Mathematical Surveys and Monographs 2002; 259 pp; softcover Volume: 91 ISBN10: 0821841653 ISBN13: 9780821841655 List Price: US$76 Member Price: US$60.80 Order Code: SURV/91.S
 Subriemannian geometries, also known as CarnotCarathéodory 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 twodimensional 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. Readership Graduate students and research mathematicians interested in geometry and topology. Reviews "Very comprehensive and elegantly written book."  Mathematical Reviews Table of Contents Geodesics in subriemannian manifolds  Dido meets Heisenberg
 Chow's theorem: Getting from A to B
 A remarkable horizontal curve
 Curvature and nilpotentization
 Singular curves and geodesics
 A zoo of distributions
 Cartan's approach
 The tangent cone and Carnot groups
 Discrete groups tending to Carnot geometries
 Open problems
Mechanics and geometry of bundles  Metrics on bundles
 Classical particles in YangMills fields
 Quantum phases
 Falling, swimming, and orbiting
Appendices  Geometric mechanics
 Bundles and the Hopf fibration
 The Sussmann and AmbroseSinger theorems
 Calculus of the endpoint map and existence of geodesics
 Bibliography
 Index
