Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1972-0355899-1
MathSciNet review: 0355899
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math., no. 55, Springer-Verlag, Berlin and New York, 1968. MR 37 #4751. MR 0229177 (37:4751)
  • [2] Robert Hermann, Differential geometry and the calculus of variations, Math. in Sci. and Engineering, vol. 49, Academic Press, New York, 1968. MR 38 #1635. MR 0233313 (38:1635)
  • [3] Shôshichi Kobayashi, Fixed points of isometries, Nagoya Math J. 13 (1958), 63-68. MR 21 #2276. MR 0103508 (21:2276)
  • [4] J. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634. MR 0163331 (29:634)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20

Retrieve articles in all journals with MSC: 53C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0355899-1
Keywords: Killing vector field, cut locus, geodesic orbit, sectional curvature
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society