On the Subanalyticity of Carnot–caratheodory Distances

Let M be a C∞ Riemannian manifold, dimM = n. A distribution on M is a smooth linear subbundle of the tangent bundle TM . We denote by q the fiber of at q ∈M ; q ⊂ TqM . The number k = dim q is the rank of the distribution. We assume that 1 < k < n. The restriction of the Riemannian structure to is a sub-Riemannian structure. Lipschitz integral curves of the distribution are called admissible paths; these are Lipschitz curves t → q(t), t ∈ [0,1], such that q̇(t) ∈ q(t) for almost all t . We fix a point q0 ∈M and study only admissible paths starting from this point, i.e. meeting the initial condition q(0) = q0. Sections of the linear bundle are smooth vector fields; we set