Asymptotic and Binormal Directions on Surfaces Sitting in the Four-Dimensional Euclidean Space

This work was intended as an attempt at exploring, in a novel and systematic way, the geometric aspects of some local invariants and the analytic features of asymptotic and binormal directions on the surfaces immersed in the four-dimensional Euclidean space by looking more closely at the normal connection and extrinsic geometry of the surfaces, and also investigating the curvature ellipse, of a given surface, and its director circle. In fact, our local constructions lead to some global results on the existence of singularities of the surfaces.