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Author : Wensheng Liu
Release : 1995
Genre : Geodesics
Kind : eBook
Book Rating : 436/5 ( reviews)
Book Synopsis Shortest Paths for Sub-Riemannian Metrics on Rank-two Distributions by : Wensheng Liu
Download or read book Shortest Paths for Sub-Riemannian Metrics on Rank-two Distributions written by Wensheng Liu. This book was released on 1995. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Wensheng Liu
Release : 1995
Genre : Mathematics
Kind : eBook
Book Rating : 049/5 ( reviews)
Book Synopsis Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions by : Wensheng Liu
Download or read book Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions written by Wensheng Liu. This book was released on 1995. Available in PDF, EPUB and Kindle. Book excerpt: A sub-Riemannian manifold ([italic capitals]M, E, G) consists of a finite-dimensional manifold [italic capital]M, a rank-two bracket generating distribution [italic capital]E on [italic capital]M, and a Riemannian metric [italic capital]G on [italic capital]E. All length-minimizing arcs on ([italic capitals]M, E, G) are either normal extremals or abnormal extremals. Normal extremals are locally optimal, i.e., every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal was recently settled by R. G. Montgomery, who exhibited a counterexample. The present work proves that regular abnormal extremals are locally optimal, and, in the case that [italic capital]E satisfies a mild additional restriction, the abnormal minimizers are ubiquitous rather than exceptional. All the topics of this research report (historical notes, examples, abnormal extremals, Hamiltonians, nonholonomic distributions, sub-Riemannian distance, the relations between minimality and extremality, regular abnormal extremals, local optimality of regular abnormal extremals, etc.) are presented in a very clear and effective way.
Author : Andre Bellaiche
Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 101/5 ( reviews)
Book Synopsis Sub-Riemannian Geometry by : Andre Bellaiche
Download or read book Sub-Riemannian Geometry written by Andre Bellaiche. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Ovidiu Calin
Release : 2009-04-20
Genre : Mathematics
Kind : eBook
Book Rating : 300/5 ( reviews)
Book Synopsis Sub-Riemannian Geometry by : Ovidiu Calin
Download or read book Sub-Riemannian Geometry written by Ovidiu Calin. This book was released on 2009-04-20. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.
Author : Gianna Stefani
Release : 2014-06-05
Genre : Mathematics
Kind : eBook
Book Rating : 32X/5 ( reviews)
Book Synopsis Geometric Control Theory and Sub-Riemannian Geometry by : Gianna Stefani
Download or read book Geometric Control Theory and Sub-Riemannian Geometry written by Gianna Stefani. This book was released on 2014-06-05. Available in PDF, EPUB and Kindle. Book excerpt: 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.
