Killing vector fields of constant length on locally symmetric Riemannian manifolds (Q941080)

Among others, the following results are obtained. {\parindent7mm \begin{itemize}\item[(1)] If \(M\) is a symmetric Riemannian manifold and \(X\) is a Killing field of constant length on \(M\), then the one-parameter isometry group generated by \(X\) consists of Clifford-Wolf translations. If, in particular, \(M\) is of positive sectional curvature, then the flow of \(X\) induces a free isometric \(S^1\)-action on \(M\). \item[(2)] If \(X\) is a Killing field of constant length on a symmetric Riemannian manifold, then \(X\) is either regular or it has no closed integral curves. (This is a corollary of (1).) \item[(3)] Sufficient conditions for the existence of quasi-regular Killing fields of constant length on a non simply connected Riemannian manifold are deduced. \item[(4)] A criterion is obtained for the existence of quasi-regular Killing fields of constant length on a locally Euclidean space. \item[(5)] A homogeneous Riemannian manifold of constant positive sectional curvature does not admit quasi-regular Killing fields of constant length, if and only if, \(M\) is either a Euclidean sphere or a real projective plane. \end{itemize}} Also, some open problems are listed.