arXiv subjects
Publications and source records attributed to Abdelghani Zeghib.
We prove a rigidity of the lightcone in Minkowski space. It is essentially the unique space endowed with a degenerate Riemannian metric, of lightlike type, and supporting an isometric non-proper action of a semi-simple group.
We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the metric has constant sectionnal curvature, or is a left invariant metric on the Heisenberg group or the Sol group. These geometries, on each of the latter two groups are characterized by having a non-compact isotropy without being flat. Recall, for the need of his formulation of the geometrization conjecture, W. Thurston counted the 8 maximal Riemannian geometries in dimension 3. Here, we count only 4 maximal Lorentz geometries, but ignoring those which are at the same time Riemannian. Also, all such manifolds are geodesically complete, except the previous non flat left invariant Lorentz metric on the SOL group.
We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature.
The above title is the same, but with "semisimple" instead of "simple," as that of a notice by N. Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her premature death. Here, using a different, geometric approach, we generalize her results to the semisimple case, and give proofs of all her announced results.
We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is then possible to define an "amazing" notion of holomorphic functions on them, and show how rigid it is in some cases.
This paper continues the investigation of constant mean curvature (CMC) time functions in maximal globally hyperbolic spatially compact spacetimes of constant sectional curvature, which was started in math.DG/0604486. In that paper, the case of flat spacetimes was considered, and in the present paper, the remaining cases of negative curvature (i.e. anti-de Sitter) spacetimes and postitive curvature (i.e. de Sitter) spacetimes is dealt with. As in the flat case, the existence of CMC time functions is obtained by using the level sets of the cosmological time function as barriers. A major part of the work consists of proving the required curvature estimates for these level sets. The nonzero curvature case presents significant new difficulties, in part due to the fact that the topological structure of nonzero constant curvature spacetimes is much richer than that of the flat spacetimes. Further, the timelike convergence condition fails for de Sitter spacetimes, and hence uniqueness for CMC hypersurfaces fails in general. We characterize those de Sitter spacetimes which admit CMC time functions (which are automatically unique), as well as those which admit CMC foliations but no CMC time function.
This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they are causally incomplete. The proof, which is based on using the level sets of the cosmological time function as barriers, is conceptually simple and will provide the basis for future work on constant mean curvature time functions in general constant curvature spacetimes, as well for an analysis of the asymptotics of constant mean foliations.
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.
We prove that any maximal globally hyperbolic spacetime locally modelled on the anti-de Sitter space of dimension 3, and admitting a closed Cauchy surface, admits a time function $τ$, such that every fiber $τ^{-1}(t)$ is a spacelike surface with constant mean curvature $t$.
All the actions considered here are (real) analytic. Consider a subgroup of finite index of SL(n, Z). We prove, in particular, the (global) homotopical rigidity, for both its standard affine action on the torus of dimension n > 2, and its standard projective action on the sphere of dimesnion n-1>3.
Here, we classify Lie groups acting isometrically on compact Lorentz manifolds, and in particular we describe the geometric structure of compact homogeneous Lorentz manifolds.
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 to enrich a plane field, as this is often practised in control theory, by adding brackets of the vector fields tangent to it, and then, look for singular solutions of the obtained distribution. We apply this to the isometry problem of rigid geometric structures.
This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; that is, they are (at least locally) warped products of constant curvature Lorentz manifolds by Riemannian manifolds.
This is the first part of a series on non-compact groups acting isometrically on compact Lorentz manifolds. This subject was recently investigated by many authors. In the present part we investigate the dynamics of affine, and especially Lorentz transformations. In particular we show how this is related to geodesic foliations. The existence of geodesic foliations was (very succinctly) mentioned for the first time by D'Ambra and Gromov, who suggested that this may help in the classification of compact Lorentz manifolds with non-compact isometry groups. In the Part II of the series, a partial classification of compact Lorentz manifolds with non-compact isometry group will be achieved with the aid of geometrical tools along with the dynamical ones presented here.