A tour of subriemannian geometries, their geodesics and applications / Richard Montgomery.

Format:

Book

Author/Creator:

Montgomery, R. (Richard), 1956- author.

Series:

Mathematical surveys and monographs ; no. 91.

Mathematical surveys and monographs, 0076-5376 ; volume 91

Language:

English

Subjects (All):

Geometry, Riemannian.

Geodesics (Mathematics).

Physical Description:

1 online resource (282 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Summary:

Subriemannian geometries, also known as Carnot-Caratheodory geometries, can be viewed as limits of Riemannian geometries. They also arise in physical phenomenon involving ``geometric phases'' or holonomy. Very roughly speaking, a subriemannian geometry consists of a manifold endowed with a distribution (meaning a $k$-plane field, or subbundle of the tangent bundle), called horizontal together with an inner product on that distribution. If $k=n$, the dimension of the manifold, we get the usual Riemannian geometry. Given a subriemannian geometry, we can define the distance between two points just as in the Riemannian case, except we are only allowed to travel along the horizontal lines between two points. The 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 the author mentions an elementary exposition of Gromov's surprising 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 (diffeomorphism types) of distributions. There is also a chapter devoted to open problems. The second part of the book is devoted to applications of subriemannian geometry. In particular, the author describes in detail the following four physical problems: 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: that of a principal bundle endowed with $G$-invariant metrics. Reading the book requires introductory knowledge of differential geometry, and it can serve as a good introduction to this new, exciting area of mathematics. This book provides an introduction to and a comprehensive study of the qualitative theory of ordinary differential equations. It begins with fundamental theorems on existence, uniqueness, and initial conditions, and discusses basic principles in dynamical systems and Poincare-Bendixson theory. The authors present a careful analysis of solutions near critical points of linear and nonlinear planar systems and discuss indices of planar critical points. A very thorough study of limit cycles is given, including many results on quadratic systems and recent developments in China. Other topics included are: the critical point at infinity, harmonic solutions for periodic differential equations, systems of ordinary differential equations on the torus, and structural stability for systems on two-dimensional manifolds. This books is accessible to graduate students and advanced undergraduates and is also of interest to researchers in this area. Exercises are included at the end of each chapter.

Contents:

""Contents""; ""Introduction""; ""Acknowledgments""; ""Part 1. Geodesies in Subriemannian Manifolds""; ""Chapter 1. Dido Meets Heisenberg""; ""1.1. Dido's problem""; ""1.2. A vector potential""; ""1.3. Heisenberg geometry""; ""1.4. The definition of a subriemannian geometry""; ""1.5. Geodesic equations""; ""1.6. Chow's theorem and geodesic existence""; ""1.7. Geodesic equations on the Heisenberg group""; ""1.8. Why call it the Heisenberg group?""; ""1.9. Proof of the theorem on normal geodesies""; ""1.10. Examples""; ""1.11. Notes""; ""Chapter 2. Chow's Theorem: Getting from A to B""

""2.1. Bracket-generating distributions""""2.2. A heuristic proof of Chow's theorem""; ""2.3. The growth vector and canonical flag""; ""2.4. Chow and the ball-box theorem""; ""2.5. Proof of the theorem on topologies""; ""2.6. Privileged coordinates""; ""2.7. Proof of the remaining ball-box inclusion""; ""2.8. Hausdorff measure""; ""Chapter 3. A Remarkable Horizontal Curve""; ""3.1. A rigid curve""; ""3.2. Martinet's genericity result""; ""3.3. The minimality theorem""; ""3.4. The minimality proof of Liu and Sussmann""; ""3.5. Failure of geodesic equations""

""3.6. Singular curves in higher dimensions""""3.7. There are no H[sup(1)]-rigid curves""; ""3.8. Towards a conceptual proof?""; ""3.9. Notes""; ""Chapter 4. Curvature and Nilpotentization""; ""4.1. The curvature of a distribution""; ""4.2. Dual curvature""; ""4.3. The derived ideal and the big growth vector""; ""4.4. The sheaf of Lie algebras""; ""4.5. Nilpotentization and Carnot groups""; ""4.6. Non-regular nilpotentizations""; ""4.7. Notes""; ""Chapter 5. Singular Curves and Geodesies""; ""5.1. The space of horizontal paths""; ""5.2. A microlocal characterization""

""5.3. Singularity and regularity""""5.4. Rank-two distributions""; ""5.5. Distributions determined by their curves""; ""5.6. Fat distributions""; ""5.7. Notes""; ""Chapter 6. A Zoo of Distributions""; ""6.1. Stability and function counting""; ""6.2. The stable types""; ""6.3. Prolongation""; ""6.4. Goursat distributions""; ""6.5. Jet bundles""; ""6.6. Maximal growth and free Lie algebras""; ""6.7. Symmetries""; ""6.8. Types (3,5), (2,3,5), and rolling surfaces""; ""6.9. Type (3, 6): the frame bundle of M[sup(3)]""; ""6.10. Type (4,7) distributions""; ""6.11. Notes""

""Chapter 7. Cartan's Approach""""7.1. Overview""; ""7.2. Riemannian surfaces""; ""7.3. G-structures""; ""7.4. The tautological one-form""; ""7.5. Torsion and pseudoconnections""; ""7.6. Intrinsic torsion and torsion sequence""; ""7.7. Distributions: torsion equals curvature""; ""7.8. The Riemannian case and the o(n) lemma""; ""7.9. Reduction and prolongation""; ""7.10. Subriemannian contact three-manifolds""; ""7.11. Why we need pseudo in pseudoconnection""; ""7.12. Type and growth (4,7)""; ""Chapter 8. The Tangent Cone and Carnot Groups""; ""8.1. Nilpotentization""

""8.2. Metric tangent cones""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 247-255) and index.

Description based on print version record.

ISBN:

1-4704-1318-3