摘要
本文用距离几河的方法证明了主要定理：对曲率为K的n维常曲率空间，其内任意n＋1个n－1维球Si（i＝1，2，…，n＋1），它们中的任一个都与其它球不交，则与Si交角为βi（i＝1，2，…，n＋1）的n－1维球一般有2（n＋1）个．当n为偶数时，它们的测地线曲率之交错和为零；当n为奇数时，此结论不成立该定理包括非欧情形，而当n＝2，βi＝1（i＝1，2，…，n＋1）时，就是WilkerJB在[1]中所证明的Krause定理。
Suppose we are given three diSjoint circles in the Euclidean plane with the property that none of them contains the other two. Then there are eight distinct circles tangent to the given three. R.M. Krause had shown that a certain alternating sum of the curvatures of these eight circles must vanish. J.B. Wilker had expressed this result in an inversively invariant way. This paper expresses this result in a n-dimensional space of constant curvature K and generalizes the tangent circles to other spheres which intersect the given spheres in any given angles. We will show that this result is tenable for even n and untenable for odd n.
出处
《应用数学学报》
CSCD
北大核心
1999年第3期376-382,共7页
Acta Mathematicae Applicatae Sinica
