A g.o. Finsler manifold with K≥0 and (FP) is compact, satisfies rk g ≤ rk h +1 when G/H with compact Lie algebra, and even-dimensional cases are coset spaces with positive curvature metrics.
Helgason, Differential Geometry, Lie Groups and Symm etric Spaces, 2nd edn
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Geodesic orbit Finsler space with $K\geq0$ and the (FP) condition
A g.o. Finsler manifold with K≥0 and (FP) is compact, satisfies rk g ≤ rk h +1 when G/H with compact Lie algebra, and even-dimensional cases are coset spaces with positive curvature metrics.