Carnot-Carathéodory spaces seen from within

Let V be a smooth manifold where we distinguish a subset H in the set of all piecewise smooth curves c in V. We assume that H is defined by a local condition on curves, i.e. if c is divided into segments c 1,…, c k , then $$ c\, \in \,\mathcal{H} \Leftrightarrow \,{c_i} \in \mathcal{H},\,i = 1, \ldots ,k. $$

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