Spherically Symmetric Metrics of Scalar Curvature

Abstract

In Riemannian geometry, one has the concept of sectional curvature. Its analogue in Finsler geometry is called flag curvature. It is one of important problems in Finsler geometry to study and characterize Finsler metrics of scalar (flag) curvature because Finsler metrics of scalar curvature and dimension $m\geqslant 3$ are the natural extension of Riemannian metrics of constant sectional curvature. A Finsler metric F is said to be of scalar (flag) curvature if the flag curvature κ at a point x is independent of the tangent plane P ⊆ T _xM. In general case, the flag curvature κ = κ(P, y) is a function of tangent planes P = span{y, v}⊂ T _xM and direction y ∈ P∖{0}.