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We characterize those space forms of nonpositive curvature that admit one-dimensional Riemannian foliations. The hyperbolic ones are essentially the trivial line bundles over the flat ones. In particular, any such space admits a flat metric. Let M denote a complete Riemannian manifold. A (nonsingular) foliation S on M is said to be Riemannian or metric if the leaves remain locally at constant distance from each other. When the leaves are one-dimensional, S is often referred to as a flow. On generic manifolds, this rarely occurs because of strong restrictions on the curvature. The simply connected space forms of constant curvature K < 0, on the other hand, admit an abundance of such foliations. When the ambient space is no longer simply connected, some restrictions apply: on compact flat manifolds, for example, any Riemannian foliation is totally geodesic, and vice-versa [Wa]. The situation for compact hyperbolic manifolds is radically different, in that they do not admit foliations of either type [W], [Wa]. A natural question therefore is to determine which space forms of nonpositive curvature can be foliated in such a way. The purpose of this note is to establish the following Theorem. Let Mn be an n-dimensional space of constant curvature K < 0, S a metric flow on M. (1) If K = 0, then M is isometric to R xr -Rn-I, where F = rI1(M) acts diagonally by rigid motions. (2) If K <0, then M is diffeomorphic to R x (R xrRn-2), with F as in (1). In particular, M admits a flat metric. The result applies in particular to space forms that admit a totally geodesic foliation by hypersurfaces, since in this case, the orthogonal complement to the leaves is a Riemannian flow. Moreover, both statements in the above theorem have converses: for example, if M is diffeomorphic to R x (R xr Rln-2) with F as above, then there exists a hyperbolic metric on M and a flow s which is Riemannian with respect to that metric. Received by the editors March 17, 1994. 1991 Mathematics Subject Classification. Primary 53C20, 30F40.

Metric flows in space forms of nonpositive curvature