Sub Riemannian Geometry

Author : Andre Bellaiche,Jean-Jaques Risler
Publisher : Birkhäuser
Page : 404 pages
File Size : 53,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034892100

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Sub-Riemannian Geometry by Andre Bellaiche,Jean-Jaques Risler Pdf

Sub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic 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: control theory classical mechanics Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) diffusion on manifolds analysis of hypoelliptic operators Cauchy-Riemann (or CR) geometry. Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub- Riemannian geometry has been recognized as a possible common framework for all these topics. This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists: Andr Bellache: The tangent space in sub-Riemannian geometry Mikhael Gromov: Carnot-Carathodory spaces seen from within Richard Montgomery: Survey of singular geodesics Hctor J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers Jean-Michel Coron: Stabilization of controllable systems.

Author : Andrei Agrachev,Davide Barilari,Ugo Boscain
Publisher : Cambridge University Press
Page : 765 pages
File Size : 46,7 Mb
Release : 2019-10-31
Category : Mathematics
ISBN : 9781108476355

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A Comprehensive Introduction to Sub-Riemannian Geometry by Andrei Agrachev,Davide Barilari,Ugo Boscain Pdf

Provides a comprehensive and self-contained introduction to sub-Riemannian geometry and its applications. For graduate students and researchers.

Author : Ludovic Rifford
Publisher : Springer Science & Business Media
Page : 146 pages
File Size : 40,9 Mb
Release : 2014-04-03
Category : Mathematics
ISBN : 9783319048048

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Sub-Riemannian Geometry and Optimal Transport by Ludovic Rifford Pdf

The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.

Author : Ovidiu Calin,Der-Chen Chang
Publisher : Cambridge University Press
Page : 371 pages
File Size : 54,8 Mb
Release : 2009-04-20
Category : Mathematics
ISBN : 9780521897303

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Sub-Riemannian Geometry by Ovidiu Calin,Der-Chen Chang Pdf

A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.

Author : Frédéric Jean
Publisher : Springer
Page : 112 pages
File Size : 50,7 Mb
Release : 2014-07-17
Category : Science
ISBN : 9783319086903

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Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning by Frédéric Jean Pdf

Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.

Author : Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson
Publisher : Springer Science & Business Media
Page : 224 pages
File Size : 51,8 Mb
Release : 2007-08-08
Category : Mathematics
ISBN : 9783764381332

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An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem by Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson Pdf

This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

Author : Richard Montgomery
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 42,8 Mb
Release : 2002
Category : Geodesics
ISBN : 9780821841655

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A Tour of Subriemannian Geometries, Their Geodesics and Applications by Richard Montgomery Pdf

Subriemannian geometries can be viewed as limits of Riemannian geometries. They arise naturally in many areas of pure (algebra, geometry, analysis) and applied (mechanics, control theory, mathematical physics) mathematics, as well as in applications (e.g., robotics). This book is devoted to the study of subriemannian geometries, their geodesics, and their applications. It starts with the simplest nontrivial example of a subriemannian geometry: the two-dimensional isoperimetric problem reformulated as a problem of finding subriemannian geodesics. Among topics discussed in other chapters of the first part of the book are an elementary exposition of Gromov's idea to use subriemannian geometry for proving a theorem in discrete group theory and Cartan's method of equivalence applied to the problem of understanding invariants of distributions. The second part of the book is devoted to applications of subriemannian geometry. In particular, the author describes in detail Berry's phase in quantum mechanics, the problem of a falling cat righting herself, that of a microorganism swimming, and a phase problem arising in the $N$-body problem. He shows that all these problems can be studied using the same underlying type of subriemannian geometry. The reader is assumed to have an introductory knowledge of differential geometry. This book that also has a chapter devoted to open problems can serve as a good introduction to this new, exciting area of mathematics.

Author : Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti
Publisher : Springer
Page : 385 pages
File Size : 47,5 Mb
Release : 2014-06-05
Category : Mathematics
ISBN : 9783319021324

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Geometric Control Theory and Sub-Riemannian Geometry by Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti Pdf

Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.

Author : Peter Petersen
Publisher : Springer Science & Business Media
Page : 443 pages
File Size : 50,5 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475764345

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Riemannian Geometry by Peter Petersen Pdf

Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialise in Riemannian geometry. Instead of variational techniques, the author uses a unique approach, emphasising distance functions and special co-ordinate systems. He also uses standard calculus with some techniques from differential equations to provide a more elementary route. Many chapters contain material typically found in specialised texts, never before published in a single source. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research - including sections on convergence and compactness of families of manifolds. Thus, this book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.

Author : Edwin J. Beggs,Shahn Majid
Publisher : Springer Nature
Page : 809 pages
File Size : 49,5 Mb
Release : 2020-01-31
Category : Science
ISBN : 9783030302948

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Quantum Riemannian Geometry by Edwin J. Beggs,Shahn Majid Pdf

This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators. The book also covers elements of Connes’ approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.

Author : Luther Pfahler B 1876 Eisenhart
Publisher : Hassell Street Press
Page : 328 pages
File Size : 41,7 Mb
Release : 2021-09-09
Category : Electronic
ISBN : 1013373251

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Riemannian Geometry by Luther Pfahler B 1876 Eisenhart Pdf

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Author : Leonor Godinho,José Natário
Publisher : Springer
Page : 476 pages
File Size : 55,6 Mb
Release : 2014-07-26
Category : Mathematics
ISBN : 9783319086668

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An Introduction to Riemannian Geometry by Leonor Godinho,José Natário Pdf

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

Author : D.N. Kupeli
Publisher : Springer
Page : 0 pages
File Size : 52,9 Mb
Release : 2010-12-05
Category : Mathematics
ISBN : 9048146895

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Singular Semi-Riemannian Geometry by D.N. Kupeli Pdf

This book is an exposition of "Singular Semi-Riemannian Geometry"- the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. The main topic of interest is those cases where the metric tensor is assumed to be nondegenerate. In the literature, manifolds with degenerate metric tensors have been studied extrinsically as degenerate submanifolds of semi Riemannian manifolds. One major aspect of this book is first to study the intrinsic structure of a manifold with a degenerate metric tensor and then to study it extrinsically by considering it as a degenerate submanifold of a semi-Riemannian manifold. This book is divided into three parts. Part I deals with singular semi Riemannian manifolds in four chapters. In Chapter I, the linear algebra of indefinite real inner product spaces is reviewed. In general, properties of certain geometric tensor fields are obtained purely from the algebraic point of view without referring to their geometric origin. Chapter II is devoted to a review of covariant derivative operators in real vector bundles. Chapter III is the main part of this book where, intrinsically, the Koszul connection is introduced and its curvature identities are obtained. In Chapter IV, an application of Chapter III is made to degenerate submanifolds of semi-Riemannian manifolds and Gauss, Codazzi and Ricci equations are obtained. Part II deals with singular Kahler manifolds in four chapters parallel to Part I.

Author : Jean-Jacques Risler
Publisher : Birkhauser
Page : 393 pages
File Size : 42,5 Mb
Release : 1996-01-01
Category : Global differential geometry
ISBN : 0817654763

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Sub-Riemannian Geometry by Jean-Jacques Risler Pdf

Author : Bernhard Riemann
Publisher : Birkhäuser
Page : 172 pages
File Size : 43,9 Mb
Release : 2016-04-19
Category : Mathematics
ISBN : 9783319260426

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On the Hypotheses Which Lie at the Bases of Geometry by Bernhard Riemann Pdf

This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.