Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics

Abstracts of talks, week two, July 18 - 22, 2011

We discuss the problem of description of homogeneous Lorentzian manifolds M = G/H of a semisimple Lie group G. The starting point is the classical result by Nadine Kowalsky about nonproper action of a simple Lie group G on a Lorentzian manifolds M and its recent generalizations by M. Deffaf, K. Melnick and A. Zeghib, which reduce the classification of homogeneous Lorentz manifolds M=G/H of a semisimple Lie group G to the case when the stabilizer H is compact.

A homogeneous manifold M = G/H with a compact (connected) stabilizer H is called minimal admissible if it admits an invariant Lorentzian metric, but any homogeneous manifold M' = G/H' with a bigger (connected) stability group H' ⊂ H does not admits such a metric. We classify minimal admissible compact homogeneous manifolds M=G/H. In particular, if the group G is simple, any such manifold is a circle bundle over a minimal adjoint orbit of the group G. We reduce the classification of minimal admissible manifold M=G/H of a simple non-compact Lie group G to description of minimal orbits of the isotropy representation of the associated non-compact symmetric space S = G/K and give a list of such manifolds of dimension ≤ 11. We discuss also the problem of description of nonproper homogeneous Lorentzian manifolds of non semisimple Lie group G.

Cartan geometries build a bridge between geometry in the sense of F. Klein's Erlangen program and geometric structures in the sense of differential geometry. The basic idea is that manifolds endowed with certain types of geometric structures can be viewed as "curved analogs" of a given homogeneous space, called the homogeneous model of the structure. In my talk, I will outline this general concept and discuss several examples of Cartan geometries. As an application of this approach, I will describe how to analyze infinitesimal automorphisms and deformations of Cartan geometries.

Let J\H be a homogeneous space. If Γ is a discrete subgroup of H such that J\H/Γ is a compact manifold, we call it a compact Clifford-Klein form. Following on work of Labourie-Mozes-Zimmer, I will prove that when H is simple, J non-compact reductive and J\H admits an action by a higher-rank simple group, then there is no compact Clifford-Klein form of J\H. The proof uses cocycle superrigidity, Ratner's theorem, and tools from hyperbolic dynamics and measure rigidity.

Motivated by Gromov's "open-dense orbit theorem", we try to understand whether a rigid geometric structure on a compact connected manifold which is locally homogeneous on an open dense set, has to be locally homogenous everywhere. We have a positive answer for lorentz 3-maifolds and for unimodular connections on surfaces.

We look into discrete-time Hamiltonian systems whose state spaces factor into an observable part and a complementary part that is known only up to a specified probability distribution. This leads to a class of Markov chains naturally arising in a differential-geometric or mechanical setting. We describe general facts and special case studies concerning the systems' Markov operators, their stationary distributions, the interplay between their spectral properties and geometric features of the underlying systems, central limit theorems, and the significance of these ideas in classical statistical mechanics.

The notion of Cartan geometry yields a unified framework to study classical geometric structures such as pseudo-Riemannian metric, conformal, projective and CR-structures.... We will be especially interested in those Cartan geometries infinitesimally modelled on the boundaries of the different hyperbolic spaces. The aim of the talk is to show how very weak assumptions on the dynamics of the automorphism group imply global rigidity phenomena. We recover in particular classical results due to J. Ferrand and R. Schoen.

This will be an elementary and mainly expository talk which, following some motivation, reviews the construction of the standard conformal tractor bundle its invariant connection and related objects. It will be indicated how this may be applied to the study of equations satisfying symmetry type equations of which the infinitesimal conformal automorphism equation is typical. Although the discussion focusses on conformal geometry, similar ideas apply to a host of other structures.

If an (m+2)-manifold M is locally modelled on R^m+2 with coordinate changes lying in the subgroup G=R^m+2 ⋊ (O(m+1,1) × R⁺) of the affine group Aff(m+2), then M is said to be a Lorentzian similarity manifold. A Lorentzian similarity manifold is also a conformally flat Lorentz manifold because the above group G is isomorphic to the stabilizer of the Lorentz group PO(m+2,2) of the Lorentz model S^2n+1,1. It contains a class of Lorentzian flat space forms. We shall discuss the properties of Lorentzian similarity manifolds using developing maps and holonomy representations.

I will discuss certain locally symmetric spaces, namely complete anti-de Sitter 3-manifolds (or complete Lorentz 3-manifolds of constant negative curvature). I will explain how the locally symmetric structure of these manifolds is related to the representation theory of surface groups into SL₂(R). I will also discuss the discrete spectrum of the Laplacian on such manifolds. This is joint work with François Guéritaud and with Toshiyuki Kobayashi.

Parallel G₂-structures are known from Riemannian geometry, G₂ ⊂ SO(7) is one of the exceptional holonomies occurring in Berger's list. Riemannian manifolds with holonomy contained in G₂ are Ricci-flat. In particular, there are no non-flat symmetric spaces among them. The group G₂ has a non-compact counterpart G₂₍₂₎ ⊂ SO(4,3). A parallel G₂₍₂₎-structure on a pseudo-Riemannian manifold of signature (4,3) is a reduction of the bundle of orthonormal frames to a subgroup of G₂₍₂₎ ⊂ SO(4,3). We will see that, contrary to the Riemannian situation, there are symmetric spaces that admit a parallel G₂₍₂₎-structure and that there are even indecomposable ones. We will study such spaces and we will see how they can be classified using the method of quadratic extensions.

We look for conditions under which a group acting on the plane has a fixed point. If the group is Z, the Brouwer plane translation theorem says that a bounded orbit implies a global fixed point. We prove an analogous result for arbitrary groups by requiring that orbits be sufficiently bounded on a ball of sufficient size. The philosophy of the proof is that the dynamics of group actions on the plane can be understood through actions on the circle.

We give a complete list of normal forms for the 2-dimensional metrics that admit a Lie algebra of projective (i.e. geodesic-preserving) symmetries whose dimension is greater or equal to 2. This solves completely a problem posed by Sophus Lie in 1882.

It's a joint work with Mickael Crampon. Hilbert geometry is the study of open convex subset Ω of the REAL projective space. There is a natural measure and a natural distance (called Hilbert's distance) on such open set. So, it's a perfect place to do some geometry. Assuming that Ω is strictly convex and with C¹ boundary (which mean assuming that the metric space Ω has a "negatively curved behaviour"), I will give necessary and sufficient condition on the action of a group Γ on Ω by projective transformation so that the quotient Ω/Γ is of finite volume. The road will make us go throw a Kazhdan-Margulis lemma for Hilbert geometry and geometrically finite action.

Projective structure on a manifold is a class of affine connections having the same (unparameterized) geodesics. h-projective structure on a Kähler manifold is a class of affine connections having the same h-planar curves (I explain what it is in my talk).

Projective and h-projective admitting symmetries is easy to describe. Under the assumption that the projective or h-projective structure came from a metric, the existence of its symmetry is a very strong condition and in many cases the symmetry must automatically preserve the Levi-Civita connection of the metric. In my talk I give a proof of Lichnerowicz-Obata and Obata-Yano conjectures (the last is the joint result with S. Rosemann) stating that on a closed connected Riemannian manifold the group of the projective (L-O) or h-projective (O-Y) transformations coincides with the connected component of the group of isometries.

This talk will be an overview of several powerful theorems originally due to Zimmer and Gromov describing the automorphism group of a rigid geometric structure, with the formulations of these theorems in the setting of Cartan geometries. The results to be presented include the Frobenius and open-dense theorems, the embedding theorem, the centralizer theorem, and the π₁-representation theorem.

We present a new method to deal with higher symmetries of differential operators, through the Laplacian (Eastwood 2002) and its conformal powers (Gover, Silhan 2009), on conformally flat manifolds (M,g). Our approach relies first on the conformally equivariant quantization, that allows to characterize the spaces of higher symmetries in terms of their principal symbols. Secondly, we resort to symplectic reduction of the cotangent bundle of M (by the Hamiltonian geodesic flow) to identify their algebraic structure, namely a quotient of the universal envelopping algebra of the conformal Lie algebra. This connects higher symmetries of the Laplacian with quantization of the minimal nilpotent coadjoint orbit of the conformal group (Roger, Zierau 1991; Astashkevich, Brylinski 2002), and elucidate the appearance of the Joseph ideal.

There is an interesting class of vector distributions, which can be equivalently described as certain types of parabolic geometries. Having introduced these types of distributions, we will show how one can deduce just from the existence of their equivalence to certain parabolic geometries strong restrictions on the possible dimensions of their automorphism groups. This talk is based on a joint work with Andreas Cap.

A conformally flat spacetime M is a (G,X)-manifold where X is the universal cover of the space-time Ein (which is Sⁿ × R with the conformal class of the metric ds²-dt², where ds² is the standard metric on the sphere et dt² on R) and G is the identity component of his group of conformal diffeomorphisms. It turns out that this group is the universal cover of SO⁺(2,n), where n is the dimension of M. Just as in the Riemannian case, this is equivalent (in dimension n>2) to have a conformally flat lorentzian metric defined on M. The causal structure of a lorentzian manifold is conformally invariant, so we have a well defined causal structure on M.

We assume that this causal structure is globally hyperbolic. With this hypothesis when M is maximal we can prove some results about the domain of injectivity of the developping map. In particular we will show that : Given M a globally hyperbolic conformally flat maximal spacetime if there are 2 conjugates points p,q in the universal cover of M (this means that there exist 2 lightlike geodesics between p and q) then M is a finite quotient of Ein. As a conseguence of this result, we have that the developping map for similarity lorentzian manifolds is injective (as in the Riemannian case).

Let Γ₀ ⊂ O(2,1)⁰ be a Schottky group. An affine deformation is a subgroup of Aff(R^2,1) whose linear part equals Γ₀. Equivalence classes of affine deformations are parametrised by H¹(Γ⁰,R^2,1). The Margulis invariant is a function on this cohomology group that detects non-properness of an affine deformation. We discuss Labourie's extension to geodesic currents of the normalised Margulis invariant.

One of the most obvious ways to study special geometries is to consider their behaviour along certain distinguished curves. The best known case is the affine geometry. Here in the flat homogenous model, the special curves are the lines and they turn to be the geodesics on the manifolds with affine connections. For general Cartan geometries, there is a straightforward generalisation of this concept, and the lecture aims to explain the main features of such curves. We shall pay special attention to many known examples and we shall also touch the links between the special curves via Fefferman type constructions.

The automorphism group of a complex torus of dimension n consists of translations together with a discrete group Γ ⊂ SL_n(C). Generically Γ is trivial, and in the most symmetric cases, Γ is a lattice in SL_n(C), or SL_n(R), e.g. Γ = SL_n(Z). We classify compact Kähler manifolds of dimension n admitting a holomorphic action of a lattice Γ in SL_n(R) or SL_n(C), or more generally in a Lie group of real rank (n-1). They all derive from tori by a Kummer construction. This is a joint work with Serge Cantat.