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 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 are 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.