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The tangent space in sub-Riemannian geometry. (English) Zbl 0862.53031
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, 1-78 (1996).
The paper provides a precise self-contained introduction into the subject including simple proofs of the classical Sussmann, Chow, and Hopf-Rinow theorems with instructive examples. Moreover, introducing a new kind of adapted coordinates and Gromov’s definition of tangent vectors to metric spaces, a theorem of fundamental importance is proved: tangent spaces to sub-Riemannian (sR) manifolds themselves are sR manifolds equipped with the structure of nilpotent Lie groups with dilatations. This result opens wide perspectives.
In more detail: Let \(X_1, \dots, X_m\) be vector fields on an \(n\)-dimensional Riemannian manifold \(M\). Then the sR distance of two points \(p,q \in M\) is defined as the infimum of lengths of paths \(x(t)\) joining \(p\) to \(q\) and such that the tangent \(dx/dt\) is a linear combination of \(X_1, \dots, X_m\). Classical Sussman and Chow theorems ensure the accessibility of \(p\) from \(q\) if the iterated brackets \([X_i, X_j]\), \([X_i, [X_j, X_k]], \dots\) span the tangent spaces to \(M\) at every point of \(M\). Assuming this case, the sR distance is finite and we obtain the sR-manifold, by definition. The author’s subtle technical tools cannot be discussed here but the main result states that for every \(p\in M\), a graded Lie algebra \({\mathfrak g}_p\) generated by components of degree 1, and a graded subalgebra \({\mathfrak h}_p \subset {\mathfrak g}_p\) exist such that the tangent space \(T_pM\) can be identified with the factor \(\exp {\mathfrak g}_p/ \exp {\mathfrak h}_p\). (At regular points is \({\mathfrak h}_p = 0\) hence \(T_pM = \exp {\mathfrak g}_p.)\) Finally, using Gromov’s definition, \(T_pM\) can be identified with the tangent space to the underlying sR-manifold.
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