arXiv subjects

Lilia Mehidi

Publications and source records attributed to Lilia Mehidi.

9 recordsLinked to original sources

arXiv2026

Locally conformally homogeneous Lorentzian spaces

We study locally conformally homogeneous Lorentzian manifolds of dimension at least $3$, admitting an essential pseudo-group of local conformal transformations. Generalizing a recent result of Alekseevsky and Galaev, we show that any such manifold $(M,g)$ is either conformally flat, or locally conformally equivalent to a homogeneous plane wave. When the manifold is non-conformally flat, we show the existence of a codimension-one lightlike foliation of Heisenberg type, which leads to the plane wave structure. Our approach relies on tools from Gromov's theory of rigid transformations. Finally, we observe that the plane wave metric in the conformal class coincides with the Penrose limit of $(M,g)$ along some null geodesic.

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 and dynamics of compact Brinkmann spacetimes

Brinkmann Lorentz manifolds are those admitting an isotropic parallel vector field. We prove geodesic completeness of the compact and also compactly homogeneous Brinkmann spaces. We also prove, partially, that their parallel vector field generates an equicontinuous flow.

math.DG↗

arXiv2025

Conformal quotients of plane waves, and Lichnerowicz conjecture in a locally homogeneous setting

In the first part of the paper, we study conformal groups that act properly discontinuously and cocompactly on simply connected, non-flat homogeneous plane waves. We show that proper cocompact similarity actions that are not isometric can occur, in contrast to the behavior of Riemannian and Lorentzian affine similarity actions. In the second part, we consider the Lorentzian conformal Lichnerowicz conjecture, which states that if the conformal group of a compact Lorentzian manifold acts without preserving any metric in the conformal class, then the manifold must be conformally flat. We prove the conjecture in a locally homogeneous setting.

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↗

arXiv2025

Introduction to Kundt spaces

This paper provides an introduction to Kundt spaces, clarifying several important properties, many of which are typically scattered across the mathematical literature or presented without explicit reference to Kundt terminology. While not exhaustive, our approach aims to offer a pedagogical introduction, using a more geometric language and focusing on key concepts directly related to these spaces, such as lightlike totally geodesic foliations.

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.

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arXiv2021

Maximal simply connected Lorentzian surfaces with a Killing field and their completeness

In the first part of this paper, we give a global description of simply connected maximal Lorentzian surfaces whose group of isometries is of dimension 1 (i.e. with a complete Killing field), in terms of a 1-dimensional generally non-Hausdorff manifold (the space of Killing orbits) and a smooth function defined there. In the second part, we study the completeness of such surfaces and prove in particular that under the hypothesis of bounded curvature, completeness is equivalent to null completeness. We also give completeness criterions involving the topological structure of the space of Killing orbits, or both the topology of this space and the geometry of the surface.

math.DG↗

arXiv2019

On the existence and stability of two-dimensional Lorentzian tori without conjugate points

Infinitely many new examples of compact Lorentzian surfaces without conjugate points are given. Further, we study the existence and the stability of this property among Lorentzian metrics with a Killing field. We obtain a new obstruction and prove that the Clifton- Pohl torus and some of our examples are as stable as possible. This shows that in constrast with the Riemannian Hopf theorem, the absence of conjugate points in the Lorentzian setting is neither "special" nor rigid.

math.DG↗