Hiroshima Math. J. 44 (2014), 267–274 Topological properties of some flat Lorentzian manifolds of low cohomogeneity Reza Mirzaie (Received October 7, 2013) (Revised March 19, 2014) Abstract. We give a topological description of orbit spaces and orbits of some flat Lorentzian G-manifolds. 1. Introduction One of the important approaches to di¤erential geometry is the di¤erential geometry of G-manifolds, that is, a manifold M with a group of di¤eomor- phisms. Of particular importance is the situation when M is a Riemannian or semi-Riemannian manifold and G is a closed and connected subgroup of IsoðMÞ, the Lie group of all isometries of M. When the maximum dimension of the orbits of the action of G on M is dim M  k, then the orbit space GnM is a topological space of dimension k, and the action is said to be of cohomogeneity k. Throughout this paper, we use the symbol GðxÞ as the G-orbit in M through a point x A M. If k ¼ 0 and M is a connected Riemannian manifold, then there exists x A M such that dim GðxÞ¼ dim M. Since GðxÞ is a submanifold without boundary then it is an open submanifold of M, and since G is closed in IsoðMÞ then GðxÞ is closed in M. Thus, we get from connectivity of M that GðxÞ¼ M. So, G acts transitively on M and M is a homogeneous G-manifold. If M is a homogeneous flat Riemannian manifold then it is di¤eomorphic to R n 1  T n 2 , n 1 þ n 2 ¼ dim M [12]. If M is a connected cohomogeneity one flat Riemannian G-manifold, the orbit space is homeomorphic to one of the spaces S 1 , R or ½0; 1Þ [4]. The orbits of cohomogeneity one complete and connected flat Riemannian manifolds are studied in [10]. There is a characterization of orbits and orbit spaces of connected cohomogeneity two flat Riemannian manifolds in the series of papers [7, 8, 9]. In the present paper, we give similar results for some flat Lorentzian G-manifolds. Properness of the actions on Riemannian manifolds, plays an important role in the study of orbits and orbit spaces. In the semi-Riemannian 2010 Mathematics Subject Classification. 53C30, 57S25. Key words and phrases. Lie group, Isometry, Manifold.