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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sufficient conditions for periodicity of a Killing vector field


Author: Walter C. Lynge
Journal: Proc. Amer. Math. Soc. 38 (1973), 614-616
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-1973-0317230-3
MathSciNet review: 0317230
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Abstract: Let $ X$ be a complete Killing vector field on an $ n$-dimensional connected Riemannian manifold. Our main purpose is to show that if $ X$ has as few as $ n$ closed orbits which are located properly with respect to each other, then $ X$ must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold $ M$. We give a generalization of this characterization which applies to arbitrary complete vector fields on $ M$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0317230-3
Keywords: Killing vector field, isometry, flow, Lie algebra, Riemannian metric, Riemannian manifold, closed orbit, minimizing geodesic
Article copyright: © Copyright 1973 American Mathematical Society