Curvas asintóticas suavemente encajadas en R4

Abstract

We will conduct a local study of curves in R4, using concepts such as curves parameterized by arc length, and in doing so, we will construct the Frenet tetrahedron of a curve using the Gram-Schmidt orthonormalization process, which will facilitate the local study of curves. We will then define a smoothly embedding of surface in R4 with codimension 2. After defining the surfaces, we will study the concept of surface curvature to understand its relationship with respect to the second fundamental form. Then, using the classical definition of an asymptotic curve of a surface, we will demonstrate that the characterization is preserved with respect to normal curvature. We will also conduct a study of an asymptotic curve to show that the classical definition of an asymptotic curve is consistent with the characterization of asymptotic curves using ellipse curvature. Finally, elliptical, hyperbolic, and parabolic points are studied to see if they retain the properties found on a 3-dimensional surface.