The Ricci Curvature Of Spherically Symmetric Finsler Metrics Of Randers Type

In this paper,we derived the general form of spherically symmetric Finsler met-rics of Randers type,and proved that the Zelmelo navigation data of a Randers metric which is spherically symmetric is spherically symmetric,too.Further,we completed the classification theorem of spherically symmetric Randers metrics of constant flag curvature.By computing the general form of the spherically symmetric Riemann-Einstein metrics,we proved that the spherically symmetric Riemann-Einstein metrics are of con-stant sectional curvature.We then concluded that the spherically symmetric Einstein-Randers metrics must be of constant flag curvature when the dimension n?3.We also gave a classification theorem of spherically symmetric Einstein-Randers metrics when the dimension n?3.Finally,we calculated the infinitesimal vector field of the Schwarzschild metric,which is Ricci flat but not spherically symmetric in Riemannian geometry.Then we gave an example of non-spherically symmetric Einstein-Randers metric by Zermelo navigation.