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Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along asmooth regular curve $\gamma$ in $\mathbb{R}^{m+n}$, we consider the followingproblem: to find an $m$-dimensional rank-one submanifold of $\mathbb{R}^{m+n}$,that is, an $(m-1)$-ruled submanifold with constant tangent space along therulings, such that its tangent bundle along $\gamma$ coincides with$\mathscr{D}$. In particular, we give sufficient conditions for the localwell-posedness of the problem, together with a parametric description of thesolution.

The geometric Cauchy problem for rank-one submanifolds