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3 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 47 We say that the control system in (38) is in triangular chained form, in fact a block triangular form. In the equation for Zj only variables having a weight < Wj appear in the right hand side. So, it is possible to compute the Zj one after the other, only by computing primitives, once given the control functions U 1 ( t), ... , u m (t) . 19. We have i = 1, ... ,m, (39) where Xi is homogeneous of order -1 and Ri is of order 2:: 0 at p. In privileged coordinates, the system m i = I::UiXi(X) i=l takes the following form m Zj = LUi [/ij(Zl"",Zn Wj _ 1 ) +O(llzIIWj)] j = 1, ...

Since this set is compact and dp(p, q) is continuous on ]Rn, it follows that there exist numbers C, C', positive and finite, such that for iiqii = 1. Using homogeneity, we get (48). • As it is simpler to prove the estimates we have in mind in the case of tangent spaces (or Carnot groups and homogeneous spaces), we will consider this case first. The proof of corresponding estimates in the manifold M will not depend on the results obtained in the case of tangent spaces, but it will follow, more or less, the same lines.

Since the algebraic structure of Carnot groups is moreover similar to that of Euclidean spaces, it is really tempting to call them non-abelian vector spaces, or nonholonomic vector spaces, or nonholonomic Euclidean spaces if one wants to take the metric into account. There is nevertheless one major difference between Euclidean spaces and Carnot groups: they are many algebraically non isomorphic Carnot groups having the same dimension n, uncountably many for n 2: 6, as there may be modules in their classification.

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3-D Shapes by Marina Cohen

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