Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics

Abstracts of talks, week one, July 11 - 15, 2011

J. Alt (Univ. Wittwersrand, South Africa)
"Some locally transitive actions on the Möbius sphere related to conformal holonomy"

Friday, June 11, 11:45-12:45

Recent results giving restrictions on the possibilities for irreducibly acting conformal holonomy groups H ⊂ O(p+1,q+1) in various signatures (p,q) identify two candidates for special conformal holonomy groups which haven't been studied yet: (A) the irreducibly acting SO(2,1) ⊂ O(3,2) for conformal Lorentzian 3-manifolds; and (B) Spin0(1,8) ⊂ O(8,8) for conformal manifolds of signature (7,7). The groups have a natural action on the Moebius spheres S2,1 and S7,7, respectively, which is globally transitive in the latter case, but only locally transitive in the former. In this talk, we report on recent joint work with A. J. Di Scala and T. Leistner (on case A), and with F. Leitner (on case B). In particular, the results of this work show that case A cannot occur as a conformal holonomy group, while in case B the associated Fefferman-type construction cannot yield the conformal holonomy group.

O. Biquard (Univ. Paris 7, France)
"Einstein metrics and parabolic geometries"

Tuesday, July 12, 9:30 - 10:30

I shall give an overview of a joint work with Rafe Mazzeo, seeing certain parabolic geometries as boundaries at infinity of Einstein metrics. This is a program began several years ago, and I will explain what is known, and what remains to be proved.

J. Case (Princeton Univ., USA)
"Quasi-Einstein metrics and the tractor calculus""

Wednesday, July 13, 14:00 - 14:30

Quasi-Einstein metrics are a general class of metrics which include conformally Einstein metrics, static metrics, gradient Ricci solitons, and more generally, metrics on the base of a conformally Einstein warped product metric. We will describe how, save for gradient Ricci solitons, these metrics can be naturally formulated in the language of conformal geometry, and moreover, how this leads to a new perspective on many aspects of the Ricci flow. In particular, we will discuss a natural prolongation of the quasi-Einstein condition using the tractor calculus, and some consequences of this perspective.

A. Derdzinski (Ohio State Univ., USA)
"Two-jets of conformal fields along their zero sets, in any metric signature"

Monday, July 11, 11:30 - 12:30

The connected components of the zero set of any conformal vector field v, in a pseudo-Riemannian manifold (M,g) of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former components consist of pointsat which v is essential, that is, cannot be turned into a Killing field by a local conformal change of the metric. In a component of the latter type, points at which v is nonessential form a relatively-open dense subset that is at the same time a totally umbilical submanifold of (M,g). An essential component is always a null totally geodesic submanifold of (M,g), and so is the set of those points in a nonessential component at which v is essential (unless this set, consisting precisely of all the singular points of the component, is empty). Both kinds of null totally geodesic submanifolds arising here carry a one-form, defined up to multiplications by functions without zeros, and satisfying a projective version of the Killing equation. The conformal-equivalence type of the 2-jet of v is locally constant along the nonessential submanifold of a nonessential component, and along an essential component on which the distinguished one-form is nonzero.

M. Eastwood (ANU, Australia)
"Conformal foliations and CR geometry"

Monday, July 11, 10:00 - 11:00

In Euclidean three-space, there are some special foliations called conformal. They enjoy remarkable properties related to CR geometry in five dimensions. I shall explain this construction and its geometric interpretation via twistor theory. The ideas are well-known but this exposition is joint work with Paul Baird.

C. Frances (Univ. Paris Sud)
"Essential singularities for higher dimensional conformal maps"

Tuesday, July 12, 11:00 - 12:00

Conformal immersions between Riemannian manifolds of same dimension at least 3 are a natural higher dimensional analogue of holomorphic maps. Like in the holomorphic setting, there is a notion of removable and essential singularities for conformal immersions. The aim of the talk is to give a classification of all essential singularities having a sufficiently small Hausdorff dimension, and to emphasize their link to Kleinian groups.

R. Graham (Univ. Washington, USA)
"Ambient extension of parallel tractors"

Wednesday, July 13, 9:30 - 10:30

Many interesting holonomy classes of conformal structures are characterized by the existence of parallel tractors. The question of whether a parallel tractor has a parallel extension to a tensor on the ambient space which is parallel for an ambient metric will be discussed.
An application to Nurowski's conformal structures associated to generic 2-plane distributions on 5-manifolds produces an infinite-dimensional family of pseudo-Riemannian metrics whose holonomy is the split real form of the exceptional Lie group G2. This is a report on joint work with Travis Willse.

M. Hammerl (Univ. Vienna, Austria)
"Holonomy-reductions of Cartan connections"

Tuesday, July 12, 15:30 - 16:15

We discuss the notion of holonomy for Cartan geometries and the geometric implications of reduced holonomy. At the core of the reduction-theorem lies a geometric comparison theorem which shows that the main properties of a holonomy reduction can already be read off from the corresponding reduction of the homogeneous model. The distinctive property of a holonomy reduction of a Cartan connection is that it naturally decomposes the manifold into curved analogs of group orbits, each of which inherits a reduced Cartan geometry.
As an interesting area of application we show how this reduction-procedure can be used to study the zero-locus and geometric properties of normal solutions to the first BGG-equation. This talk is based on joint work with A. Cap (Univ. Vienna) and A.R. Gover (Univ. Auckland), [arXiv:1103.4497]

A. Juhl (Univ. Uppsala, Sweden)
"Explicit formulas for GJMS-operators and Q-curvatures"

Wednesday, July 13, 11:00 - 12:00

It is well-known that in two dimensions the Laplace-Beltrami operator is conformally covariant. The Yamabe operator extends this fact to higher dimensions. It arises by correcting the Laplacian by a multiple of scalar curvature. More generally, one can also correct powers of the Laplacian by lower-order terms as to obtain conformally covariant operators. Although these operators were constructed already in 1990 by Graham, Jenne, Mason and Sparling (using the Fefferman-Graham ambient metric construction), the structure of the lower order correction terms remained notoriously complicated. We describe how one can resolve the structure of GJMS-operators in a surprising and esthetically appealing way. These results also lead to a new formula for Q-curvature.

S. Krysl (Charles Univ., Czech Republic)
"First order operators in projective contact geometry - a classification"

Wednesday, July 13, 16:30 - 17:00

In the sense of Cartan, the contact projective geometry is a curved version of the Klein homogeneous space G/P, where G=Sp(2n,R) and P is the isotropy group of the 'natural' action of Sp(2n,R) on PR2n-1.

Its underlying structure consists of a manifold M2n-1, a contact subbundle of the tangent bundle TM and a set of unparameterized geodesics going in the contact directions.

After recalling the definition of the contact projective geometry, we define first order invariant differential operators for a general Cartan geometry, and mention the result of Soucek and Slovak on first order invariant differential operators acting between the sections of irreducible finite rank bundles over parabolic geometries.

We will prove a similar result for a countable family of bundles over a projective contact geometry, each of them being associated by an infinite dimensional irreducible representation of the double cover of the symplectic group. The inducing representations we shall use are symplectic analogues of the spinor-tensor representations of the orthogonal groups. The Segal-Shale-Weil representation is an example of one of them.


C. Luebbe (Univ. Leicester, UK)
"Predicting / prescribing the location of conformal infinity"

Wednesday, July 13, 17:15 - 17:45

On an Einstein manifold a conformal geodesic has a canonical conformal factor associated to it, which is quadratic in the parameter of the curve and whose roots give the location of conformal infinity. This talk will introduce a set of conformal curves that closely mimics the behaviour of conformal geodesics. In particular the canoncial quadratic conformal factor can be retained. This allows to predict / prescribe the location of conformal infinity. We will discuss some applications.

M. Markowitz (Inform. Sec. Corp., USA)
"Intrinsic Pseudodistances for Parabolic Geometries"

Wednesday, July 13, 15:45 - 16:15

This paper generalizes the Kobayashi intrinsic pseudodistance construction to arbitrary parabolic geometries. Let C(G,P) denote the category of Cartan geometries modeled on a semisimple homogenous space G/P with P parabolic. We define an intrinsic Kobayashi pseudodistance on the objects of C(G,P) with respect to which connection preserving morphisms are nonexpansive. In general this construction depends on a suitable choice of an admissable class C of canonical curves through the origin of G/P. In contact projective geometry, for example, we have two natural choices for C: the horizontal contact projective geodesics (Cg-1) and the chains (Cg-2). A theorem of H. Wu implies that the pseudodistance associated to Cg-1 is a true distance when the underlying projective class of torsionfree affine connections contains one with negatively bounded Ricci tensor. The Kobayashi pseudodistances introduced here provide new coarse measures of incompleteness for various types of canonical curves in parabolic geometries. As such, they may prove valuable in the study of the morphisms in these categories.


V. Matveev (Univ. Jena, Germany)
"Conformal and isometric transformations of finsler manifolds: Wang theorem, Matsumoto question, Deng-Hu conjecture, conformal invariants of finsler metrics and Lichnerowicz-Obata conjecture"

Tuesday, July 12, 14:00 - 15:00

The talk is based on the joint paper with Marc Troyanov. We introduce a new construction basing on the convex geometry that associates a Riemannian metric gF (called the Binet-Legendre metric) to a given Finsler metric F on a smooth manifold M. The transformation mapping F to gF is C0-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric gF also behave nicely under conformal, isometric or bilipshitz deformation of the Finsler metric F$ that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Minkowski spaces. We describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hu on locally symmetric Finsler spaces.


B. McKay (Trinity College, Ireland)
"Hartogs extension theorem for parabolic geometries"

Thursday, July 14, 15:30 - 16:30

I will explain the proof that holomorphic parabolic geometries exhibit the Hartogs extension phenomenon: any holomorphic parabolic geometry defined on a domain in a Stein manifold extends to a unique holomorphic parabolic geometry on the envelope of holomorphy of that domain.

J. Merker (Univ. Orsay, France)
"Effective Cartan-Tanaka connections on C6-smooth strongly pseudoconvex hypersurfaces M3 in C2"

Wednesday, July 13, 16:30 - 17:15

In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type when the distribution is equipped with an integrable complex structure, so as to derive the principal curvature invariants and (co)frame(s) associated to strongly pseudoconvex real hypersurfaces M3 in C2, an approach which is alternative and complementary to the `hyperspherical' connection of Elie Cartan, and to the so-called calculi of Fefferman, of Chern-Moser, of Webster. By choosing an initial frame for $TM$ which is explicit in terms of a local graphing function v=f(x,y,u) for M, we provide a Cartan-Tanaka connection all elements of which are completely explicit in terms of f(x,y,u), assuming only C6-smoothness of M. The Gaussian requirement for systematic computational effectiveness then shows - a bit unexpectedly - that the two main curvatures are rational differential expressions in the sixth-order jet of f(x,y,u), the lengths of which are several hundred pages long on a computer - just for the simplest instance of local embedded CR geometry. Large parts of the memoir aim at formulating general statements that will be useful for further constructions of Cartan-Tanaka connections related to the equivalence problem for (local) embedded CR manifolds whose CR-automorphism group is not semi-simple, cf. e.g. some model lists by Beloshapka.


A. Taghavi-Chabert (Masaryk Univ. Czech Republic)
"The Goldberg-Sachs theorem in higher dimensions"

Wednesday, July 13, 14:45 - 15:15

In four dimensions, the Goldberg-Sachs theorem gives necessary and sufficient conditions on the Weyl tensor and Cotton-York tensor for the existence of a locally integrable distribution of complex null 2-planes on a real or complex (pseudo)-Riemannian manifold. We show how the theorem generalises to higher dimensions in the holomorphic category, and time-permitting, we discuss its real versions.

B. Warhurst (Univ. New South Wales, Australia)
"Ultrarigid tangents of sub-Riemannian nilpotent groups"

Monday, July 11, 17:00 - 17:45

Margulis and Mostow showed that if two equiregular sub-Riemannian manifolds are quasiconformally equivalent, then almost eve- rywhere they have isomorphic Gromov tangent cones. In other words, the tangent cone is a quasiconformal invariant. Their work extends a result of Pansu which says that two Carnot groups are quasiconformally equivalent if and only if they are isomorphic. In this talk I will present a joint result with E. Le Donne and A. Ottazzi showing that the converse of the theorem of Margulis and Mo- stow fails in a strong sense. In particular, we show that there exist two nilpotent Lie groups equipped with left invariant sub-Riemannian metrics, whose tan- gent cones are isomorphic at every point, but which are not quasiconformally equivalent. This result relies on studying those Carnot groups whose quasiconformal maps can only be translations and dilations. We shall refer to groups with this property as ultrarigid groups. One of the tools in our method is to provide an algebraic characterization of ultrarigidity.

V. Zadnik (Masaryk Univ., Czech Republic)
"On split signature conformal metrics induced by half-dimensional projective structures"

Thursday, July 14, 16:45 - 17:30

Constructions of conformal metrics of split signature (2,2) from 2-dimensional projective structures are known due to the works of Nurowski-Sparling and Dunajski-Tod. We plan to revisit this topic from the parabolic point of view, discuss an analogy in general dimension and, hopefully, extend the characterization of the resulting conformal structures.


I. Zelenko (Texas A&M Univ., USA)
"Tanaka prolongation and symplectification procedure for filtered structures on manifolds"

Friday, July 15, 11:00 - 12:00

The talk is devoted to local equivalence problem for vector distributions (subbundles of tangent bundles). First I will review the classical approaches to this problem, making special emphasis to the algebraic version of Cartan's method of equivalence developed by N. Tanaka in 1970s. The central object in the Tanaka approach is the notion of a symbol of a distributions at a point, which is a graded nilpotent Lie algebra. The prolongation procedure (i.e. the procedure of getting a canonical frame) can be described in terms of natural algebraic operation in the category of graded Lie algebras. Through this review of Tanaka theory I will motivate the recent approach of B. Doubrov and myself to this problem. Our approach is a combination of a kind of a symplectification of the problem (taking its origin in Pontryagin theory in Optimal Control) and various Tanaka type prolongations. This approach allowed us to make a unified construction of canonical frames for distribution of arbitrary rank independently of their Tanaka symbols, avoiding the problem of classification of graded nilpotent Lie algebras with given number of generators.