One-Dimensional Level Sets of $hc$-Differentiable Mappings of Carnot–Carathéodory Spaces

We study continuously $hc$-differentiable mappings from the Carnot–Carathéodory space $\mathcal{M}$ such that $\dim H_g \mathcal{M} = \dim T_g \mathcal{M} -1 = N$ in every $g \in \mathcal{M}$ into the Euclidean $N$-dimensional space with the property that $hc$-differential of the mapping is surjective. We establish that the level set of such mapping is a curve that has Hausdorff dimension 2 in sub-Riemannian metric. We obtain area formulas for curves of that kind.