Riemannian and sub-Riemannian geometry on Lie groups and homogeneous spaces

Geometry Analysis and Dynamics on sub-Riemannian manifolds

**14:00-14:50**Alexey Bolsinov*(Loughborough University, UK)*

Title:*Integrable geodesic flows of Riemannian and sub-Riemannian metrics on homogeneous spaces*

Abstract: We discuss general algebraic methods for constructing integrable geodesic flows of Riemannian and sub-Riemannian metrics on homogeneous spaces and Lie groups. Our approach is based on the concept of non-commutative integrability and the classical idea of dual Poisson algebras suggested by Sophus Lie.**14:50-15:20**Coffee break**15:20-16:10**Claudio Gorodski*(Universidade de Sao Paulo, Brazil)*

Title:*A metric approach to representations of compact Lie groups*

Abstract: Consider a (finite dimensional) real representation of a compact Lie group and the associated orbit space with its canonical metric space structure. In this talk, we would like to address the following question: "How much of the representation can be recovered from the quotient metric space?". The prototypical example of such situation is the adjoint representation of a compact connected Lie group on its Lie algebra, whose orbit space is identified with a Coxeter orbifold. Our work generalizes some results about adjoint representations to other representations. (Joint work with A. Lytchak (Koeln)).**16:10-17:00**Enrico Le Donne*(University of Jyvaskyla, Finland)*

Title:*Isometries of homogeneous spaces equipped with intrinsic distances*

Abstract: We consider homogeneous spaces equipped with distances for which every pair of points can be joined with an arc with length equal to the distance of the two points. These distances are generalizations of Riemannian distances. They are completely described as subFinsler structures, by the work of Gleason, Montgomery, Zippin, and Berestowski. We are interested in studying the isometries of such metric spaces. As for the Riemannian case, we show that an isometry is uniquely determined by the blown-up map at a point. The blown-up map is an isometry between the tangent metric spaces, which in this case are particular groups called Carnot groups. Generalizing a result of U. Hamenstadt, we also show that an isometry between open sets of Carnot groups are affine maps. A key point in the argument is in showing smoothness of such isometries.

The work is in collaboration with L. Capogna and A. Ottazzi.

**9:00-9:50**Valentino Magnani*(Pisa University, Italy)*

Title:*Surface measure in SR Geometry and shape of the unit ball*

Abstract: Motivated by recent results in Geometric Measure Theory, we present a new approach to the surface measure in homogeneous groups. We will see how the geometry of the metric unit ball plays a prominent role to establish the validity of a suitable area formula for the spherical Hausdorff measure. In the case the distance is sub-Riemannian, we find a bridge between this formula and the regularity properties of the sub-Riemannian ball. We will focus our attention on the perimeter measure in Carnot groups.**9:50-10:20**Coffee break**10:20-11:10**Piotr Mormul*(Institute of Mathematics, University of Warsaw, Poland)*

Title:*All corny extremals in Goursat Monster Tower with corner points tangential are not local SR minimizers*

Abstract: In the year 1997 the author proved that the corny extremals in the SR structure in ${\mathbb R}^5(x_1,\dots,\,x_5)$ with the orthonormal basis $\bigl(\partial_1\,,\ \partial_2 + x_1\partial_3 + x_3\partial_4 + x_3^{\,2}\partial_5\bigr)$ are not local SR minimizers -- an improvement over a similar example in [1], Section 9.6, constructed in dimension 6. Our example lived in the third stage of the Goursat Monster Tower (GMT) --- the first stage featuring singularities of Gorsat distributions, discovered by Kumpera & Ruiz not earlier than in 1978. We came back to this topic during a winter school/last winter/Krynica, Poland. Later we ran across a draft text [4] dealing with the well-known Agrachev & Gauthier' example of 2012.

By merging the techniques developed during the Krynica school with that of [4] we now get rid of all extremals' corners [for the Goursat structures] which are tangential points, in any fixed stage of the GMT. (The definition of a tangential point has been given in [2], Definition 4. Tangential points build up tangential strata which could be of arbitrarily high codimensions, in sufficiently high stages of the GMT.)

Corners show up when two horizontal curves meet: (1) a vertical curve and (2) critical curve which is a prolongation of a vertical one. (The notions of vertical and critical curves in the GMT have been introduced in [3], Definitions 2.16 and 2.17.) A critical curve in (2) could be an arbitrarily long sequence of prolongations of its vertical ancestor lying many stages down in the GMT. At a concatenation corner point one swaps between (1) and (2) losing not curve's overall extremality/abnormality.

In the initial example of 1997 the (1) curve was the line $\{(t,0,0,0,0)\}$, and (2) was the line $\{(0,t,0,0,0)\}$, with the corner point $(0,0,0,0,0)$.

[1] W.Liu, H.J.Sussmann; Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions, Memoirs AMS 564 (1995).

[2] P. Mormul; Goursat distributions not strongly nilpotent in dimensions not exceeding seven, Lecture Notes in Control and Information Sciences 281 (2003), 249--261.

[3] R. Montgomery, M. Zhitomirskii; Points and Curves in the Monster Tower, Memoirs AMS 956 (2010).

[4] E. Le Donne et al; A short proof of the non-minimality of corners in a 4-dimensional subriemannian structure, November 2013 (a draft).**11:10-12:00**Davide Barilari*(Université Paris 7, France)*

Title:*Invariants and conjugate points for sub-Riemannian structures on 3D Lie groups*

Abstract: In this talk I will describe the two functional invariants of a 3D contact sub-Riemannian structure, discussing their role in the classification of left-invariant sub-Riemannian structures on three dimensional Lie groups. I will then discuss some results about comparison theorems for conjugate points along sub-Riemannian geodesics. As a byproduct we obtain a new result about existence of conjugate points along geodesics on left-invariant sub-Riemannian structures on unimodular Lie groups.

**14:00-14:50**Philippe Jouan*(Université de Rouen)*

Title:*Almost-Riemannian geometry and linear vector fields on Lie groups*

Abstract: A vector field on a connected Lie group G is said to be linear if its flow is a one parameter group of automorphisms (also called infinitesimal automorphism), and a controlled system on G is said to be linear if the drift vector field is linear, and the controlled ones are right invariant. The motivation for dealing with such systems is twofold. On the one hand they are natural extensions of invariant systems on Lie groups. On the other one they can be generalized to homogeneous spaces and appear as models for a wide class of systems, on account of the Equivalence Theorem of [1] .

It is natural to define an Almost-Riemannian Structure, ARS in short, on a connected Lie group G by n=dim G invariant or linear vector fields, considered as an orthonormal frame. This kind of ARS can also be defined on homogeneous spaces, and we show that a general ARS that generates a finite dimensional Lie algebra is diffeomorphic to an ARS on a Lie group or a homogeneous space. This equivalence is local or global according to some technical assumptions. Excepted for the equivalence statement, we will describe ARSs on Lie groups defined by one linear vector field and n-1 invariant ones. This description includes the singular locus, the lifting to a sub-Riemannian problem on a (n+1)-dimensional Lie group, the Hamiltonian equations of the geodesics, and is applied to examples.

This is a joint work with Victor AYALA.

[1] Ph. Jouan, Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces, ESAIM: Control Optimization and Calculus of Variations, 16 (2010) 956-973.**14:50-15:40**Antonio Lerario*(Université Claude Bernard Lyon 1, France)*

Title:*Geodesics and horizontal path spaces in Carnot Groups*

Abstract: I will discuss quantitative and qualitative aspects of the structure of geodesics on subriemannian manifolds on the infinitesimal scale, i.e. on Carnot groups. I will introduce a topological invariant (defined on the space of horizontal paths) which allows to pass information on the structure of geodesics from the infinitesimal to the local scale. (joint work with A. A. Agrachev and A. Gentile)**15:40-16:10**Coffee break**16:10-17:00**Remco Duits*(Eindhoven University of Technology, The Netherlands)*

Title:*Best exponential curve fits, locally adaptive frames, sub-Riemannian geodesics within SE(d), and their applications in medical imaging*

Abstract: In image analysis and cortical modeling, image processing on the joint space of positions and orientations is beneficial. This joint space is embedded as a Lie group quotient $\R^{d} \rtimes S^{d-1}:= SE(d)/(\{0\} \times SO(d\!-\!1))$ in $SE(d)$. It allows to include context of lines via stochastic line propagation PDE's yielding generic crossing-preserving flows acting on rich data representations $U:\R^{d}\rtimes S^{d-1} \to \mathbb{C}$, typically obtained via a wavelet- or coherent state transform, or via orientation lifts. Processing on $U$ must be left-invariant. Therefore it primarily relies on the frame of left-invariant vector fields, as done in previous works by August & Zucker, Boscain & Rossi, Citti & Sarti, Duits & Franken, Felsberg, Gautier, Petitot, Sachkov, Siddiqi etc..

This differential frame is left-invariant, but it lacks adaptivity. Therefore we generalize the notion of locally adaptive frames on images (see e.g. previous works by Blom, Florack, Koenderink, Lindeberg), to locally adaptive frames into the rich data-representations $U$. We set up theory and algorithms for such gauge frames steered by local and optimal exponential curve fits and show clear benefits in 2D \& 3D medical image analysis.

Finally, we present and analyze sub-Riemannian geodesics within SE(3) and analyze where their spatial projections exhibit cusps. Numerical Jacobian computations clearly indicate that there are no conjugate points before such cusps appear, and stationary curves whose spatial projections do not exhibit cusps seem to be globally optimal, as was already formally proven for sub-Riemannian geodesics within SE(2).

Dmitri Alekseevsky *(Brno)* and Yuri Sachkov *(Pereslavl)*