Line Congruences on singular surfaces

TL;DR

This paper extends Kummer's theory to line congruences on singular surfaces, analyzing the relationship between principal and developable surfaces in the context of proper frontals.

Contribution

It introduces a framework for line congruences on singular surfaces, linking principal surfaces to developable surfaces via a multiplicative factor related to singularities.

Findings

Normal congruences relate principal and developable surfaces through a multiplicative factor.

The multiplicative factor is associated with the singular set of the frontal.

The theory extends classical results to singular surface cases.

Abstract

This paper is a first step in order to extend Kummer's theory for line congruences to the case $\lbrace x, \xi \rbrace $, where $x: U \rightarrow \mathbb{R}^3$ is a smooth map and $\xi: U \rightarrow \mathbb{R}^3$ is a proper frontal. We show that if $\lbrace x, \xi \rbrace$ is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the multiplicative factor is associated to the singular set of $\xi$.