We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset. It follows from [B1] (see also [BP1, Theorem 1]) that for each partition of the n-dimensional space R into n pieces one of the pieces contains an unbounded centrally symmetric subset. On the other hand, R admits a partition...

Let (M, g) be a spacetime, where M is a smooth, connected, Hausdorff four-dimensional manifold and g is smooth Lorentzian metric of signature (+ -) defined on M . The manifold M and the metric g are assumed smooth (C). A smooth vector field ξ is said to preserve a matter symmetry [1] on M if, for each smooth local diffeomorphism φt associated with ξ, the tensors T and φ∗tT are equal on the doma...

We investigate a self-gravitating thick domain wall for a λΦ potential. The system of scalar and Einstein equations admits two types of non-trivial solutions: domain wall solutions and false vacuum-de Sitter solutions. The existence and stability of these solutions depends on the strength of the gravitational interaction of the scalar field, which is characterized by the number ǫ. For ǫ ≪ 1, we...

The gravitational properties of the only static plane-symmetric vacuum solution of Einstein’s field equations without cosmological term (Taub’s solution, for brevity) are presented: some already known properties (geodesics, weak field limit and pertainment to the Schwarzschild family of spacetimes) are reviewed in a physically much more transparent way, as well as new results about its asymptot...