Classification of rank-one submanifolds

TL;DR

This paper extends the classification of developable surfaces to all flat, ruled submanifolds in Euclidean space, introducing a degree function to analyze their singularities and structure.

Contribution

It introduces a degree function for ruled submanifolds, enabling a comprehensive classification of rank-one submanifolds beyond classical developable surfaces.

Findings

Singularities occur along specific submanifolds depending on degree

Rank-one submanifolds are composed of cylindrical, conical, and tangent regions

The classification generalizes classical results to higher dimensions

Abstract

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.