双曲空间中的Crofton公式

Crofton formula in hyperbolic space form

欧氏平面的Crofton公式揭示了过一条平面曲线上所有点的直线测度与曲线长度之间的关系,从而给出了一种求平面曲线弧长的近似方法.研究了n维实双曲空间中关于任意一条参数曲线段的Crofton公式.首先,将n维实双曲空间Hn+(-1)视为n+1维Minkowski空间Rn1+1中全体h-单位类时向量的集合.然后,利用n维定向线性子空间与其h-单位法向量的一一对应关系,把Hn+(-1)中的n-1维完备全测地超平面的集合转换成Rn1+1中h-单位类空向量的集合.最后,通过计算所有与一条空间曲线相交的双曲超平面的h-单位法向量所构成的集合的不变测度,得到n维实双曲空间中关于任意一条参数曲线段的Crofton公式.

The Crofton formula in the Euclidean plane reveals the relationship between the length of a plane curve segment and the measure of the set of lines intersecting with it. It can be used to compute the approximated value of the length of a complicated plane curve segment. The Crofton formula for arbitrary curve segment in the n dimensional real hyperbolic space is concerned. At first, the n dimensional real hyperbolic space H＋^n（-1） is considered as the set of all h-unit time-like vectors in the n＋1 dimensional Minkowski space R1^n＋1. Then, the set of the n-1 dimensional complete totally geodesic hypersurfaces of H^- （-- 1） is transfered to the set of the h-unit space-like vectors in R^1 via the one to one correspondence between the n dimensional oriented linear subspace and its h-unit normal vector. Finally, the Crofton formula for an arbitrary curve segment in the n dimensional real hyperbolic space is obtained via computing the invariant measure of the set of all h-unit normal vectors of the n dimensional linear spaces in n ＋ 1 dimensional Minkowski space intersecting with the given curve segment.