Sur la complétude de certaines variétés pseudo-riemanniennes localement symétriques

Tholozan, Nicolas

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1921-1952.

Nous prouvons que certains espaces pseudo-riemanniens symétriques n’admettent pas d’ouvert strict divisible par l’action d’un groupe discret d’isométries. Autrement dit, si une variété pseudo-riemannienne compacte est localement isométrique à un tel espace, et si son application développante est injective, alors la variété est géodésiquement complète, et donc isométrique à un quotient de l’espace modèle tout entier. Ces résultats étendent, sous une hypothèse supplémentaire (l’injectivité de l’application développante), les théorèmes de Carrière et Klingler selon lesquels les variétés lorentziennes compactes de courbure constante sont géodésiquement complètes.

We prove that certain pseudo-Riemannian symmetric spaces do not admit a proper domain which is divisible by the action of a discrete group of isometries. In other words, if a closed pseudo-Riemannian manifold is locally isometric to such a model, and if its developing map is injective, then the manifold is actually geodesically complete, and therefore isometric to a quotient of the whole model space. Those results extend, under an additional assumption (the injectivity of the developing map), the theorems of Carrière and Klingler stating that closed Lorentz manifolds of constant curvature are geodesically complete.

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