On the number of geodesic segments connecting two points on manifolds of non-positive curvature
On the number of geodesic segments connecting two points on manifolds of non-positive curvature
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关于非正曲率流形上连接两点的测地线段数
DOI：
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发表时间：
1995
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通讯作者：
D. Morrison
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文献类型：
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作者：
Jianguo Cao;R. Hain;J. Harer;D. Morrison
1.设M是n ≥ 2维的紧致黎曼流形，其截面曲率度量的上界为χ ≤ 0（非正曲率）。本文证明了在负曲率（χ < 0）的情况下，存在由至少2n + 1条几何上不同的测地线线段连接的点对（即长度极小化）。在这类中提供例子的一类点是位于距离等于流形直径的点。该方法的一个简化版本允许我们证明，在非正曲率（χ = 0）的情况下，任何点都存在另一个点和n + 1个几何上不同的测地线段连接它们。证明中的基本要素是非正曲率和负曲率空间的基本度量性质，即它们的距离函数是凸的，并且在本文中将解释的某种意义上，甚至是几乎严格凸的。这些结果也可以被看作是对非正曲率流形的截轨迹中的点的“阶“的估计。在正曲率的情况下，情况发生了变化：对于IR 3中具有不同长度轴的椭球，最大距离处的点由两个测地线段连接，但对于球体，由无限多个测地线段连接。对于由不是由两个正交向量生成的晶格作为IR 2的商获得的平坦环面，切割轨迹中的点的最大“阶“是3。有趣的是凸多面体的情况在红外
1. Introduction Let M be a compact manifold Riemannian manifold of dimension n ≥ 2, with a metric of sectional curvature bounded above by χ ≤ 0 (non-positive curvature). In this paper we prove that in the case of negative curvature (χ < 0) on such manifolds there exist pairs of points connected by at least 2n + 1 geometrically distinct geodesic segments (i.e. length minimizing). A class of points which provide examples in this class are the points situated at distance equal to the diameter of the manifold. A simplified version of the method allows us to show that in the case of non-positive curvature (χ = 0) for any point there exist another point and n + 1 geometrically distinct geodesic segments connecting them. The essential ingredient in the proofs is the basic metric property of the spaces of non-positive and negative curvature to have their distance function convex and, in a sense that will be explained in the paper, even almost strictly convex. The results can also be seen as estimates for the " order " of the points in the cut locus for manifolds of non-positive curvature. In the case of positive curvature the situation changes: for the ellipsoid in IR 3 with axes of different lengths, the points at maximal distance are connected by two geodesic segments, but for the sphere by infinitely many geodesic segments. For the flat torus obtained as a quotient of IR 2 by a lattice not generated by two orthogonal vectors, the maximal " order " of the points in the cut locus is 3. Interesting is the situation for convex polyhedra in IR