Killing vector fields of constant length on Riemannian manifolds

In this paper nontrivial Killing vector fields of constant length and corresponding flows on smooth complete Riemannian manifolds are investigated. It is proved that such a flow on symmetric space is free or induced by a free isometric action of the circle $S^1$. The properties of the set of all points with finite (infinite) period for general isometric flow on Riemannian manifolds are described. It is shown that this flow is generated by an effective almost free isometric action of the group $S^1$ if there are no points of infinite or zero period. In the last case the set of periods is at most countable and naturally generates an invariant stratification with closed totally geodesic strata; the union of all regular orbits is open connected everywhere dense subset of complete measure. Examples of unit Killing vector fields generated by almost free but not free actions of $S^1$ on Riemannian manifolds close in some sense to symmetric spaces are constructed; among them are "almost round" odd-dimensional spheres, homogeneous (non simply connected) Riemannian manifolds of constant positive sectional curvature, locally Euclidean spaces, and unit vector bundles over Riemannian manifolds. Some curvature restrictions on Riemannian manifolds admitting nontrivial Killing vector fields of constant length are obtained. Some unsolved questions are formulated.