积分几何中两个定理的新证明

A New Proof of Two Theorems in Integral Geometry

使用李群的结构常数刻画出李群以及齐性空间上运动密度存在的条件.利用外微分理论中的嘉当同伦公式以及李群上不变微分式的性质重新证明了李群上存在运动密度的充要条件.通过直接使用外微分运算得到齐性空间上运动密度存在的充要条件的新证明.

The conditions for the existence of the kinematic density on the Lie groups and the homogenous spaces can be represented by the structure constants of the Lie groups.By using the Cartan homotopy formula and a property of the right invariant differential forms in the theory of exeter differential forms,a new proof of the kinematic density existence theorem of the Lie groups is given.Moreover,by direct differentiation,the existence of the kinematic density of a homogenous space is reproved.