Riemannian Foliations and the Topology of Lorentzian Manifolds

2010/10/11 by Lärz, Kordian
#53C29 #53C50 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1010.2194

Abstract

A parallel lightlike vector field on a Lorentzian manifold X naturally defines a foliation F of codimension one. If either all leaves of 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 ≤ b₁ ≤ dim X if X is compact and 0 ≤ b₁ ≤ 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.

Related

• On Lorentzian manifolds with highest first Betti number
• Lightlike foliations on Lorentzian manifolds with weakly irreducible holonomy algebra
• Nonexistence of spacelike foliations and the dominant energy condition in Lorentzian geometry
• Lorentzian foliations on 3-manifolds
• Spacelike Foliations on Lorentz manifolds