Geodesic completeness and stability
Geodesic completeness and stability
复制标题
测地线完整性和稳定性
DOI：
10.1017/s0305004100067347
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发表时间：
1987
影响因子：
0.8
通讯作者：
P. Ehrlich
中科院分区：
文献类型：
--
作者：
J. Beem;P. Ehrlich
（连通的）黎曼流形M是测地线完备的（即每个测地线都可以扩展为具有（−∞,+∞）域的测地线）。如果作为黎曼距离函数诱导下的度量空间，M是柯西完备的（[12],p. 138）。此外，所有紧黎曼流形都是完备的。另一方面，存在非测地线完备的紧致伪黎曼流形。例如Fierz和Jost[8]构造了2dx dy + h dy2在T2上的不完全度量。此外，Williams[16]证明了测地线完备性对于伪黎曼流形可能不稳定。如果M的度量集合具有这个性质是开放的，那么这个性质就是稳定的。这种稳定性的失效对于紧致和非紧致流形都可能发生。特别地，Williams发现M = S1 x S1 = 1 /(2πZ x 2πZ)，它可以给定完整的洛伦兹度量g = 2dx dy，也允许洛伦兹度量gn = 2dx dy+(sin(x)/n) dy2，它是测地线不完全的，但对于大n，它在Whitney精细Cr拓扑中任意接近g。在这个例子中，集合S = {(0, y)|0≤y≤2π}表示（M, g）和所有（M, gn）的封闭零测地线的像。然而，对于度量（M, gn），这个封闭的零测地线具有仿射参数，而仿射参数是不完全的，因为这个零测地线的速度矢量在s的每一次完整循环后返回到它自身的标量倍。通常，如果一个封闭的零测地线在其图像的每一次循环后其速度矢量完全返回到它自己，那么它就是完整的。
A (connected) Riemannian manifold M is geodesically complete (i.e. each geodesic may be extended to a geodesic with domain ( −∞, + ∞)) iff, as a metric space under the induced Riemannian distance function, M is Cauchy complete ([12], p. 138). Furthermore, all compact Riemannian manifolds are complete. On the other hand, compact pseudo-Riemannian manifolds exist which are not geodesically complete. For example Fierz and Jost[8] have constructed incomplete metrics of the form 2dx dy + h dy2 on T2. Furthermore, Williams [16] has shown that geodesic completeness may fail to be stable for pseudo-Riemannian manifolds. Here a property is said to be stable if the set of metrics for M with this property is open. This failure of stability may occur for both compact and non-compact manifolds. In particular, Williams has found that M = S1 x S1 = ℝ/(2πZ x 2πZ), which may be given the complete Lorentzian metric g = 2dx dy, also admits Lorentzian metrics gn = 2dx dy+(sin(x)/n) dy2 which are geodesically incomplete yet which for large n are arbitrarily close to g in the Whitney fine Cr topologies. In this example the set S = {(0, y)|0 ≤ y ≤ 2π} represents the image of a closed null geodesic for (M, g) and also for all (M, gn). However, for the metrics (M, gn) this closed null geodesic has affine parametrizations which are incomplete because the velocity vector of this null geodesic returns to a scalar multiple of itself after each complete circuit of S. In general, a closed null geodesic will be complete iff its velocity vector returns exactly to itself after each trip around its image.
DOI：
10.1017/9781009253161
发表时间：
2023-02
期刊：
--
影响因子：
--
作者：
S. Hawking;G. Ellis
通讯作者：
S. Hawking;G. Ellis