arXiv subjects

Tito Alexandro Medina Tejeda

Publications and source records attributed to Tito Alexandro Medina Tejeda.

3 recordsLinked to original sources

arXiv2022

Line Congruences on singular surfaces

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 $ξ$.

math.DG↗

arXiv2021

Extendibility and boundedness of invariants on singularities of wavefronts

We investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities on fronts. We also study the convergence to infinite limits of these geometrical invariants and show how this is tightly related to a particular property of uniform approximation of fronts by parallel surfaces.

math.DG↗

arXiv2019

The fundamental theorem for singular surfaces with limiting tangent planes

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundamental forms with local properties in the differential of its parametrization and decompositions in the matrices associated to the fundamental forms. In particular we introduce new types of curvatures which can be used to characterize wave fronts. The only restriction on the parametrizations that is assumed in several occasions is that the singular set has empty interior.

math.DG↗