Randers Metrics of Sectional Flag Curvature

A Finsler metric is of sectional flag curvature if its flag curvature depends only on the section. In this article, we characterize Randers metrics of sectional flag curvature. It is proved that any non-Riemannian Randers metric of sectional flag curvature must have constant flag curvature if the dimension is greater than two. 0. Introduction Finsler geometry has a long history dated from B. Riemann and P. Finsler. Recently, the study of Finsler geometry has taken on a new look. Many curvatures have been discovered, both Riemannian and non-Riemannian [Bao et al. 2000, Shen 2004]. Among them, the most important one is always the flag curvature, which is a natural generalization of the sectional curvature in Riemannian geometry. A flag planted at a base point x on the manifold M , consists of a flagpole y ∈ TxM and a section Π containing y. Typically, for Π spanned by {y, V }, the flag curvature of (y,Π) is K(x, y, V ). Since it is independent of the choice of V in Π, we also write it as K(x, y,Π). A Finsler metric is said to be of scalar flag curvature if the flag curvature depends only on the flagpole K = K(x, y). In [Mo-Shen 2005], it is verified that every closed Finsler manifold of negative scalar flag curvature must be of Randers type. Later on, Randers metrics of scalar flag curvature were characterized by [Shen-Yildirim 2005]. Now, what happens if the flag curvature is independent of the flagpole?(asked by Professor Zhongmin Shen) Such a metric is called of sectional flag curvature since the flag curvature depends only on the section, and its flag curvature can then be written as K(x,Π). 2000 Mathematics Subject Classification. 53B40, 53C60.