Line Congruences on singular surfaces

This paper is a first step in order to extend Kummer’s theory for line congruences to the case $\{x,\xi \}$, where $x:U\to {\mathbb{R}}^3$ is a smooth map and $\xi :U\to {\mathbb{R}}^3$ is a proper frontal. We show that if $\{x,\xi \}$ 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 $\xi$.