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 homogeneous 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 compact 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.