Conformal transformations of Cahen-Wallach spaces

Abstract

We study conformal transformations of indecomposable Lorentzian symmetric spaces of non-constant sectional curvature, the so-called Cahen-Wallach spaces. When a Cahen-Wallach space is conformally curved, its conformal transformations are homotheties. Using this we show that a conformal transformation of a conformally curved Cahen-Wallach space is essential if and only if it has a fixed point. Then we explore the possibility of properly discontinuous groups of conformal transformations acting with a compact orbit space on a conformally curved Cahen-Wallach space. We show that any such group cannot centralise an essential homothety and that for Cahen-Wallach spaces of imaginary type must be contained within the isometries.