By R. H. Bing, Ralph J. Bean

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322 (1996), no. 4, 377–384 [2] A. Bella¨ıche, The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, A. -J. , Progress in Mathematics, 144, Birkh¨auser, (1996), 4 – 78 [3] P. Bieliavsky, E. Falbel, C. Gorodski, The classification of simply-connected contact sub-Riemannian symmetric spaces,Pacific J. Math. , 188 (1999), no. 1, 65–82 [4] M. Buliga, Sub-Riemannian geometry and Lie groups. MG/0210189, (2002) [5] M. Buliga, Tangent bundles arXiv:mathMG/0307342, (2003) to sub-Riemannian groups, e-print [6] M.

We get a strictly positive definite metric g¯. We are going to define now the group G(σ). , m we have F (V0 + ... + Vk ) = V0 + ... + Vk (b) for any u0 ∈ V0 we have F (u0 ) = u0 . (c) for any u0 ∈ V0 and any u1 ∈ V1 we have F [u0 , u1 ] = [u0 , F (u1 )] that is F commutes with the representation Q. This implies that F (V1 ) = V1 . (d) the restriction of F on V1 is a g isometry. 1 For any F ∈ G(σ) and any sufficiently small ε > 0 we have P(F σ)(ε) = P(σ)(ε). The proof is a straightforward computation.

Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, AMS Providence, Rhode Island, (2000) [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces,Geom. Funct. ,9 (1999), no. 3, 428–517 [10] M. Gromov, Groups of polynomial growth and expanding maps. Inst. Hautes ´ Etudes Sci. Publ. Math. No. 53, 53–73, (1981) [11] M. Gromov, Carnot-Caratheodory spaces seen from within, in: Sub-Riemannian Geometry, A. -J. , Progress in Mathematics, 144, Birkh¨auser, (1996), 79 – 323 [12] M.