arXiv subjects

Malek Hanounah

Publications and source records attributed to Malek Hanounah.

6 recordsLinked to original sources

arXiv2026

Completeness of compact Locally symmetric Lorentz manifolds

The geodesic completeness of compact locally symmetric Lorentz manifolds has been established in several important cases, namely, the constant curvature, indecomposable, and Brinkmann settings. In this paper, we prove geodesic completeness in all remaining cases, thereby confirming the conjecture that all compact, locally symmetric Lorentz manifolds are geodesically complete. Along the way, we use the completeness result to give a comprehensive overview on four-dimensional compact locally symmetric Lorentz manifolds.

math.DG↗

arXiv2026

Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

Let $\mathbb{R}^{2,2}$ denote the model space of flat pseudo-Riemannian manifolds of signature $(2,2)$. We prove that the only domain divisible by a discrete subgroup of the isometry group of $\mathbb{R}^{2,2}$ is $\mathbb{R}^{2,2}$ itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products $R \ltimes G$, where $G$ is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carrière and Dal'bo.

math.DG↗

arXiv2025

On completeness of certain locally symmetric pseudo-Riemannian manifolds of signature $(2,2)$

We show geodesic completeness of certain compact locally symmetric pseudo-Riemannian manifolds of signature $(2,n)$. Our model space $\mathbf{X}$ is a $1$-connected, indecomposable symmetric space of signature $(2,n)$, that admits a unique (up to scale) parallel lightlike vector field. This class of spaces is the natural generalization of the class of Cahen--Wallach spaces to signature $(2,n)$. In dimension $4$ we show that $\mathbf{X}$ has no proper domain $Ω$ which is divisible by the action of a discrete group $Γ$ of $\operatorname{Isom}(\mathbf{X})$, i.e. $Γ$ acts properly and cocompactly on $Ω$. Therefore, we deduce geodesic completeness in the aforementioned situation. In arbitrary dimension we show geodesic completeness of compact locally symmetric space modeled on $\mathbf{X}$ under the assumption that the transition maps of $M$ are restrictions of transvections of $\mathbf{X}$. Along the way, we establish a new case in the Kleinian $3$-dimensional Markus's conjecture for flat affine manifolds. Moreover, we classify geometrically Kleinian compact manifolds that are modeled on the hyperbolic oscillator group endowed with its bi-invariant metric. Finally we discuss a natural dynamical problem motivated by the Lorentz setting (Brinkmann spacetimes). Specifically, we show that the parallel flow on $M$ is equicontinuous in dimension $4$, even in our non-Lorentz setting.

math.DG↗

arXiv2025

On homogeneous plane waves

Plane waves are a special class of Lorentzian spaces with a parallel null vector field. They are of great importance in Geometry (e.g. Lorentzian holonomy) and in Physics (General Relativity as well as alternative gravity theories). Our contribution in the present paper aims at a rigorous mathematical treatment focusing on completeness of Killing fields, and globality of coordinates. Equivalence of different approaches to plane waves is by no means easy to handle. We use here cohomogeneity one Heisenberg actions to introduce a point of view from which one can see plane waves as a deformation of Minkowski spacetime. We determine the identity component of the isometry group of a 1-connected non-flat homogeneous plane wave, which establishes a correspondence between these spaces and certain 1-parameter groups of automorphisms of the Heisenberg group. The extendibility of spacetimes (when incomplete) is a natural, important and delicate question. One of our main results is the proof of the $C^2$-inextendibility of non-flat homogeneous plane waves. We also prove that they are geodesically complete if and only if the null parallel vector field is preserved by the identity component of the isometry group. Finally, we show that a 1-connected homogeneous plane wave admits global Brinkmann coordinates.

math.DG↗

arXiv2025

On completeness of foliated structures, and null Killing fields

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.

math.DG↗

arXiv2024

Topology and Dynamics of compact plane waves

We study compact locally homogeneous plane waves. Such a manifold is a quotient of a homogeneous plane wave $X$ by a discrete subgroup of its isometry group. This quotient is called standard if the discrete subgroup is contained in a connected subgroup of the isometry group that acts properly cocompactly on $X$. We show that compact quotients of homogeneous plane waves are ``essentially" standard; more precisely, we show that they are standard or `semi-standard'. We find conditions which ensure that a quotient is not only semi-standard but even standard. As a consequence of these results, we obtain that the flow of the parallel isotropic vector field of a compact locally homogeneous plane wave is equicontinuous.

math.DG↗