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Geodesic completeness and stability

Published online by Cambridge University Press:  24 October 2008

John K. Beem  and
Paul E. Ehrlich
John K. Beem
Affiliation:
Mathematics Department, University of Missouri, Columbia, MO 65211, U.S.A.
Paul E. Ehrlich
Affiliation:
Mathematics Department, University of Missouri, Columbia, MO 65211, U.S.A.
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 319 - 328
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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