The topology and the Lorentz-invariant pseudo-Riemannian metric of the manifold of directions in the physical space

In the mathematical model of the special theory of relativity the two-dimensional Minkowski subspace is interpreted as a one-dimensional direction of the physical space. The manifold of such planes is naturally provided with the structure of a pseudoriemann manifold, the group of isochronous Lorentz transformations on which acts by isometries transitively. The present paper is dedicated to the study of the topology and metric geometry of this manifold.