Essential isometry groups of noncompact two-dimensional flat lorenzian orbifolds

Actuality and goals. Lorentzian geometry finds widespread application in physics and is radically different from Riemannian geometry. As it is known an every smooth orbifold admits a Riemannian metric. The existence of a Lorentzian metric on an orbifold imposes restrictions on its structure. The isometry group of a Lorentzian orbifold is called inessential if it acts properly, otherwise the isometry group of a Lorentzian orbifold is called essential. The goal of this work is the investigation of the structure of noncompact smooth two-dimensional orbifolds admitting a complete flat Lorentzian metric with an essential isometry group. Methods. Using the bundle of pseudo-orthogonal frames some canonical covering map for two-dimensional Lorentzian orbifolds is constructed and applied. The existence of such map shows that any two-dimensional Lorentzian orbifold is very good. Results. It is proved that there are only two (up to isomorphisms in the category of orbifolds) two-dimensional smooth noncompact orbifolds admitting complete flat Lorentzian metrics with an essential isometry group. They are the plane and the $Z_2$-cone. Unlike compact orbifolds, the metric can be any from the class of flat complete Lorentzian metrics. Examples are constructed. Conclusions. Only four two-dimensional smooth orbifolds allow complete flat Lorentzian metrics with the essential isometry group: the plane, the torus, the $Z_2$-cone and the “pillow”.