Lorentzian manifolds close to Euclidean space

We study the Lorentzian manifolds $M_1$, $M_2$, $M_3$, and $M_4$ obtained by small changes of the standard Euclidean metric on $\mathbb{R}^4$ with the punctured origin $O$. The spaces $M_1$ and $M_4$ are closed isotropic space-time models. The manifolds $M_3$ and $M_4$ (respectively, $M_1$ and $M_2$) are geodesically (non)complete; $M_1$ and $M_4$ are globally hyperbolic, while $M_2$ and $M_3$ are not chronological. We found the Lie algebras of isometry and homothety groups for all manifolds; the curvature, Ricci, Einstein, Weyl, and energy-momentum tensors. It is proved that $M_1$ and $M_4$ are conformally flat, while $M_2$ and $M_3$ are not conformally flat and their Weyl tensor has the first Petrov type.