Completeness of closed flat kleinian pseudo-Riemannian manifolds of signature (2,2)

Carrière proved in 1989 that every compact flat Lorentzian manifold (i.e. pseudo-Riemannian of signature (n,1)) is complete. Beyond this Lorentzian case, no completeness is known in higher signature. In collaboration with Farid Diaf and Malek Hanounah, we prove the completeness of compact flat Kleinian pseudo-Riemannian manifolds of signature (2,2). This result falls into the context of the study of completeness of compact affine manifolds with a parallel volume form (Markus' conjecture). I will present a sketch of the proof of completeness in signature (2,2), and then focus on proving a key lemma, which is a reduction result for certain closed kleinian affine manifolds.