We study invariant
Finsler metrics on polar homogeneous manifolds. After establishing existence
results, we prove that an invariant Finsler metric on a nonsymmetric polar
homogeneous manifold of a simply connected compact simple Lie group is
Berwaldian if and only if it is Riemannian. As an application, we prove that on
each such manifold with generalized rank of at least 2, there exist infinitely
many invariant Finsler metrics that are reversible, non-Berwaldian and of
vanishing S-curvature; this kind of space is sought after in an open problem of
Shen. Finally, using one type of polar homogeneous manifold, we give a
classification of homogeneous Randers spaces with positive constant flag
curvature.
