度量方程应用于Krause定理的推广

本文用距离几何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变，则与Ｓｉ交角为βｉ（ｉ＝１，２，…，ｎ＋１）的ｎ－１维一般有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ｉｉｌｋｅｒＪＢ在「１」中所

METRIC EQUTION APPLIED TO GENERALIZED FORM OF KRAUSE'S THEOREM

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.