A Tour of Subriemannian Geometries, Their Geodesics and Applications

Abstract

A subriemannian, or Carnot-Cartheodory, geometry is a nonintegrable distribution, or subbundle of the tangent bundle of a manifold, which is endowed with an inner product. Part I presents the basic theory and examples, focussing on the geodesies. Chapters explaining the ideas of Cartan and Gromov are included. Part II presents applications to physics. These include Berry's quantum phase and an explanation of how a falling cat rights herself to land on her feet. Library of Congress Cataloging-in-Publication Data Montgomery, R. (Richard), 1956-A tour of subriemannian geometries, their geodesies and applications / Richard Montgomery. p. cm.-(Mathematical surveys and monographs,

Yet another introduction to rough paths

“…In the general case, this problem is called the Dido problem, and the solutions are known to be arcs of circle (see for example [Str87,Mon02]), but they are less practical to use than our construction with circles and loops (see below in the proof of Proposition 4).…”

“…The Heisenberg group has been widely studied, as appears in subRiemannian geometry, quantum physics, ... (see for example [Fol89,Mon02,Bau04]). …”

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

“…We shall assume that D satisfies the bracket generating condition, i.e., D(1) generates g; by the Chow-Rashevskii theorem this condition is necessary and sufficient for any two points in G to be connected by a D-curve (see, e.g., [21]). The length of a D-curve x(·) is given by (x(·)) = t1 0 g(ẋ(t),ẋ(t)) dt.…”

Geodesics in the Heisenberg Group

“…Also, the structure of every geodesic in a form of an explicit parameterization is known. The proofs in the case of H can be found in [2,5,7,11], and the general case of H n is treated in [1,3,13].…”

“…If n = , the structure of geodesics can be obtained via the two dimensional isoperimetric inequality (see [2,5,11]). Consider a horizontal curve Γ = (γ, t) in H connecting the origin to some point q = ( , , T) with T ≠ .…”