Abdelghani Zeghib

Group Actions on Lorentz Spaces, Mathematical Aspects: A Survey

From a purely mathematical viewpoint, one can say that most recent works in Lorentz geometry, concern group actions on Lorentz manifolds. For instance, the three major themes: space form problem of Lorentz homogeneous spacetimes, the completeness problem, and the classification problem of large isometry groups of Lorentz manifolds, all deal with group actions. However, in the first two cases, actions are “zen” (e.g., proper), and in the last, the action is violent (i.e., with strong dynamics). We will survey recent progress in these themes, but we will focus attention essentially on the last one, that is, on Lorentz dynamics.

Remarks on Lorentz symmetric spaces

We consider homogeneous Lorentz spaces of dimension at least 3. We prove that if such a space has ‘big’ isotropy (that is, a non-precompact and irreducible isotropy group), then this space must have constant sectional curvature. As a corollary, we obtain a new direct proof of the fact that irreducible Lorentz symmetric spaces have constant curvature, which was known via (algebraic) classification.

Leafwise holomorphic functions

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

On Gromov's theory of rigid transformation groups: a dual approach

Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual approach, which consists of enriching a plane field, as this is often practised in control theory, by adding brackets of the vector fields tangent to it and, then, looking for singular solutions of the obtained distribution. We apply this to the isometry problem of rigid geometric structures. 0. Content In §1 we exhibit a natural class of plane fields for which the accessibility behaviour, as studied in control theory, possesses, essentially, the same nice properties as in the analytic case. In §2, we observe that there is a control theory approach to the local isometry problem of affine manifolds (e.g. pseudo-Riemannian manifolds), which is dually equivalent, to the usual differential systems (i.e. partial differential relations) approach. We then apply the results of §1 to deduce a celebrated corollary of Gromovs theory on rigid transformation groups. In fact the developments of §1 suggest how to proceed in order to recover essentially most of the part of Gromovs theory related to Corollary 2.1, together with some independent results. However, we will not follow this because our primary goal here is to be as elementary as possible. Further extensions and applications of our approach will be developed elsewhere.

Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques

We introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C 1 -submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.

On Lorentz dynamics: From group actions to warped products via homogeneous spaces

We show a geometric rigidity of isometric actions of non-compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.

Ensembles invariants des flots géodésiques des variétés localement symétriques

We study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.

Dynamics on the space of harmonic functions and the foliated Liouville problem

We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem. Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on the manifold, is the function leafwise constant? We give a number of positive results and also show a general class of examples for which the Liouville property does not hold. The connection between the Liouville property and the dynamics on the space of harmonic functions as well as general properties of this dynamical system are explored. It is shown among other properties that the Z-action generated by hyperbolic or parabolic elements of PSL(2,R) is chaotic.

Actions of semisimple Lie groups preserving a degenerate Riemannian metric

We prove a rigidity of the lightcone in Minkowski space. It is (essentially) the unique space endowed with a lightlike metric and supporting an isometric nonproper action of a semisimple Lie group.

On the Isometry Group of Lorentz Manifolds

We will first review a construction in [10] to establish the Lie group structure of the group of isometries of a semi-Riemannian manifold. The problem is cast in the language of G-structures. In the second part of this chapter, we will review some recent results on the classification of groups acting isometrically on compact Lorentz manifolds and on the geometry of compact manifolds whose isometry group is non compact.