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Carnot-Carathéodory spaces seen from within. (English) Zbl 0864.53025
Bellaïche, André (ed.) et al., Sub-Riemannian geometry. Proceedings of the satellite meeting of the 1st European congress of mathematics ‘Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique’, Paris, France, June 30–July 1, 1992. Basel: Birkhäuser. Prog. Math. 144, 79-323 (1996).
Carnot-Carathéodory (C-C) spaces (or: sub-Riemannian manifolds) are by definition Riemannian spaces \(V\) equipped with a submodule \(H\subset T(V)\) of the tangent bundle. Then a piecewise smooth curve in \(V\) is called horizontal if its tangent vectors are lying in \(H\), and we introduce the C-C distance \(d(v_1,v_2)\) between points \(v_1,v_2\in V\) as the infimum of lengths of all horizontal curves joining \(v_1\) and \(v_2\).
The author raises two main problems: to develop an internal C-C language which enables to capture the external characteristics, and to develop an external technique for evaluation of the internal geometry of C-C space. The paper, however, contains much more and may be regarded as an usually complete survey of concepts, theorems, techniques, open problems, including a large amount of examples and references with interesting comments.
It is impossible to adequately present the contents here. After an introductory part (with precise basic definitions), the following more advanced sections are devoted to C-C balls (with a thorough discussion of the Chow theorem), hypersurfaces in C-C spaces (involving isoperimetric and Sobolev inequalities), C-C geometry of contact manifolds (with Rumin homologies), Pfaffian systems (devoted to various \(h\)-principles), and the theory of anisotropic connections (curvature and monodromy). The paper is self-contained and may be used as convenient introduction into this beautiful area of differential geometry.
For the entire collection see [Zbl 0848.00020].
Reviewer: J.Chrastina (Brno)

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A17 Pfaffian systems