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Theses

Points conjugués des tores lorentziens

Abstract : In the first part of this thesis, we give a description of simply connected maximal Lorentzian surfaces whose group of isometries is of dimension 1 (i.e. with a complete Killing field), in terms of a 1-dimensional generally non-Hausdorff Riemannian manifold and a smooth function defined there. Next, we study the geodesic completeness of such surfaces. In the second part which is the main part of this thesis, we give infinitely many new examples of compact Lorentzian surfaces without conjugate points. Further, we study the existence and the stability of this property among Lorentzian metrics with a Killing field. We obtain a new obstruction and prove that the Clifton- Pohl torus and some of our examples are as stable as possible. This shows that in constrast with the Riemannian Hopf theorem, the absence of conjugate points in the Lorentzian setting is neither "special" nor rigid.
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Lilia Mehidi. Points conjugués des tores lorentziens. Géométrie différentielle [math.DG]. Université de Bordeaux, 2019. Français. ⟨NNT : 2019BORD0295⟩. ⟨tel-02481025⟩

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