Invariants and bonnet-type theorem for surfaces in $\r^4$

TL;DR

This paper develops invariants and a Bonnet-type theorem for surfaces in four-dimensional space, characterizing surfaces via invariant functions and geometric figures like tangent indicatrices and curvature ellipses.

Contribution

It introduces a Weingarten-type invariant map, constructs a moving frame, and proves a fundamental theorem linking invariants to surface determination in -space.

Findings

Eight invariant functions characterize surfaces up to motion.

Geometric classes are described via tangent indicatrix and curvature ellipse.

Constructed a family of surfaces with flat normal connection.

Abstract

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map of Weingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. 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 two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.