On the Asymptotic Geometry of Nonpositively Curved Manifolds
On the Asymptotic Geometry of Nonpositively Curved Manifolds
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DOI：
10.1007/s000390050025
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发表时间：
1997-08
影响因子：
2.2
通讯作者：
Gerhard Knieper
中科院分区：
文献类型：
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作者：
Gerhard Knieper
在本文中，我们推导了所有承认紧商的哈达玛流形的新渐近性质。我们研究了测地球体体积的增长函数，推广了负曲率情况下 Margulis 的工作。我们证明，当且仅当哈达玛流形的秩为 1 时，增长才属于纯指数类型。一般来说，与纯指数行为存在多项式偏差，这以意想不到的方式取决于流形的秩。此外，我们获得了关于紧致 1 级空间上闭合测地线增长率的新结果。
In this paper we derive new asymptotic properties of all Hadamard manifolds admitting compact quotients. We study the growth function of the volume of geodesic spheres, generalizing the work of Margulis in the case of negative curvature. We show that the growth is of purely exponential type if and only if the Hadamard manifold is of rank 1. In general, there is a polynomial deviation from purely exponential behavior, depending in an unexpected way on the rank of the manifold. Furthermore, we obtain new results on the growth rate of closed geodesics on compact rank 1 spaces.