摘要
本文利用Ｒｎ＋Ｎ中ｎ－维紧致、连通，可定向的嵌入子流形Ｉ（Ｍｎ）的Ｋｉｌｉｎｇ－ｌｉｐｓｃｈｉｔｚ曲率Ｇ（Ｐ，Ｖ），定义一个Ｍｏｒｓｅ多项式Ｍｔ（Ｍｎ，Ｉ）。再利用高度函数导出Ｍｔ（Ｍｎ，Ｉ）的系数与Ｉ（Ｍｎ）上的Ｍｏｒｓｅ函数的临界点及指数间的内在联系的关系式，从而得到一个主要定理：存在非负实系数多项式Ｑ（ｔ）使得：Ｍｔ（Ｍｎ，Ｉ）－Ｐｔ（Ｍｎ，Ｉ）＝（Ｉ＋ｔ）Ｑ（ｔ）．
For any compact、oriented、n-dimensional C ∞-manifold M n embedded into R n+N , we construct a normal fibre bundle.By structure equations and Killing-Lipschitz curvature, we can introduce:a k=1ω(n+N-1)∫ B k |G(p,v(p))| d v∧∑ N-1 (where ω(n+N-1)) is the area of n+N-1-dimensional sphere, B k is the subset of S v such that the structure matrix (A v sj (I(p))) n×n ) have type number k).M t(M n,I)=∑nk=0a kt k (be called Morse Polynomial) With high level function, we generalized the crofton formula which is the bridge from the Morse polynomial of topological form to the Geometric form and we have: Main Theorem: There exist a polynomial with nonnegative real coefficient Q(t), Such thatM t(M n,I)-P t(M n,I)=(1+t)Q(t) Where P t(M n,I)=∑nk=0β kt k, β k= dim H k(I(M n),R).
