Download Sub-Riemannian Geometry by André Bellaïche (auth.), André Bellaïche, Jean-Jacques PDF

By André Bellaïche (auth.), André Bellaïche, Jean-Jacques Risler (eds.)

Sub-Riemannian geometry (also referred to as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been an entire examine area for fifteen years, with motivations and ramifications in different components of natural and utilized arithmetic, namely:
• regulate concept • classical mechanics • Riemannian geometry (of which sub-Riemannian geometry constitutes a typical generalization, and the place sub-Riemannian metrics might seem as restrict circumstances) • diffusion on manifolds • research of hypoelliptic operators • Cauchy-Riemann (or CR) geometry.
Although hyperlinks among those domain names have been foreseen by way of many authors long ago, it is just in recent times that sub- Riemannian geometry has been well-known as a potential universal framework for most of these themes.
This ebook presents an creation to sub-Riemannian geometry and provides the cutting-edge and open difficulties within the box. It contains 5 coherent and unique articles by means of the best specialists:
• André Bellaïche: The tangent house in sub-Riemannian geometry • Mikhael Gromov: Carnot-Carathéodory areas noticeable from inside of • Richard Montgomery: Survey of singular geodesics • Héctor J. Sussmann: A cornucopia of 4-dimensional irregular sub-Riemannian minimizers • Jean-Michel Coron: Stabilization of controllable systems

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Example text

Taking this into account, and replacing).. by c:- l , one can rewrite (46) as (47) In fact, R = 1 suffices. Thus, the existence of TpM means simply that small balls B(p, c:) in M (renormalized to radius 1) get more and more alike when c: ----* 0. When M is a C l Riemannian manifold, one thus recovers in a purely metric way the tangent space TpM with its Euclidean metric. , TpM endowed with its natural sub-Riemannian structure. Until then, we shall denote the former space by Gromov-TpM, if needed.

11, we obtain n = ljk+lljk + L Vj lj, j=l where the order of Vj at p is 2: Wj - Wjk - Wjk+l' Now, we replace ljk ljk+1 in the product ljl ... ljq by the right hand side in the last equality. This yields ljl ... ljq = ljl .. ·ljk+lljk .. ljq +L n ljl .. ·ljk-l (Vj lj )ljk+2 .. ljq, j=l n L L(Y)'l '" YACVj)Y/Ll ... ljq, where in the last sum (A, JL) runs on the set of all partitions of the sequence (j1, ... ,jk-1) into subsequences A = (A1, ... ,Ae) and JL = (JLI, . , JLk-I-e). To compute the order of Y A1 ...

The theorem is therefore proved. _ Remark. The coordinates Yj having weight 1 need not be changed. In case r = 2, no change at all is needed, the coordinates Yj (with weight 1 and 2) form already a system of privileged coordinates, as it follows immediately from the definition. Notice that the coordinates Zl, ... 15 are given from original coordinates by expressions of the form Zl = Yl Z2 = Y2 + POI(Yl) Zn = Yn + POI(Yl , ... , Yn-l) where pol denotes a polynomial, without constant or linear term.

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