On the Theory of Surfaces in the Four-dimensional Euclidean Space

TL;DR

This paper develops a theoretical framework for analyzing surfaces in four-dimensional Euclidean space, introducing invariants and characterizations for various classes of surfaces, including minimal and rotational surfaces.

Contribution

It introduces an invariant linear map of Weingarten type, defines invariants k and kappa, and characterizes different surface classes in four-dimensional space.

Findings

Surfaces with k = kappa = 0 are flat points.

Minimal surfaces satisfy kappa^2 = k.

Rotational surfaces have flat normal connection.

Abstract

For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality kappa^2=k. The class of the surfaces with flat normal connection is characterized by the condition kappa = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas. We apply our theory to the class of the rotational surfaces, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.