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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Zero points of Killing vector fields, geodesic orbits, curvature, and cut locus


Author: Walter C. Lynge
Journal: Trans. Amer. Math. Soc. 172 (1972), 501-506
MSC: Primary 53C20
MathSciNet review: 0355899
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Abstract: Let $ (M,g)$ be a compact, connected, Riemannian manifold. Let $ X$ be a Killing vector field on $ M$. $ f = g(X,X)$ is called the length function of $ X$. Let $ D$ denote the minimum of the distances from points to their cut loci on $ M$. We derive an inequality involving $ f$ which enables us to prove facts relating $ D$, the zero ponts of $ X$, orbits of $ X$ which are closed geodesics, and, applying theorems of Klingenberg, the curvature of $ M$. Then we use these results together with a further analysis of $ f$ to describe the nature of a Killing vector field in a neighborhood of an isolated zero point.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0355899-1
PII: S 0002-9947(1972)0355899-1
Keywords: Killing vector field, cut locus, geodesic orbit, sectional curvature
Article copyright: © Copyright 1972 American Mathematical Society