Riemannian Foliations and the Topology of Lorentzian Manifolds

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 $1\le b_1\le \dim X$ if $X$ is compact and $0\le b_1\le \dim X-1$ 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.

16 pages, comments are welcome

References in corpus (2)

• Metrics that realize all Lorentzian holonomy algebras
• Minimal entropy and geometric decompositions in dimension four