On Lorentzian manifolds with highest first Betti number

Schliebner, Daniel

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1423-1436.

On considère des variétés lorentziennes avec un champ de vecteurs de genre lumière parallèle. Comme il est parallèle et de genre lumière, son complément orthogonal induit un feuilletage de codimension un. Si on suppose les feuilles compactes et la courbure de Ricci positive (ou nulle) sur les feuilles, on sait que le premier nombre de Betti est borné par la dimension de la variété ou de la feuille, suivant la compacité ou non-compacité de la variété. Nous démontrons que dans le cas où le nombre de Betti est maximal, quitte à la remplacer par un revêtement fini, toute telle variété lorentzienne est soit difféomorphe au tore (si elle est compacte) ou alors au produit d’une droite avec un tore (autrement), et que sa courbure est très dégénérée, c’est-à-dire le tenseur de courbure induit sur les feuilles est de genre lumière.

We consider Lorentzian manifolds with parallel light-like vector field $V$. Being parallel and light-like, the orthogonal complement of $V$ induces a codimension one foliation. Assuming compactness of the leaves and non-negative Ricci curvature on the leaves it is known that the first Betti number is bounded by the dimension of the manifold or the leaves if the manifold is compact or non-compact, respectively. We prove in the case of the maximality of the first Betti number that every such Lorentzian manifold is – up to finite cover – diffeomorphic to the torus (in the compact case) or the product of the real line with a torus (in the non-compact case) and has very degenerate curvature, i.e. the curvature tensor induced on the leaves is light-like.

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