Some Considerations on the Cut Locus of a Riemannian Manifold
Some Considerations on the Cut Locus of a Riemannian Manifold
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关于黎曼流形切割轨迹的一些思考
DOI：
10.2969/aspm/00310029
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
Jin
中科院分区：
文献类型：
--
作者：
Jin
设（M，g）是n维紧致连通黎曼流形，且固定M的一点p.设rx是从p发出的测地线，其初始方向为单位X ∈ TpM.我们定义p沿着r x的截点为r x上最后一点，测地线使到该点的距离最小。p的所有割点的轨迹C（p）称为p的割轨迹，根据上述定义，M是由C（p）通过附加一个n-胞腔得到的，割轨迹包含了关于M的拓扑的基本信息。现在，确定切割轨迹的结构的问题与奇点理论有关。近年来，M。Buchner（[2] [3] [4]）.但是，由于他们的作品呼吁强大的一般理论（Hironaka的或马瑟的理论），具体结构的削减轨迹没有明确给出。另一方面，S. B. Myers（[8]）、T. Sakai（[11] [12]）和M. Takeuchi（[13]）.但对于任意度量，割轨迹可能非常复杂，例如H。Gluck，D. Singer（[5]）证明了在任何流形上都存在一个度量，它的割迹是不可三角化的。本文主要研究某些莫尔斯函数的正指标临界点的不稳定流形的并与截轨的关系。首先在第一节中，我们用关于CO-拓扑的莫尔斯函数来逼近p的距离函数，并定义集合Cl（p）为莫尔斯函数具有正指标的临界点的所有不稳定流形的极限集。则C1（p）包含在p的割轨迹中，并在一定条件下继承了M的拓扑本质.我们称C1（p）为基本割轨迹.总的来说，
Let (M, g) be a compact connected Riemannian manifold of dimension n and fix a point p of M. Let r x be a geodesic emanating from p with the unit initial direction X E TpM. We define the cut point of p along r x as the last point on r x to which the geodesic minimizes the distance. The locus C(p) of all cut points of p is called the cut locus of p. By the above definition M is obtained from C(p) by attaching an n-cell and the cut locus contains the essential informations on the topology of M. Now the problem of determining the structure of the cut locus is interesting in connection with the singularity theory. Recently in case of analytic Riemannian structures or in generic case much progress has been made by M. Buchner ([2] [3] [4]). But since their works appeal to the powerful general theory (Hironaka's or Mather's theory), the concrete structure of the cut locus is not given explicitly. On the other hand the above problem is answered for the 2-dimensional analytic case by S.B. Myers ([8]), symmetric spaces and Berger's spheres by T. Sakai ([11] [12]) and M. Takeuchi ([13]). But with respect to an arbitrary metric, the cut locus may be very complicated, for example, H. Gluck, D. Singer ([5]) showed that there exists a metric on any manifold whose cut locus is not triangurable. The main purpose of the present paper is to study the relation between the cut locus and the union of all unstable manifolds of critical points with positive index of some Morse functions. Firstly in Section 1 we approximate the distance function from p by Morse function with respect to CO-topology and define the set Cl(p) as the limit set of all unstable manifolds of critical points with positive index of Morse functions. Then C 1(p) is contained in the cut locus of p and inherits the essence of the topology of M under some conditions. We call C 1(p) the essential cut locus. In general it seems that the structure of