The geometric Cauchy problem for rank-one submanifolds

Abstract

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 developable submanifold of R m+n , that is, a 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 well-posedness of the problem, together with a parametric description of the solution.

Nonrigidity of flat ribbons

“…The proof of the theorem is based on a well-known lemma [5, p. 195-197]. See [17] for a more general result and an alternative proof.…”

“…In order to analyze the dependence of the bending energy on the initial condition, we calculate dE(N (q))/dq and set it equal to zero. This leads to (17) A sin(q) cos(q) + B sin(q) 2 − cos(q) 2 = 0, where…”