In 1980 Johnson and Whitt (see ) proved that any Killing field preserves a codimension-one totally geodesic foliation by compact leaves. In 1983 Oshikiri (see ) proved a similar result when the manifold is compact and later (see )he generalized this result to Killing fields with bounded length.
Our goal in this paper is a result of preservation for codimension..one totally geodesic foliations, having at least one compact leaf. We give a proof of this result in Section 3, but we first need to study these foliations carefully. Thus, in Section 1 we are concerned with some general properties about Killing fields on such foliations; and use the universal covering of a manifold with an induced foliation. We also consider in this section the dynamical topology of foliations having at least one compact leaf. In Section 2 we find dosed geodesics associated to any element of the fundamental group of the compact leaf, which, with the assumption of having non-trivial holonomy, will be used in Section 3 in proving the main result. Finally in Section 4 some examples of such foliations are given.
I would like to express my gratitude to G.-I. Oshikiri for his useful comments and suggestions about this work.
1. PRELIMINARY RESULTS
This section is concerned with some general properties of complete Riemannian manifolds with a codimension-one totally geodesic foliation. For further details, see . The first part is devoted by study Killing vector fields on such manifolds and the second to their dynamical topology.
From now on (M, A¢) will denote a complete Riemannian manifold with a codimension-one totally geodesic foliation A a. As usual, the Riemannian metric will be written as g. We recall (see or ) that if (2~, 70 denotes the universal covering of M, 7~ the canonical projection and # the canonical lifting of A a, then 2~t is isometric to a trivially foliated Riemannian manifold L x R, where L is the universal covering of any leaf of .W and the metric is given by = ds~ + f2 dt 2. *Partially supported by CAICYT 1085-84.