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arXiv · 1010.2194

Riemannian Foliations and the Topology of Lorentzian Manifolds

Kordian Lärz
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Abstract

A parallel lightlike vector field on a Lorentzian manifold $X$ naturally defines a foliation $\mathcal{F}$ of codimension one. If either all leaves of $\mathcal{F}$ are compact or $X$ itself is compact admitting a compact leaf and the (transverse) Ricci curvature is non-negative then a Bochner type argument implies that the first Betti number of $X$ is bounded by $1 \leq b_{1} \leq \dim X$ if $X$ is compact and $0 \leq b_{1} \leq \dim X -1$ otherwise. We show that these bounds are optimal and depending on the holonomy of $X$ we obtain further results. Finally, we classify the holonomy representations for those $X$ admitting a compact leaf with finite fundamental group.

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Kordian Lärz. 2010-10-11. Riemannian Foliations and the Topology of Lorentzian Manifolds. https://arxiv.org/abs/1010.2194

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