?
СУЩЕСТВЕННЫЕ ГРУППЫ ИЗОМЕТРИЙ НЕКОМПАКТНЫХ ДВУМЕРНЫХ ПЛОСКИХ ЛОРЕНЦЕВЫХ ОРБИФОЛДОВ
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”.