Classification of rank-one submanifolds

We study ruled submanifolds of Euclidean space. First, to each (parametrized) ruled submanifold $\sigma$, we associate an integer-valued function, called degree, measuring the extent to which $\sigma$ fails to be cylindrical. In particular, we show that if the degree is constant and equal to $d$, then the singularities of $\sigma$ can only occur along an $(m-d)$-dimensional “striction” submanifold. This result allows us to extend the standard classification of developable surfaces in ${\mathbb{R}}^3$ to the whole family of flat and ruled submanifolds without planar points, also known as rank-one: an open and dense subset of every rank-one submanifold is the union of cylindrical, conical, and tangent regions.