Lorentzian spacetimes with constant curvature invariants in four dimensions

In this paper we investigate four dimensional Lorentzian spacetimes with constant curvature invariants ($CSI$ spacetimes). We prove that if a four dimensional spacetime is $CSI$, then either the spacetime is locally homogeneous or the spacetime is a Kundt spacetime for which there exists a frame such that the positive boost weight components of all curvature tensors vanish and the boost weight zero components are all constant. We discuss some of the properties of the Kundt-$CSI$ spacetimes and their applications. In particular, we discuss $\mathcal{I}$-symmetric spaces and degenerate Kundt $CSI$ spacetimes.