Sobre el conjunto focal en la geometría diferencial afín

Abstract

In this work we study the focal set from the affine space, for immersed surfaces in the 3-space. Initially, we consider the whole study of curves under the action of the affine group, length of affine arc, affine curvature and evolute, the latter gives us an idea of ​​what the affine focal set on surfaces. For our study, in Euclidean space, the group of isometries preserves distances and the metrics are maintained, conversely, in an affine space we don't have a distance invariance, therefore, we think of a metric that preserves the volume that analogously in the affine arc length preserves the area. In that order, we define the first shape affine fundamental and the third affine fundamental form, where we introduce the definitions more relevant (the affine operator, affine curvatures and main directions) without the use local coordinates, since the geometric content of these is clear, however, for both computational and theoretical purposes, it is important for us to express these concepts in local coordinates, and in this way develop the examples. Finally, we come to define and analyze the affine focal set, as described in its equation, which in which case turns out to degenerate to curves, points, or surfaces that can be or not singular, and consequently, if the singular set is studied, it turns out to coincide with the Ridge. In this work, we develop some examples for the Euclidean case, since it is of sum It is important to address our already known theory in the course of surface geometry, since it gives us some notions to understand and conceptually adapt the geometric meaning to a related space. To address the theory, we make use of Maple Software, in which we work all the calculation in local coordinates and we present the graphs associated with the examples.