We study the differential geometry of codimension-one totally geodesic foliations admitting Killing field and as applications we prove, among others, that any Killing field preserves a codimension-one totally geodesic foliation of a manifold of dimension 2 or 3, under certain topological conditions on the leaves.
In this paper we consider a harmonic Kähler foliation F and study the infinitesimal automorphisms of F which are either transversally holomorphic or transversally Killing. A special study is made for the case of foliations with constant transversal scalar curvature.