Randers Metrics of Weakly Isotropic Flag Curvature

Abstract

It is still an open problem to classify Randers metrics of scalar flag curvature. However, if the flag curvature is weakly isotropic, one can determine the local metric structure. By definition, a Randers metric F = α+β on an n-dimensional manifold M is of weakly isotropic flag curvature if its flag curvature is a scalar function on TM in the following form:

$$ K = \frac{{3\theta }} {F} + \sigma , $$

((7.1))

where θ = t _i (x)y ⁱ is a 1-form and σ = σ(x) is a scalar function on M. The main method is to express a Randers metric F = α + β using navigation data (h, W). This method can be also used to investigate weak Einstein Randers metrics.