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

TL;DR

This paper provides a geometric interpretation of surface invariants in R^4 using the tangent indicatrix and the normal curvature ellipse, linking invariant classes to properties of these figures.

Contribution

It introduces a new geometric framework connecting surface invariants in R^4 with the properties of tangent indicatrix and curvature ellipse.

Findings

Basic classes of surfaces characterized by invariants

Geometric conditions linked to tangent indicatrix and curvature ellipse

Examples illustrating the classification of surfaces

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.