)-curvature is a non-Riemannian quantity (Theorem 6.1)