Abstract. We give the ®rst examples of contact metric spaces which are weakly locally 9-symmetric (that is, rR 0 on horizontal vectors), but not strongly (that is, not all re¯ections with respect to the characteristic lines are local isometries). These examples are three-dimensional non-unimodular Lie groups with a left-invariant contact metric structure. We exhibit additional symmetries on these spaces.1991 Mathematics Subject Classi®cation: 53B20, 53C15, 53C25.

Examples of a three- and a four-dimensional Lorentz manifold are presented which are curvature homogeneous up to order one, without being locally homogeneous, in contrast to the situation in the Riemannian case, where a curvature homogeneity up to order one implies local homogeneity in the three- and four-dimensional cases. It is further shown that these manifolds satisfy the property that all scalar curvature invariants vanish identically, i.e. are those of a flat Lorentz manifold. As an immediate consequence, we also obtain examples of Lorentz manifolds whose curvature invariants are all constant, but which are not locally homogeneous, again in contrast to the Riemannian case where such manifolds are always locally homogeneous.

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A class of perfect fluid metrics with flat threedimensional hypersurfacesThe main aim of this paper is the study of (nonhomogeneous) three-dimensional Riemannian manifolds with constant principal Ricci curvatures pL=p2#p3. An error in a recent paper by McManus is pointed out and corrected, and it is shown that the techniques introduced by McManus provide a very simple method to obtain the complete local classification of these manifolds, which was first given by Kowalski.