Why do all the curvature invariants of a gravitational wave vanish ?

We prove the theorem valid for (Pseudo)-Riemannian manifolds $V_n$: “Let $x \in V_n$ be a fixed point of a homothetic motion which is not an isometry then all curvature invariants vanish at $x$.” and get the Corollary: “All curvature invariants of the plane wave metric $$ds\text{sp}2=2dudv+a\text{sp}2(u)dw\text{sp}2+b\text{sp}2(u)dz\text{sp}2$$ identically vanish.” Analysing the proof we see: The fact that for definite signature flatness can be characterized by the vanishing of a curvature invariant, essentially rests on the compactness of the rotation group $SO(n)$. For Lorentz signature, however, one has the non-compact Lorentz group $SO(3,1)$ instead of it. A further and independent proof of the corollary uses the fact, that the Geroch limit does not lead to a Hausdorff topology, so a sequence of gravitational waves can converge to the flat space-time, even if each element of the sequence is the same pp-wave.