Zero points of Killing vector fields, geodesic orbits, curvature, and cut locus
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- by Walter C. Lynge PDF
- Trans. Amer. Math. Soc. 172 (1972), 501-506 Request permission
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.References
- D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
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- Shoshichi Kobayashi, Fixed points of isometries, Nagoya Math. J. 13 (1958), 63–68. MR 103508
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 172 (1972), 501-506
- MSC: Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9947-1972-0355899-1
- MathSciNet review: 0355899