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Fixed Points of Isometries

Published online by Cambridge University Press:  22 January 2016

Shoshichi Kobayashi
Shoshichi Kobayashi*
Affiliation:
Institute for Advanced Study, Princeton
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The purpose of this paper is to prove the following

Theorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ Vi, where the Vi’s are the connected components of F. Then (assuming F to be non-empty)

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 13 , June 1958 , pp. 63 - 68
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1958

References

1) The result of (2) is due to A. Dold and R. Thom. The proof presented here is a modification of theirs.

2) (Added in proof) We shall prove elsewhere that every totally geodesic submanifold of a homogeneous Riemannian manifold is homogeneous Riemannian.