Rank inequality in homogeneous Finsler geometry

This is a survey on some recent progress in homogeneous Finsler geometry. Three topics are discussed, the classification of positively curved homoge[1]neous Finsler spaces, the geometric and topological properties of homogeneous Finsler spaces satisfying K ≥ 0 and the (FP) condition, and the orbit number of prime closed geodesics in a compact homogeneous Finsler manifold. These topics share the same similarity that the same rank inequality, i.e., rankG ≤ rankH + 1 for G/H with com[1]pact G and H, plays an important role. In this survey, we discuss in each topic how the rank inequality is proved, explain its importance, and summarize some relevant results.