Superficies de rotación en el espacio afín

Abstract

In this work the affine parabolic set of affine hyperbolic rotation surfaces immersed in R3 will be characterized, based on article [3] where we find a relationship between the Euclidean space with the affine space, through an application between the rotation surface and the corresponding conormal surface. To do this, we will begin by defining basic concepts in affine differential geometry such as affine arc length, affine curvature, affine parabolic points, affine operator, affine principal directions, among other concepts that will help us study the geometry of the rotation surfaces in the affine space. It is known that by definition, the surfaces of rotation in Euclidean space are obtained by rotating a regular curve around a circumference, which is a flat curve with positive constant curvature, the same is also done with curves of constant curvature equal to 0, the latter are known as ruled surfaces [4]. So, in the affine context, generalized rotations are described where the same phenomenon described for Euclidean geometry is considered but now, we will study the surfaces that are obtained by rotating a regular curve around an ellipse instead of a circle, parabolas instead of lines and Now in this context hyperbolas, we will call these surfaces surfaces of affine rotation, to later show that in each of these surfaces the coordinate curves are lines of curvature.