Workshop on

Equivalence, invariants, and symmetries of vector distributions

and related structures : from Cartan to Tanaka and beyond

Institut Henri Poincare

December 10-12, 2014

Organizers: R. Bryant (Duke), Y. Chitour (Orsay), J. Merker (Orsay), I. Zelenko (Texas A&M)

This workshop is a part of the IHP trimester "Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds"

Speakers

Program

Abstracts

List of Participants

How to come to IHP

Registration

The workshop will focus on recent advances in differential geometry of various geometric structures involving nonholonomic distributions such as distirbutions themselves, sub-Riemannian structures, Cauchy-Riemann structures, parabolic geometries etc with applications to geometry of differential equations and control theory.

Speakers

Valery Beloshapka (Moscow), Сoordinate approach in CR-geometry

Gil Bor (CIMAT, Mexico), Rolling of projective planes with G2-symmetry

Robert Bryant (Duke University), Some geometric constructions of holonomic plane fields and their analysis

Andreas Cap (Vienna), Projective compactness

Boris Doubrov (Minsk) Equivalence problem of vector distributions via abnormal extremals

Maciej Dunajski (Cambridge), Conformally Einstein metrics and three dimensional path geometries

Matthias Hammerl (Greifswald, Germany), A non-normal Fefferman-type construction: Holonomy-methods and
characterization

Jun-Muk Hwang (KIAS), Cone structures with characteristic connections

Svetlana Ignatovich (Kharkiv) , On application of formal power series and free algebras in the homogeneous approximation problem for nonlinear control systems

Alexander Isaev (Canberra, Australia), Reduction of five-dimensional uniformly Levi degenerate CR-structures to absolute parallelisms and its application to establishing the affine rigidity of Levi degenerate tube hypersurfaces.

Velimir Jurdjevic (Toronto), Integrable affine-quadrartic systems on Lie groups: geometry and mechanics.

Aroldo Kaplan (Cordoba, Argentina), Non-integrable systems in algebraic geometry

Boris Kruglikov (Tromso, Norway), Tanaka theory and dimensional bounds for the symmetry algebras

Richard Montgomery (Santa Cruz, USA), A Tale of Two Towers

Tohru Morimoto (Nara, Japan), Equivalence problems of geometric structures and applications to differential equations
and subriemannian geometries.

Alessandro Ottazzi (University of Trento, Italy), Diffeomorphisms in Carnot groups and Tanaka prolongation

Colleen Robles (College Station, USA), Characteristic cohomology of the horizontal distribution on flag manifolds and flag domains

Jan Slovak (Brno) , Subriemannian Metrics and some Parabolic Geometries

Program

	December 10, Wednesday	December 11, Thursday	December 12, Friday
8:30-9:00	Registration
9:00-10:00	R. Bryant	J.-M. Hwang	A. Isaev
10:00-10:30	Coffee break	Coffee break	Coffe break
10:30-11:30	T. Morimoto	C. Robles	R. Montgomery
11:35-12:35	B. Kruglikov	B. Doubrov	A. Kaplan
12:35-14:00	Lunch	Lunch	Lunch
14:00-15:00	A. Cap	G. Bor	J. Slovak
15:05-16:05	V. Jurdjevic	S. Ignatovich	V. Beloshapka
16:05-16:35	Coffee break	Coffee break	Coffee break
16:35-17:35	A. Ottazzi	M. Dunajski	M. Hammerl

        Abstracts

Valery Beloshapka, Сoordinate approach in CR-geometry

In modern differential geometry it works the unwritten imperative: coordinate-free methods are better and more preferable than coordinate
ones. However, in CR-geometry these two different methods interact fruitfully:
-- coordinate, analytical one (R. Descartes, H. Poincaré, J. Moser, ...)
-- coordinate-free, geometrical one(F. Klein, E. Cartan, N, Tanaka, ...)
In my talk I am going to tell about some successes of first analytical approach and about problems related with its correspondence with the second
approach.

Gil Bor, Rolling of projective planes with G2-symmetry

I will report on several interrelated apparently new geometric interpretations of the well-known Cartan-Engel (2,3,5) distribution with
symmetry algebra of type G2. One of the interpretations is a projective geometric analog of rolling of riemannian surfaces. This is joint work with
Luis Hernandez and Pawel Nurowski.

Robert Bryant, Some geometric constructions of holonomic plane fields and their analysis

As is well-known, the holonomic system of a ball rolling without slipping or twisting on a plane is described by a 2-plane field with growth
vector (2,3,5), but this is not the famous plane field discovered by Cartan and Engel, whose local automorphism group is of dimension 14. However, the
case of a ball of radius 1 rolling without slipping or twisting over a ball of radius 3 does turn out to be locally isomorphic to the Cartan/Engel 2-plane field.
Recently, Nurowski and An found examples of surfaces whose holonomic system when rolling over a plane *is* the Cartan/Engel 2-plane
field, and I have shown that, for one surface rolling over another (where the two surfaces have distinct Gauss curvatures), the associated 2-plane
field cannot be of Cartan/Engel type unless at least one of the surfaces has constant Gauss curvature. I will report on this result and some other
related geometric constructions of 2- and 3-plane fields whose equivalence with the flat model is of interest.

Andreas Cap, Projective compactness

This talk is based on recent joint work with A.R. Gover (Auckland). Guided by examples arising from holonomy reductions of parabolic geometries, we develop a concept of projectively compact (complete) affine connections on the interior of a smooth manifold with boundary and obtain induced structures on the boundary. Via the Levi-Civita connection, this automatically applies to pseudo-Riemannian metrics. Thus it provides an alternative to the concept of conformal compactness, in which the geodesic structure is
emphasized.

Boris Doubrov, Equivalence problem of vector distributions via abnormal extremals

This is a joint work with Igor Zelenko. We show how ideas from geometric Control Theory lead to powerful techniques for solving the equivalence problem for a large family of vector distributiuons. Starting from the notion of abnormal extremals, we construct a natural pseudo-product structure associated with an arbitrary vector distribution. In many cases this allows us to construct a canonical coframe associated with a distribution and, thus, solve an equivalence problem. Unlike Tanaka theory, using this approach we get the full classification of all possible flat models. We illustrate this technique on a number of classical and non-classical examples, relating the geometry of vector distributions with properties of rational curves in flag varieties.

Maciej Dunajski, Conformally Einstein metrics and three dimensional path geometries

I shall discuss the necessary and sufficient conditions for a neutral or a Riemannian four-dimensional manifold with
anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar
and tensor conformal invariants, or as point invariants for a system of second order ODEs defining a three-dimensional path geometry.

Matthias Hammerl, A non-normal Fefferman-type construction: Holonomy-methods and characterization

I will discuss a particular Fefferman-type construction which is based on the inclusion $SL(n+1)->Spin(n+1,n+1)$ and which describes an alternative
approach to a construction by Dunajski-Tod. This Fefferman-type construction is non-normal for n>2 in the non-flat case, and
it has been a considerable challenge to obtain a suitable characterization of the resulting conformal structures. In this talk I will describe an
adapted holonomy-reduction which yields a complete characterization in terms of underlying conformal data. The talk is based on joint work with
Josef Silhan, Arman Taghavi-Chabert and Vojtech Zadnik from Masaryk University Brno and Katja Sagerschnig from University of Vienna.

Jun-Muk Hwang, Cone structures with characteristic connections

We will start with an introductory survey of cone  structures with characteristic connections, many examples of which arise in algebraic geometry. Then we will concentrate on a 7-dimensional
example subordinate to a degenerate distribution of rank 6, which plays a crucial role in a rigidity problem
in algebraic geometry.

Svetlana Ignatovich, On application of formal power series and free algebras in the homogeneous approximation problem for nonlinear control systems

We consider nonlinear (real analytic) control systems that are linear or affine with respect to control. We show how, in
studying of local properties of such systems, algebraic objects arise, namely, a free associative algebra, a free Lie algebra, formal
power series of non-commuting variables, shuffle product. Then we discuss homogeneous approximation problem in an appropriate algebraic
context. This allows us to give a complete classification of homogeneous approximations, show a way for construction of approximating
systems, and describe all privileged coordinates.
The talk is based on joint works with G.M.Sklyar.

Alexander Isaev, Reduction of five-dimensional uniformly Levi degenerate CR-structures to absolute parallelisms and its application to establishing
the affine rigidity of Levi degenerate tube hypersurfaces.

Owing to classical work by E. Cartan, Tanaka and Chern, the CR-structures of Levi nondegenerate CR-hyperfaces are known to
reduce to absolute parallelisms, which, in particular, yields a solution to the local CR-equivalence problem for such CR-manifolds. Until
recently, all generalizations of this fact to other classes of CR-manifolds have assumed Levi nondegeneracy. I will speak about our result
with D. Zaitsev (published in 2013), which achieves reduction to absolute parallelisms for a large class of Levi degenerate
CR-hypersurfaces. Furthermore, I will explain how the reduction implies the following affine rigidity result: every tube hypersurface in
complex Euclidean 3-space that is 2-nondegenerate and uniformly Levi degenerate of rank 1 with vanishing CR-curvature is affinely
equivalent to an open subset of the tube over the future light cone.

Velimir Jurdjevic, Integrable affine-quadrartic systems on Lie groups: geometry and mechanics.

Abstract in pdf

Aroldo Kaplan, Non-integrable systems in algebraic geometry

I will review Griffiths’ horizontal distribution that arises in the Hodge theory of algebraic varieties, apply Tanaka theory to it, and
discuss some associated parabolic geometries.

Boris Kruglikov, Tanaka theory and dimensional bounds for the symmetry algebras

I will discuss maximal and submaximal bounds on the dimension of
symmetry of a filtered geometric structure. In addition to the general
result on the gap of dimensions, I will discuss some partial progress on
finer bounds depending on curvature types, the amount of submaximal models
and the results in non-parabolic context. The talk is based on joint works
with D.The, V.Matveev, H.Winther, B.Doubrov.

Richard Montgomery, A Tale of Two Towers

Iterated Cartan prolongation, done projectively, starting off with a flat space, leads to the `Monster tower' which has been useful in understanding the class of Goursat distributions.
Iterated Nash blow-up, done a la Algebraic geometry, starting with the projective space, leads to the `Semple tower' useful in intersection theory and counting questions amongst curves intersecting non-transversally in the projective plane. Both have as inputs positive integers k and n, $k < n$, with n the dimension of the initial ambient space and k the dimension of submanifolds of interest. When $(k,n) = (1,2)$ the two towers are `equal'. Likely they are equal for $(k,n) = (1,n)$. For $k > 1$ both spaces are singular once you have climbed to around floor 3 (French ètage 2). What benefit can be gained by building bridges between these two towers? What old or new questions might be answered? This is an initial report on work begun with Alex Castro [PUC, Rio de Janeiro, Brazil], Susan Colley [Oberlin, Ohio] , Wyatt Howard [Santa Clara University and De Anza College, San Jose, CA, US], Gary Kennedy [Ohio State U., US] , Corey Shanbrom [Cal State Sacramento, CA, US], and especially Misha Zhitomirksii [Tecnion, Haifa, Israel] .

Tohru Morimoto, Equivalence problems of geometric structures and applications to differential equations and subriemannian geometries

Reviewing the equivalence problems and geometric studies of differential equations from Lie and Cartan to Kuranishi, Spencer, Sternberg, Tanaka and others, I will
address several related problems on holonomic systems of differential equations and subriemanniangeometries.

Alessandro Ottazzi , Diffeomorphisms in Carnot groups and Tanaka prolongation.

In this talk I consider some classes of diffeomorphisms in Carnot groups: isometric, conformal, quasiconformal, contact maps. I present some results in the study of rigidity of these maps, obtained in different collaations with M. Cowling, E. Le Donne and B. Warhurst. The proof of these results partially relies on Tanaka prolongation theory.

Colleen Robles, Characteristic cohomology of the horizontal distribution on flag manifolds and flag domains

Abstract in pdf

Jan Slovak, Subriemannian Metrics and some Parabolic Geometries

The talk will report on work in progress, a joint project of David M.J. Calderbank, Vladimir Soucek and myself, devoted to the
exploitation of the classical linearization principle known from the projective metrizability problem.
In the realm of parabolic geometries, this leads to the quest for subriemannian metric partial connections within the class of the Weyl
structures on a given parabolic geometry. I shall pay particular attention to the class of parabolic geometries with the irreducible
defining distributions (i.e. one cross, or exceptionally two crosses, in the corresponding Satake diagram).