A characterization of Riemannian flows

Abstract

We prove that a flow on a closed manifold is Riemannian if and only if it is locally generated by Killing vector fields for a Riemannian metric. Consider a flow F, i.e. an oriented one-dimensional foliation on a manifold M. The purpose of this note is to prove the following result. Theorem. Let M be a closed manifold. Then the flow F is Riemannian if and only if the tangent bundle of F is locally generated by Killing vector fields for a Riemannian metric g on M. If the flow F is locally generated by Killing vector fields for a metric g on M, then the flow is clearly Riemannian, and the metric g bundle-like for F in the sense of Reinhart [8]. What we wish to show is that on a closed manifold the converse also holds. It is well-known that a Riemannian flow on a closed manifold Mn+l is not necessarily defined by a global Killing vector field for a Riemannian metric on M. According to a result of Molino and Sergiescu [7], a necessary and sufficient condition for this to be the case is that HD (F) $& 0, where HD (F) is the topdimensional basic cohomolQgy vector space. Carriere [1] proved a general structure theorem for Riemannian flows, and also provided an example of a Riemannian flow on a closed oriented manifold M3 with HW (.F) = 0. Proof of the theorem. Let F be a Riemannian flow on the closed manifold M. The fundamental new fact we are going to use is the result of Dominguez [2], establishing the existence of a bundle-like metric g for which the mean curvature 1-form i, is basic. The mean curvature 1-form is dual to the mean curvature vector field, and as such vanishes on vectors tangent to F. The essential property making i, basic is that the Lie derivative O(V)s = 0 for vector fields V tangent to F. A fact pointed out in [4, (4.4)] is that under this condition ds = 0 (see also the proof in [9, (12.5)]). It follows that locally r, = df. This local function f is necessarily basic, as follows from Vf = df(V) = (V) = 0 for a vector field V tangent to F. It suffices to verify that for such a local unit vector field V the modified e-fV is a local Killing vector field for g. The property to verify is that O(e-fV)g = 0. We evaluate this bilinear form successively on the tangent bundle L of F, its g-orthogonal complement L', and Received by the editors June 19, 1996. 1991 Mathematics Subject Classification. Primary 53C12, 57R30. ?)1997 American Mathematical Society