arXiv subjects
Publications and source records attributed to Igor Chagas Santos.
In this paper, we classify the generic singularities of 2-parameter plane congruences in $\mathbb{R^4}$ and the generic singularities of affine normal plane congruences. We also study the generic singularities of the family of affine distance functions.
In this paper, we generalize the idea of equiaffine structure to the case of frontals and we define the Blaschke vector field of a frontal. We also investigate some necessary and sufficient conditions that a frontal needs to satisfy to have a Blaschke vector field and provide some examples. Finally, taking the theory developed here into account we present a fundamental theorem, which is a version for frontals of the fundamental theorem of affine differential geometry.
This paper is a first step in order to extend Kummer's theory for line congruences to the case $\lbrace x, ξ\rbrace $, where $x: U \rightarrow \mathbb{R}^3$ is a smooth map and $ξ: U \rightarrow \mathbb{R}^3$ is a proper frontal. We show that if $\lbrace x, ξ\rbrace$ 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 $ξ$.
In this paper, we give the generic classification of the singularities of 3-parameter line congruences in $\mathbb{R}^4$. We also classify the generic singularities of Blaschke (affine) normal congruences.