Geometric interpretation of the invariants of a surface in R^4 via the tangent indicatrix and the normal curvature ellipse

Abstract

At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures. We give some non-trivial examples of surfaces from the classes in consideration.