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 = ∪ V i , where the V i ’s are the connected components of F. Then (assuming F to be non-empty )
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Shôshichi Kobayashi (1958) studied this question.