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Abdelghani Zeghib

Publications and source records attributed to Abdelghani Zeghib.

At least 37 records · Page 2Linked to original sources
arXiv2023

Pseudo-Conformal Actions of the M{ö}bius Group

We study compact connected pseudo-Riemannian manifolds $(M,g)$ on which the conformal group $\operatorname{Conf}(M,g)$ acts essentially and transitively. We prove, in particular, that if the non-compact semi-simple part of $\operatorname{Conf}(M,g)$ is the M{ö}bius group, then $(M,g)$ is conformally flat.

math.DG↗
arXiv2022

Kundt Three Dimensional Left Invariant Spacetimes

Kundt spacetimes are of great importance to General Relativity. We show that a Kundt spacetime is a Lorentz manifold with a non-singular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kundt structures on Lie groups. Finally, we classify all left invariant Kundt structures on three dimensional simply connected unimodular Lie groups.

math.DG↗
arXiv2020

Rigidity of pseudo-Hermitian homogeneous spaces of finite volume

Let $M$ be a pseudo-Hermitian homogeneous space of finite volume. We show that $M$ is compact and the identity component $G$ of the group of holomorphic isometries of $M$ is compact. If $M$ is simply connected, then even the full group of holomorphic isometries is compact. These results stem from a careful analysis of the Tits fibration of $M$, which is shown to have a torus as its fiber. The proof builds on foundational results on the automorphisms groups of compact almost pseudo-Hermitian homogeneous spaces. It is known that a compact homogeneous pseudo-Kähler manifold splits as a product of a complex torus and a rational homogeneous variety, according to the Levi decomposition of $G$. Examples show that compact homogeneous pseudo-Hermitian manifolds in general do not split in this way.

math.DG↗
arXiv2019

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of $\mathfrak{g}$ is an infinitesimal isometry for $\langle\cdot,\cdot\rangle $. Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume. We prove a strong invariance property for nil-invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non-compact type of $\mathfrak{g} $ act by infinitesimal isometries for $\langle\cdot,\cdot\rangle $. Moreover, we study properties of the kernel of $\langle\cdot,\cdot\rangle $ and the totally isotropic ideals in $\mathfrak{g} $ in relation to the index of $\langle\cdot,\cdot\rangle $. Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most two. Examples show that the theory becomes significantly more complicated for index greater than two.

math.DG↗
arXiv2018

Simply connected indefinite homogeneous spaces of finite volume

Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with isometry group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie group acting transitively on $M$. For metric index less than three, we find that the isometry group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decomposition into three distinct types of metric Lie algebras.

math.DG↗
arXiv2017

On Rigidity of Generalized Conformal Structures

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $\geq 3$. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined by its 2-jet at any point. We prove here a similar rigidity for generalized conformal structures defined by giving a one parameter family of metrics (instead of scalar multiples of a given one) on each tangent space.

math.DG↗
arXiv2017

On homogeneous Hermite-Lorentz spaces

We define naturally Hermite-Lorentz metrics on almost-complex manifolds as special case of pseudo-Riemannian metrics compatible with the almost complex structure. We study their isometry groups.

math.DG↗
arXiv2012

Some open questions on anti-de Sitter geometry

We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichmüller theory, as well as analogs in hyperbolic geometry.

math.DG↗
arXiv2011

Geometry of warped products

This is a survey on the geometry of warped products, without, or essentially with only soft, calculation. Somewhere in the paper, the goal was to give a synthetic account since existing approaches are rather analytic. Somewhere else, we have interpreted statements, especially by means of a physical terminology. This is essentially heuristic, but we think it might be helpful in both directions, that is, in going from a synthetic geometrical language to a relativistic one, and vice-versa.

math.DG↗
arXiv2011

Homogeneous spaces, dynamics, cosmology: Geometric flows and rational dynamics

The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimensional dynamical systems, obtained by considering homogeneous metrics.

math.DG↗
arXiv2011

Singular Riemannian metrics, sub-rigidity vs rigidity

We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?

math.DG↗
arXiv2010

Holomorphic actions, Kummer examples, and Zimmer Program

We classify compact Kähler manifolds $M$ of dimension $n\geq 3$ on which acts a lattice of an almost simple real Lie group of rank $\geq n-1$. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.

math.AG↗
arXiv2008

Finsler conformal Lichnerowicz-Obata conjecture

We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.

math.DG↗
arXiv2008

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-dimensional Minkowski space for datas that are invariant under the action of a co-compact Fuchsian group.

gr-qc↗