二维紧致具有非正曲率Riemann流形Euler数的一个计算公式

A Formula of Euler Characteristic of Two Dimensional Compact Riemann Manifolds of Nonpositive Curvature

设Ｍ是二维紧致、曲率Ｋ（Ｍ）≤０的Ｒｉｅｍａｎｎ流形．对任一ｘ Ｍ，在Ｍ上类数≥３的点集非空且只有有限个点｛α１，α２，…；αｄ｝．用Ｋｊ表示αｊ的类数，即αｊ到ｘ的最短测地线的条数．那么，Ｍ的Ｅｕｌｅｒ数Ｘ（Ｍ）可以表示为：Ｘ（Ｍ）＝（ｄ＋１）＝Ｋｊ．如果Ｍ上类数２３的点只有一个，那么这个点是Ｍ上距离ｘ最远的点．

Let M be any two dimensional compact Riemann manifolds of nonpositivecurvature. Fixing x M, there is a nonempty finite set {a1, a2,..., ad} of pointshaving type number ≥ 3. Let Kj denote the type number of aj, that is the number ofall minimal geodesics from aj to x. Then X(M) = (d + 1) where X(M)is Euler characteristic of M.