On completeness of foliated structures, and null Killing fields

TL;DR

The paper develops a comprehensive framework for foliated completeness in non-Riemannian settings by studying a compact manifold $M$ endowed with a foliation $rak{F}$ and a tangential affine connection $ abla_{rak{F}}$. It introduces and analyzes the foliated $(G,X)$-structures, with emphasis on affine unimodular lightlike geometry and its dual, proving leafwise completeness for compact $( extsf{L}_{ ext{u}}(n), eal^{n+1})$-foliations via Riemannian submersion techniques, and extending results to affine lightlike geometries and pseudo-Riemannian variants. The work also investigates global completeness in the presence of a null Killing field, constructing incomplete compact Lorentzian manifolds and detailing how equicontinuity of the Killing flow in dimension three influences completeness, including explicit counterexamples and classifications. Applications to pp-waves and homogeneous plane waves illustrate the leafwise vs global distinction and connect to the Markus conjecture in affine geometry. Overall, the results illuminate when leafwise completeness implies global completeness and when foliated structures admit incomplete leaves, enriching the understanding of Lorentzian and affine foliations in compact settings.

Abstract

We consider a compact manifold $(M,\mathfrak{F})$ with a foliation $\mathfrak{F}$, and a smooth affine connection $\nabla$ on the tangent bundle of the foliation $T\mathfrak{F}$. We introduce and study a foliated completeness problem. Namely, under which conditions on $\nabla$ the leaves are complete? We consider different natural geometric settings: the first one is the case of a totally geodesic lightlike foliation of a compact Lorentzian manifold, and the second one is the case where the leaves have particular affine structures. In the first case, we characterize the completeness, and obtain in particular that if a compact Lorentzian manifold admits a null Killing field $V$ such that the distribution orthogonal to $V$ is integrable, then it defines a (totally geodesic) foliation with complete leaves. In the second case, we give a completeness result for a specific affine structure called "the unimodular affine lightlike geometry", and characterize the completeness for a natural relaxation of the geometry. On the other hand, we study the global completeness of a compact Lorentzian manifold in the presence of a null Killing field. We give two non-complete examples, starting from dimension $3$: one is a locally homogeneous manifold, and the other is a $3$D example where the Killing field dynamics is equicontinuous.

Figures (1)

• Figure 1: The foliation tangent to $X^*$ (resp. to $Y^*$) is represented in black (resp. red)

Theorems & Definitions (90)

• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Example 1.9: pp-waves are $(\mathsf{L}_{\sf{u}}(n), \mathbb{R}^{n+1})$-foliated manifolds
• Theorem 1.10
• Theorem 1.11
• Example 1.12: weakly pp-waves are $(\mathsf{L}(n), \mathbb{R}^{n+1})$-foliated manifolds
• Theorem 1.13