The geometric Cauchy problem for rank-one submanifolds

Given a smooth distribution D of m-dimensional planes along a smooth regular curve γ in R^m+n, we consider the following problem: to find an m-dimensional rank-one submanifold of R^m+n, that is, an (m − 1)-ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along γ coincides with D. In particular, we give sufficient conditions for the local wellposedness of the problem, together with a parametric description of the solution.