Dec 01, 2018 for all that concerns general sub riemannian geometry, including almost riemannian one, the reader is referred to. This paper is the third in a series dedicated to the fundamentals of sub riemannian geometry and its implications in lie groups theory. A leftinvariant distribution is uniquely determined by a two dimensional subspace of the lie algebra of the group. A closed subgroup of a lie group g is a lie subgroup of g. In such a setting, the blowup spaces do not depend on the scaling sequences and are strati. A homogeneous space macted upon by the lie group gwith isotropy sub group g.
An invitation to cauchyriemann and subriemannian geometries. Besides computing explicitly geodesics and conjugate loci. Lie groups occupy a central position in modern di erential geometry and physics, as they are very useful for describing the continuous symmetries of a space. Local conformal flatness of left invariant structures 25 6. Mg0210189, available at this url, and tangent bundles to sub riemannian groups, math. Ultrarigid tangents of subriemannian nilpotent groups arxiv. Riemannian geometry, lie groups, and homogeneous spaces. Geometric control theory and subriemannian geometry gianna. My work mainly revolves around the regularity of geodesics and the sard problem in sub riemannian geometry, with a particular focus on the case of sub riemannian and sub finsler carnot groups. Conversely, every such quadratic hamiltonian induces a subriemannian manifold. Lie groups act on themselves via the left translations given by l gx gxfor g. Such structures are basic models for sub riemannian manifolds and as such serve to elucidate general features of sub riemannian geometry. I completed my phd in august 2019 at the university of jyvaskyla under the supervision of enrico le donne.
It will also be a valuable reference source for researchers in various disciplines. There are few other books of sub riemannian geometry available. This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Some of the fundamental topics of riemannian geometry part ii. In chapter 3 we introduce generalisations of some important di erential operators.
Curvature of metric spaces, coadjoint orbits and associated representations 2004 current version pdf this paper is the third in a series dedicated to the fundamentals of sub riemannian geometry and its implications in lie groups theory. Differential geometry and lie groups a computational. Riemannian geometry of infinitedimensional lie groups 3 but the corresponding cameronmartin subgroups are, of necessity, only dense subgroupsof gw. Riemannian structures on threedimensional simple lie groups i. The space is gk for suitable lie groups g and k compact with lie algebras g and.
Isometries of riemannian and subriemannian structures on. Subriemannian geometry on 3d simple lie groups and lens. Sub riemannian geometry and time optimal control of three spin systems. Matrices m 2c2 are unitary if mtm idand special if detm 1. Vector fields and basic pointset topology bridge into the second part of the book, which focuses on riemannian geometry. Two sub riemannian structures on the same simply connected 3d lie group are isometric if and only if they are lisometric. We assume that g admits a direct sum decomposition g v 1. Actu ally from the book one can extract an introductory course in riemannian geometry as a special case of sub riemannian one, starting from the geometry of surfaces in chapter 1. This textbook offers an introduction to differential geometry designed for readers. We give a complete classification of leftinvariant subriemannian structures on threedimensional lie groups in terms of the basic differential invariants.
Contact lie groups turn out to exhibit some behaviour different from that of symplectic ones. Proposition two sub riemannian structures on the same simply connected 3d lie group are isometric if and only if they are lisometric. Rory biggs rhodes university isometries of sr structures on lie groups 7ecm 14. 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. Topics that follow include submersions, curvature on lie groups, and the logeuclidean framework. In mathematics, a subriemannian manifold is a certain type of generalization of a riemannian manifold. Paper related content geodesics in the subriemannian. This situation is more closely related to the osculating structures as they appear in analysis. To further understand the connection between riemannian geometry and the study of harmonic morphisms we must rst introduce the concept of a foliation. Lecture 3 lie groups and geometry july 29, 2009 1 integration of vector fields on lie groups let mbe a complete manifold, with a vector eld x. If g is a lie group and h a lie subgroup then the quotient space gh has a unique smooth structure such that the map g gh. Quantum gates and coherence transfer navin khaneja,1, steffen j. Roughly speaking, to measure distances in a subriemannian manifold, you are allowed to go only along curves tangent to socalled horizontal subspaces sub riemannian manifolds and so, a fortiori, riemannian manifolds carry a natural intrinsic metric called the metric of carnot.
It is the tangent space as a group that makes it a useful approximation to the manifold as a sub riemannian metric space. Theyaredetermined by theirtangent space, gcm, at the identity. At generic points, this is a nilpotent lie group endowed with a. In 7 a complete classification of leftinvariant sub riemannian structures on 3dimensional lie groups was given. The existence of geodesics of the corresponding hamiltonjacobi equations for the sub riemannian hamiltonian is given by the chowrashevskii theorem. Isometries of invariant subriemannian structures on 3d. Mg0210189 v3 31 oct 2002 subriemannian geometry and lie groups part i seminar notes, dmaepfl, 2001 m. Pdf almostriemannian geometry on lie groups researchgate.
Paper related content geodesics in the subriemannian problem. Subriemannian geometry and lie groups part i seminar notes, dmaepfl, 2001 m. Subriemannian structures on 3d lie groups springerlink. Emma carberry october 12, 2015 recap from lecture 17. Isometries of invariant subriemannian structures on 3d lie. Remark the two sub riemannian structures are locally isometric.
A comprehensive introduction to subriemannian geometry. Introduction to riemannian and subriemannian geometry. We will make use of this method at several occasions, especially for verifying the existence of geodesics in certain sub riemannian geometries. We do not require any knowledge in riemannian geometry.
Carnot group, a class of lie groups that form sub riemannian manifolds. Lectures on lie groups and geometry imperial college london. Subriemannian geometry on 3d simple lie groups and lens spaces. Sub riemannian manifolds are manifolds with the heisenberg principle built in. Subriemannian geometry and lie groups pdf free download. However, this is the first part of three, in preparation, dedicated to this subject. Notes on differential geometry and lie groups upenn cis. An isotropy subgroup g p of a lie group g is a lie subgroup of g. In chapter 2 we present some preliminaries of semi riemannian geometry and basic lie theory, necessary for the calculations in the rest of the thesis. General theory and examples is the perfect resource for graduate students and researchers in pure and applied mathematics, theoretical physics, control theory, and thermodynamics interested in the most recent developments in sub riemannian geometry. Using these geometric ideas, explicit expressions for the minimum time required for producing these effective hamiltonians, transfer of coherence, and implementation of indirect swap gates, in a threespin network are derivedtheorems 1 and 2. Jun 01, 2016 a leftinvariant sub riemannian 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.
Honoring andrei agrachevs 60th birthday, this volume presents recent advances in the interaction between geometric control theory and sub riemannian geometry. Then we get a biinvariant riemannian metric on g, preserved by left and. It covers, with mild modifications, an elementary introduction to the field. Classi cation of non flat left invariant structures 29 7. This paper is an expository article meant to introduce the theory of lie. The basic theory of manifolds and lie groups part i. Chapters on riemannian manifolds encompass riemannian metrics, geodesics, and curvature. Pdf riemannian geometry on contact lie groups andre. These pages covers my expository talks during the seminar subriemannian geometry and lie groups organised. Local conformal equivalence of sub riemannian three dimensional structures on lie groups 11 1. Isometries of riemannian and subriemannian structures on 3d. Mis called the ow of xif, for any function fand any time. Symmetry and compatibility lemma let n be any manifold, and f. Introduction to differential manifolds and lie groups.
The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to riemannian manifolds. In order to get the corresponding heat kernel measure to live on glhs, sohs or sphs, respectively, the tangent space gcm must be given a hilbert norm. Mar 28, 2012 subriemannian structures on 3d lie groups. Geometric control theory and subriemannian geometry. Differential geometry and lie groups a computational perspective. Proposition two sub riemannian structures on the same simply connected 3d lie group. Let f 1, f 2, f p be a set of smooth vector fields on a manifold m. About almost riemannian structures on lie groups more details can be found in. Minimal and conformal foliations of codimension two on. On the one hand, geometric control theory used the differential geometric and lie algebraic language for studying controllability, motion.
Hilbertschmidt groups as lie groups and their riemannian geometry 2 1. The culmination of the concepts and results presented in this book is the theory of. The cameronmartin group and the exponential map 5 4. That is, if a2gand x a, y aare vectors at a, then gx a. These pages covers my expository talks during the seminar sub riemannian geometry and lie groups organised. Rory biggs rhodes university isometries of sr structures on lie groups.
Working from basic undergraduate prerequisites, the authors develop manifold theory and lie groups from scratch. This comprehensive text and reference begins by introducing the theory of sub riemannian manifolds using a variational approach in which all properties are obtained from minimum principles, a robust method that. We also introduce the lie groups and lie algebras relevant to us. Isometries of almostriemannian structures on lie groups. The book may serve as a basis for an introductory course in riemannian geometry or an advanced course in sub riemannian geometry, covering elements of hamiltonian dynamics, integrable systems and lie theory. On extensions of subriemannian structures on lie groups. Subriemannian geometry and time optimal control of three. 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.
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