Randers Metrics with Special Riemann Curvature Properties

Abstract

As we know, if a Finsler metric is a Berwald metric, then its spray coefficients \( G^i = \frac{1} {2}\Gamma _{jk}^i \left( x \right)y^j y^k \) are quadratic in y ∈ T x M at every point x ∈ M. In this case, by (4.1), we can see that the Riemann curvature coefficients R i k are quadratic in y. Hence the Ricci curvature Ric = R m m is quadratic in y, too. Further, by definition, if the Riemann curvature coefficients R i k are quadratic in y, then the Weyl curvature coefficients W i k are quadratic in y. A natural problem is to study and characterize Randers metrics with quadratic Riemann curvature or Ricci curvature. When a Finsler metric is Riemannian, its flag curvature is independent of the flagpole. In other words, the flag curvature depends only on the flag (section). Thus the flag curvature is called the sectional curvature in Riemannian geometry. A natural problem is to study and characterize Randers metrics of sectional flag curvature. In this chapter, we will discuss the above two problems.