Classification of Rank-One Submanifolds

We study ruled submanifolds of Euclidean space. First, to each (parametrized) ruled submanifold [Formula: see text] , we associate an integer-valued function, called degree, measuring the extent to which [Formula: see text] fails to be cylindrical. In particular, we show that if the degree is constant and equal to d, then the singularities of [Formula: see text] can only occur along an [Formula: see text] -dimensional “striction” submanifold. This result allows us to extend the standard classification of developable surfaces in [Formula: see text] 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.