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)

MSC:

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