Seminar

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Carrière proved in 1980 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). After giving a historical overview of these works, I will present a sketch of the proof of completeness in signature (2,2). Along the way, I will present two results of independent interest that apply in other contexts. The first is a geometric reduction theorem for certain divisible domains in affine space. The second concerns the existence of syndetic hulls in semidirect products of the form R \rtimes G, where G is a homothety Lie group, extending earlier developments in affine geometry due to Yves Carrière and Françoise Dal'bo.