Lectures on closed geodesics
Lectures on closed geodesics
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DOI：
10.1007/978-3-642-61881-9
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发表时间：
1978
期刊：
影响因子：
--
通讯作者：
W. Klingenberg
中科院分区：
文献类型：
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作者：
W. Klingenberg
自上世纪全球微分几何开始以来，黎曼流形上封闭测地线的存在性以及测地流中相应周期轨道的性质问题一直是深入研究的对象。最简单的情况发生在负曲率的封闭曲面上。这里，基本群非常大，并且如 Hadamard [Had] 在 1898 年所示，每个非零同伦 c10sed 曲线都可以变形为其自由同伦类中具有最小长度的 c10sed 曲线。根据参数化，这条最小曲线是唯一确定的，并且代表一条 c10sed 测地线。简单连通的闭合曲面上是否存在闭合测地线的问题要困难得多。正如庞加莱[po 1]于1905年所指出的，这个问题与受限三体问题中周期轨道的存在性问题有很多共同之处。庞加莱[lc]概述了一个证明，即在与标准球面相差不大的解析凸面上，总是存在至少一个椭圆型闭测地线，即测地流中相应的周期轨道是无穷小稳定的。
The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [lc] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, ie, the corres ponding periodic orbit in the geodesic flow is infinitesimally stable.