Sub riemannian geometry and lie groups pdf

Isometries of almostriemannian structures on lie groups. Geodesics equation on lie groups with left invariant metrics. Pseudoriemannian manifolds and isometric actions of. Rossiy abstract in this paper we study the carnotcaratheodory metrics on su2 s3, so3 and sl2 induced by their cartan decomposition and. Leftinvariant structures on lie groups are the basic models of sub riemannian manifolds and the study of such structures is the starting point to understand the general properties of sub riemannian geometry. Geometric actions of simple groups rigidity or structure results geometric actions and their symmetries killing elds xing pointsreferences pseudoriemannian manifolds and isometric actions of simple lie groups raul quirogabarranco cimat, guanajuato, mexico 7th international meeting on lorentzian geometry, 20 sao paulo, brazil. In particular, thanks to the group structure, in some of these. This paper is the third in a series dedicated to the fundamentals of sub riemannian geometry and its implications in lie groups theory. Informally, a subriemannian geometry is a type of geometry in which the trajectories. Subriemannian geometry also known as carnot geometry in france, and nonholonomic riemannian geometry in russia has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely. Sorry, we are unable to provide the full text but you may find it at the following locations. Constant mean curvature surfaces in subriemannian geometry hladky, r.

However, this is the first part of three, in preparation, dedicated to this subject. At generic points, this is a nilpotent lie group endowed with a. The spectral geometry of a riemannian manifold gilkey, peter b. It covers manifolds, riemannian geometry, and lie groups, some central.

Scienti c semester geometry, analysis and dynamics on subriemannian manifolds. The purpose of the present paper is to develop a subriemannian calculus on smooth hypersurfaces in a class of nilpotent lie groups which possess a rich geometry. For all that concerns general subriemannian geometry, including almostriemannian one, the reader is referred to. Introduction to riemannian and subriemannian geometry. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. It is shown that to each lie group one can associate a lie algebra, i. Riemannian geometry is the special case in which h tm. Subriemannian structures on nilpotent lie groups rory biggs geometry, graphs and control ggc research group department of mathematics. About almostriemannian structures on lie groups more details can be found in. Integrability of the subriemannian geodesic flow on 3d lie groups19. Browse other questions tagged liegroups riemanniangeometry or ask your own question. This paper is a continuation of 5, where it is argued that subriemannian geometry is in fact noneuclidean analysis. In chapter 1 we introduce the necessary notions and state the basis results on the curvatures of lie groups.

Although a great progress has been made lately, specially in the solvable case, the general case seems to be far from being completely understood. These pages covers my expository talks during the seminar sub riemannian geometry and lie groups organised by the author and tudor ratiu at the mathematics. Rossiy abstract in this paper we study the carnotcaratheodory metrics on su2 s3, so3 and sl2 induced by their cartan decomposition and by the killing form. These pages covers my expository talks during the seminar subriemannian geometry and lie groups organised by the author and tudor ratiu at the mathematics department, epfl, 2001. These notes entitled sub riemannian geometry on the heisenberg group are mostly based on. Introduction to riemannian and subriemannian geometry fromhamiltonianviewpoint andrei agrachev. A lie group is a group with g which is a differentiable manifold and such.

It can be quite difficult even in the simplest case of leftinvariant 3dimensional manifolds, where the complete description of the. It covers, with mild modifications, an elementary introduction to the field. Asymptotic expansion of the 3d contact exponential map20. There are few other books of subriemannian geometry available. Actu ally from the book one can extract an introductory course in riemannian geometry as a special case of subriemannian one, starting from the geometry of surfaces in chapter 1. We give a complete classi cation of leftinvariant subriemannian structures on three dimensional lie groups in terms of the basic di erential invariants. Then we get a biinvariant riemannian metric on g, preserved by left and. G, endowed with such a subriemannian structure, a k. Carnot group, a class of lie groups that form subriemannian manifolds.

Subriemannian structures on 3d lie groups springerlink. A leftinvariant distribution is uniquely determined by a two dimensional subspace of the lie algebra of the group. On extensions of subriemannian structures on lie groups. The subject is in close link with subriemannian lie groups because curvatures could be classi. Such structures are basic models for subriemannian manifolds and as such serve to elucidate general features of subriemannian geometry. Nodal geometry on riemannian manifolds chanillo, sagun and muckenhoupt, b. This paper is about noneuclidean analysis on lie groups endowed with left invariant distributions, seen as subriemannian manifolds. The study of riemannian geometry is rather meaningless without some basic knowledge on gaussian geometry i. The goal of this phd is to investigate a subriemannian setting where the control of the shape deformations is limited to a subspace of the lie algebra of a transformation group either by pca or because of the discretization. Rashevski and ballbox theorems of nitedimensional subriemannian geometry see 11,35. Riemannian geometry and lie groups in this section we aim to give a short summary of the basics of classical riemannian geometry and lie group theory needed for our proofs in following sections.

This is more easy to see when approaching the concept of a subriemannian lie group. In chapter 3 we turn instead to topology, and we investigate the structure of the bers of the endpoint map from the. Subriemannian geometry andre bellaiche, jeanjaques. Subriemannian geometry is the study of a smooth manifold equipped with a. Pdf tangent bundles to subriemannian groups semantic. If the grading is xed, we say that g is a graded lie group. A comprehensive introduction to subriemannian geometry by. Lie group action, approximate invariance and subriemannian geometry in statistics at the interface of geometry, statistics, image analysis and medicine, computational anatomy aims at analyzing and modeling the biol ogical variability of the organs shapes and their dynamics at the population level. Paper related content geodesics in the subriemannian. A sub riemannian structure on a lie group is said to be leftinvariant if its distribution and the inner product are preserved by left translations on the group. In this talk we will be interested in lie groups admitting a metric with negative ricci curvature. Lie groups and homogeneous spaces this item was submitted to loughborough universitys institutional repository by thean author. Graduable lie groups are nilpotent and in particular g is di eomorphic to g via the exponential map.

In the words of montgomery 16 carnot groups are to subriemannian geometry as euclidean spaces are to riemannian geometry. Homotopy properties of horizontal path spaces and a theorem of serre in subriemannian geometry 41 1. Subriemannian geometry is a relatively young area in mathematics 2. A comprehensive introduction to subriemannian geometry. Read subriemannian structures on 3d lie groups, journal of dynamical and control systems on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The standard reference for classical riemannian geometry is do carmos book do carmo, 1992. The intrinsic hypoelliptic laplacian and its heat kernel. This paper is the third in a series 4, 5, dedicated to the fundamentals of subriemannian geometry and its implications in lie groups theory. We do not require any knowledge in riemannian geometry. This is a an updated version, which will be modified according to the contributions of the other participants to the borel seminar. Subriemannian calculus on hypersurfaces in carnot groups. Integrable geodesic ows of riemannian and subriemannian metrics on lie groups and homogeneous spaces. Classi cation of non flat left invariant structures 29 7. Subriemannian structures on 3d lie groups, journal of.

As a starting point we have a real lie group gwhich is connected and a vectorspace d. The homogeneous spaces we are interested in can be seen as factor spaces of lie groups with left invariant distributions. Integrable geodesic ows ofriemannian and subriemannian. Taking m to be a lie group, the subriemannian structure.

Conversely, every such quadratic hamiltonian induces a subriemannian manifold. Local conformal flatness of left invariant structures 25 6. Strati ed lie groups are examples of graded lie groups. Subriemannian structures on groups of di eomorphisms. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other.

Notes on differential geometry and lie groups cis upenn. A lie group is a manifold which is also a group and is such that the group. Let f 1, f 2, f p be a set of smooth vector fields on a manifold m. Researchers have studied subriemannian geodesics in lie groups and have found them useful for the e. Such groups arise as tangent spaces of gromovhausdorff limits of riemannian manifolds 4,75, and since they. Of special interest are the classical lie groups allowing concrete calculations of many of the abstract notions on the menu. The existence of geodesics of the corresponding hamiltonjacobi equations for the subriemannian hamiltonian is given by the chowrashevskii theorem. Subriemannian geometry is the geometry of a world with nonholonomic constraints. Differential geometry, lie groups, and symmetric spaces, academic press, 1978. A leftinvariant subriemannian structure g, d, g on a lie group g consists of a nonintegrable leftinvariant distribution d on g and a leftinvariant riemannian metric g on d. This paper is the third in a series dedicated to the fundamentals of subriemannian geometry and its implications in lie groups theory. It will also be a valuable reference source for researchers in various disciplines.

These notes entitled subriemannian geometry on the heisenberg group are mostly based on. A comprehensive introduction to subriemannian geometry hal. The book may serve as a basis for an introductory course in riemannian geometry or an advanced course in subriemannian geometry, covering elements of hamiltonian dynamics, integrable systems and lie theory. The overflow blog coming together as a community to connect. Thanks to 40, selfsimilar lie groups are graduable lie groups with a grading induced by the metric dilation. These pages covers my expository talks during the seminar subriemannian geometry and lie groups organised by the author and tudor ratiu at the mathematics.