Shirley Bromberg

A homogeneous space–time model with singularities

We show the existence of a left invariant pseudo-Riemannian metric on the Oscillator Group of dimension four, such that this group becomes a space–time with singularities in the sense of Hawking and Penrose. As an application we exhibit new incomplete, nonhomeomorphic compact Lorentzian manifolds.

Geodesically Complete Lorentzian Metrics on Some Homogeneous 3 Manifolds

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.

A Note on the Completeness of Homogeneous Quadratic Vector Fields on the Plane

In this paper we give a full description of the complete homogeneous quadratic vector fields defined on the plane.