A Lorentz manifold with a group of conformal transformations containing a normal subgroup of homotheties (Q1903874)

The author considers a Lorentzian manifold \((M, g)\) of dimension \(n\) which admits a group \(H\) of homotheties such that \(H\) preserves a point \(p\in M\) and it is a normal subgroup in the full group of conformal transformations of \(M\). The main result is that such a manifold is the manifold with the plane-wave metric. In particular, \(M\) admits a fibering over \(\mathbb{R}\) or \(S^1\) with totally geodesic fibers isometric to \(\mathbb{R}^{n- 1}\) with semi-Euclidean metric. Homogeneous (with respect to isometry group) manifolds of this class are characterized and their full group of conformal transformations is described. These results generalize a result of \textit{C. Barbance} [C. R. Acad. Sci., Paris, Sér. A 291, 342-350 (1980; Zbl 0455.53045)].