Affine transformations and parallel lightlike vector fields on compact Lorentzian $3$-manifolds

Charles Boubel (1) , Pierre Mounoud (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 IMB - Institut de Mathématiques de Bordeaux

We describe the compact Lorentzian $3$-manifolds admitting a parallel lightlike vector field. The classification of compact Lorentzian $3$-manifolds admitting non-isometric affine diffeomorphisms follows, together with the complete description of these morphisms. Such a Lorentzian manifold is in some sense an equivariant deformation of a flat one.

Mots clés: parallel vector field; affine map; holonomy; Lorentzian manifold

Domaines: Géométrie différentielle [math.DG]

Charles Boubel, Pierre Mounoud. Affine transformations and parallel lightlike vector fields on compact Lorentzian $3$-manifolds. Transactions of the American Mathematical Society, 2016, 368 (3), pp.2223-2262. ⟨10.1090/tran/6645⟩. ⟨hal-00973940v3⟩