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

Abstract

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.

Projective properties of Lorentzian surfaces

“…It is clearly not an isometry. As g is not flat the only parallel endomorphisms of the tangent bundle are the multiples of the identity (see Fact page 2226 of [5] for a proof). It means that any metric having the same Levi-Civita connection as g is proportional to g. However, the vector field ∂ x is lightlike when x = 0 but not when x = 1, so τ * g is not proportional to g. Consequently τ is not an affine map.…”

Some Non-Trivial and Non-Gradient Closed Pseudo-Riemannian Steady Ricci Solitons

“…We call R 3 /Γ n a parabolic torus, as a suspension of the parabolic automorphism τ z,n (x, y, z) of R 2 /Z 2 over R/Z. See [2] for more details. Proposition 3.1.…”

“…In this section, we construct the examples of non-trivial and non-gradient closed pseudo-Riemannian steady Ricci solitons with zero scalar curvature in the neutral signature (2,2) such that the associated potential vector fields can be time-like or null.…”

“…In [10], we show the existence of non-trivial and non-gradient steady Ricci solitons on a special group of compact indecomposable Lorentzian 3-manifolds admitting a parallel light-like vector field with closed orbits. The aim of this paper is to study the Ricci soliton equation on compact indecomposable Lorentzian 3-manifolds with a parallel light-like vector field with closed orbits, that were classified recently in [2], in order to construct more examples of closed Lorentzian steady Ricci solitons with zero scalar curvature. In each case, the associated potential vector field with the Ricci soliton structure is space-like.…”

“…Section 3 is devoted to studying the Ricci soliton structure on orientable compact indecomposable Lorentzian manifolds admitting a parallel light-like vector field with closed orbits. Finally, in Section 4, we construct the examples of non-trivial and non-gradient closed pseudo-Riemannian Ricci solitons with the neutral signature (2,2) in the steady case with zero scalar curvature.…”