The standard reference for classical riemannian geometry is do carmos book do carmo, 1992. Then we get a biinvariant riemannian metric on g, preserved by left and. Introduction to riemannian and subriemannian geometry. This paper is the third in a series 4, 5, dedicated to the fundamentals of subriemannian geometry and its implications in lie groups theory. A lie group is a group with g which is a differentiable manifold and such. Homotopy properties of horizontal path spaces and a theorem of serre in subriemannian geometry 41 1. About almostriemannian structures on lie groups more details can be found in. A lie group is a manifold which is also a group and is such that the group. 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. A comprehensive introduction to subriemannian geometry hal. However, this is the first part of three, in preparation, dedicated to this subject. Notes on differential geometry and lie groups cis upenn. A comprehensive introduction to subriemannian geometry. Subriemannian structures on nilpotent lie groups rory biggs geometry, graphs and control ggc research group department of mathematics.
It is shown that to each lie group one can associate a lie algebra, i. A comprehensive introduction to subriemannian geometry by. It covers manifolds, riemannian geometry, and lie groups, some central. As a starting point we have a real lie group gwhich is connected and a vectorspace d. This paper is about noneuclidean analysis on lie groups endowed with left invariant distributions, seen as subriemannian manifolds. A leftinvariant distribution is uniquely determined by a two dimensional subspace of the lie algebra of the group. This paper is a continuation of 5, where it is argued that subriemannian geometry is in fact noneuclidean analysis. The homogeneous spaces we are interested in can be seen as factor spaces of lie groups with left invariant distributions. There are few other books of subriemannian geometry available. This paper is the third in a series dedicated to the fundamentals of subriemannian geometry and its implications in lie groups theory. Isometries of almostriemannian structures on lie groups. In chapter 1 we introduce the necessary notions and state the basis results on the curvatures of lie groups. Researchers have studied subriemannian geodesics in lie groups and have found them useful for the e.
Introduction to riemannian and subriemannian geometry fromhamiltonianviewpoint andrei agrachev. 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. Rossiy abstract in this paper we study the carnotcaratheodory metrics on su2 s3, so3 and sl2 induced by their cartan decomposition and. 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. Riemannian geometry is the special case in which h tm. 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. Informally, a subriemannian geometry is a type of geometry in which the trajectories. In chapter 3 we turn instead to topology, and we investigate the structure of the bers of the endpoint map from the. The subject is in close link with subriemannian lie groups because curvatures could be classi. Local conformal flatness of left invariant structures 25 6. Although a great progress has been made lately, specially in the solvable case, the general case seems to be far from being completely understood.
We give a complete classi cation of leftinvariant subriemannian structures on three dimensional lie groups in terms of the basic di erential invariants. 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. Taking m to be a lie group, the subriemannian structure. Conversely, every such quadratic hamiltonian induces a subriemannian manifold. The intrinsic hypoelliptic laplacian and its heat kernel. Geodesics equation on lie groups with left invariant metrics. 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.
Strati ed lie groups are examples of graded lie groups. If the grading is xed, we say that g is a graded lie group. Subriemannian geometry is the study of a smooth manifold equipped with a. G, endowed with such a subriemannian structure, a k. In particular, thanks to the group structure, in some of these. 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. Constant mean curvature surfaces in subriemannian geometry hladky, r.
Subriemannian geometry is a relatively young area in mathematics 2. Integrable geodesic ows ofriemannian and subriemannian. Subriemannian structures on groups of di eomorphisms. This paper is the third in a series dedicated to the fundamentals of sub riemannian geometry and its implications in lie groups theory. Nodal geometry on riemannian manifolds chanillo, sagun and muckenhoupt, b. 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. Such groups arise as tangent spaces of gromovhausdorff limits of riemannian manifolds 4,75, and since they. 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. Thanks to 40, selfsimilar lie groups are graduable lie groups with a grading induced by the metric dilation. It covers, with mild modifications, an elementary introduction to the field.
The spectral geometry of a riemannian manifold gilkey, peter b. For all that concerns general subriemannian geometry, including almostriemannian one, the reader is referred to. 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 is more easy to see when approaching the concept of a subriemannian lie group. Of special interest are the classical lie groups allowing concrete calculations of many of the abstract notions on the menu.
Asymptotic expansion of the 3d contact exponential map20. Pseudoriemannian manifolds and isometric actions of. Subriemannian structures on 3d lie groups, journal of. Subriemannian structures on 3d lie groups springerlink.
On extensions of subriemannian structures on lie groups. 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. Integrable geodesic ows of riemannian and subriemannian metrics on lie groups and homogeneous spaces. 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. Let f 1, f 2, f p be a set of smooth vector fields on a manifold m.
These notes entitled subriemannian geometry on the heisenberg group are mostly based on. These pages covers my expository talks during the seminar subriemannian geometry and lie groups organised by the author and tudor ratiu at the mathematics. Paper related content geodesics in the subriemannian. Subriemannian geometry is the geometry of a world with nonholonomic constraints. Subriemannian geometry andre bellaiche, jeanjaques. 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. Pdf tangent bundles to subriemannian groups semantic.
Classi cation of non flat left invariant structures 29 7. The overflow blog coming together as a community to connect. Lie groups and homogeneous spaces this item was submitted to loughborough universitys institutional repository by thean author. Rashevski and ballbox theorems of nitedimensional subriemannian geometry see 11,35.
The study of riemannian geometry is rather meaningless without some basic knowledge on gaussian geometry i. Graduable lie groups are nilpotent and in particular g is di eomorphic to g via the exponential map. At generic points, this is a nilpotent lie group endowed with a. This is a an updated version, which will be modified according to the contributions of the other participants to the borel seminar. The existence of geodesics of the corresponding hamiltonjacobi equations for the subriemannian hamiltonian is given by the chowrashevskii theorem. Browse other questions tagged liegroups riemanniangeometry or ask your own question. Such structures are basic models for subriemannian manifolds and as such serve to elucidate general features of subriemannian geometry. These notes entitled sub riemannian geometry on the heisenberg group are mostly based on. Carnot group, a class of lie groups that form subriemannian manifolds.
It can be quite difficult even in the simplest case of leftinvariant 3dimensional manifolds, where the complete description of the. Scienti c semester geometry, analysis and dynamics on subriemannian manifolds. Subriemannian calculus on hypersurfaces in carnot groups. It will also be a valuable reference source for researchers in various disciplines. Integrability of the subriemannian geodesic flow on 3d lie groups19. Sorry, we are unable to provide the full text but you may find it at the following locations. 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. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. We do not require any knowledge in riemannian geometry. In this talk we will be interested in lie groups admitting a metric with negative ricci curvature. 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 the words of montgomery 16 carnot groups are to subriemannian geometry as euclidean spaces are to riemannian geometry. 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. Differential geometry, lie groups, and symmetric spaces, academic press, 1978.
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