Affine transformations and parallel lightlike vector fields on compact Lorentzian 3-manifolds (Q2787970)

The authors give a classification of all compact Lorentzian 3-manifolds admitting a parallel light-like vector field. The complete list of all compact Lorentzian 3-manifolds possessing non-isometric affine diffeomorphisms is also exhibited.NEWLINENEWLINENEWLINEMore precisely, let \((M,g)\) be a compact Lorentzian 3-manifold and let \(\mathrm{Isom}(g)\) denote its group of isometries and \(\mathrm{Aff}(g)\) its group of diffeomorphisms preserving the Levi-Civita connection alone. Of course \(\mathrm{Isom}(g)\subseteqq\mathrm{Aff}(g)\) is a closed subgroup. It is easy to show that if \(\mathrm{Aff}(g)/\mathrm{Isom}(g)\) is not trivial, i.e., the geometry admits a non-isometric affine diffeomorphism then, up to finite cover, \((M,g)\) possesses a parallel vector field. The interesting case is when this parallel vector field is light-like (or in other words null). The first main result in the article is the classification of all compact orientable and indecomposable Lorentzian 3-manifolds possessing a parallel light-like vector field (see Theorem 7.2 together with Table 2). The second main result is the classification of all compact orientable and time-orientable Lorentzian 3-manifolds with \(\mathrm{Aff}(g)/\mathrm{Isom}(g)\) being non-trivial (see Theorem 7.3 and Table 3). In short, these results say that all these geometries are topologically tori or parabolic tori carrying flat deformations of the standard flat metric on them. These deformed metrics have however rich and subtle structures investigated in great detail in the paper.NEWLINENEWLINENEWLINEFinally, the authors consider some higher-dimensional examples (Section 8).